Subgroups — C₆₇×#60142 #30739393

Order 17152.

2714 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3449	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2935	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2933	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2930	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2929	C₂	1	4	yes	maximal, complemented
8	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2879	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	#568×C₆₇	—	2	8	no	—
15	C₂×#131×C₆₇	—	2	8	no	—
16	C₂×#131×C₆₇	C₂²	1	4	yes	—
17	C₂×#131×C₆₇	C₂²	1	4	yes	—
18	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
24	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	complemented
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
27	#537×C₆₇	C₄	1	4	yes	—
28	#536×C₆₇	C₄	1	2	yes	—
29	#536×C₆₇	C₄	1	2	yes	—
30	#535×C₆₇	C₄	1	2	yes	—
31	#535×C₆₇	C₄	1	2	yes	—
32	#534×C₆₇	C₄	1	4	yes	—
33	#532×C₆₇	C₂²	1	4	yes	—
34	#525×C₆₇	C₂²	1	4	yes	—
35	#525×C₆₇	C₂²	1	4	yes	—
36	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
37	#436×C₆₇	C₂²	1	16	yes	complemented
38	#435×C₆₇	C₂²	1	16	yes	complemented
39	#435×C₆₇	C₂²	1	16	yes	complemented
40	C₈×D₄×C₆₇	C₂²	1	16	yes	complemented
41	#424×C₆₇	C₂²	1	2	yes	complemented
42	#424×C₆₇	C₂²	1	2	yes	complemented
43	#424×C₆₇	C₂²	1	4	yes	complemented
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
45	#408×C₆₇	C₂²	1	2	yes	—
46	#408×C₆₇	C₂²	1	2	yes	—
47	#408×C₆₇	C₂²	1	4	yes	—
48	#408×C₆₇	C₂²	1	4	yes	complemented
49	#408×C₆₇	C₂²	1	4	yes	complemented
50	#407×C₆₇	C₂²	1	4	yes	complemented
51	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
52	#405×C₆₇	C₂²	1	4	yes	complemented
53	#405×C₆₇	C₂²	1	4	yes	complemented
54	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
55	C₂×#41×C₆₇	—	2	4	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	4	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×#41×C₆₇	C₂³	1	2	yes	—
64	C₂×#41×C₆₇	C₂³	1	2	yes	—
65	C₂×#41×C₆₇	C₂³	1	2	yes	—
66	C₂×#41×C₆₇	C₂³	1	2	yes	—
67	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	#131×C₆₇	—	2	16	no	—
77	#131×C₆₇	—	2	16	no	—
78	#131×C₆₇	—	2	16	no	—
79	#131×C₆₇	—	2	16	no	—
80	C₂×M₁₆×C₆₇	—	2	4	no	—
81	C₂×M₁₆×C₆₇	—	2	4	no	—
82	C₂×M₁₆×C₆₇	—	2	16	no	—
83	C₂×M₁₆×C₆₇	—	2	8	no	—
84	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
85	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
88	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
89	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
90	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
91	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
92	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
93	C₂²×C₈×C₆₇	—	2	4	no	—
94	C₂²×C₈×C₆₇	—	2	8	no	—
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
98	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
99	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
100	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
101	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
102	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
103	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
104	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
105	#127×C₆₇	C₂×C₄	1	4	yes	—
106	#124×C₆₇	C₂×C₄	1	4	yes	—
107	#124×C₆₇	C₂×C₄	1	4	yes	—
108	#123×C₆₇	C₂×C₄	1	8	yes	—
109	#123×C₆₇	C₂×C₄	1	8	yes	—
110	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
111	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
112	#121×C₆₇	C₂×C₄	1	8	yes	—
113	#121×C₆₇	C₂×C₄	1	8	yes	—
114	#121×C₆₇	C₂×C₄	1	16	yes	—
115	#120×C₆₇	C₂×C₄	1	8	yes	—
116	#120×C₆₇	C₂×C₄	1	8	yes	—
117	C₄×D₄×C₆₇	C₂³	1	16	yes	—
118	#117×C₆₇	C₂×C₄	1	2	yes	—
119	#117×C₆₇	C₂×C₄	1	2	yes	—
120	#117×C₆₇	C₂³	1	2	yes	—
121	#117×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
126	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
127	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
128	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
129	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
130	#97×C₆₇	C₂³	1	8	yes	—
131	C₄×C₈×C₆₇	C₂³	1	8	yes	—
132	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	#41×C₆₇	—	2	8	no	—
136	#41×C₆₇	—	2	8	no	—
137	#41×C₆₇	—	2	8	no	—
138	#41×C₆₇	—	2	8	no	—
139	#41×C₆₇	—	2	4	no	—
140	#41×C₆₇	—	2	4	no	—
141	#41×C₆₇	—	2	4	no	—
142	#41×C₆₇	—	2	4	no	—
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	—	2	4	no	—
145	#41×C₆₇	—	2	4	no	—
146	#41×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	2	4	no	—
148	C₂×Q₈×C₆₇	—	2	4	no	—
149	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	8	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
158	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
159	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
161	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
163	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
164	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
165	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₆₇	#41	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	M₁₆×C₆₇	—	2	4	no	—
186	M₁₆×C₆₇	—	2	4	no	—
187	M₁₆×C₆₇	—	2	4	no	—
188	M₁₆×C₆₇	—	2	4	no	—
189	M₁₆×C₆₇	—	2	16	no	—
190	M₁₆×C₆₇	—	2	16	no	—
191	M₁₆×C₆₇	C₂×D₄	1	8	yes	—
192	M₁₆×C₆₇	C₂×D₄	1	8	yes	—
193	C₂×C₈×C₆₇	—	2	2	no	—
194	C₂×C₈×C₆₇	—	2	2	no	—
195	C₂×C₈×C₆₇	—	2	4	no	—
196	C₂×C₈×C₆₇	—	2	4	no	—
197	C₂×C₈×C₆₇	—	2	4	no	—
198	C₂×C₈×C₆₇	—	2	4	no	—
199	C₂×C₈×C₆₇	—	2	16	no	—
200	C₂×C₈×C₆₇	—	2	16	no	—
201	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
202	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
203	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
204	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
205	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
206	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
208	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
209	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
210	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
211	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
212	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
213	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
214	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
215	C₄²×C₆₇	C₂⁴	1	4	yes	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	—	4	4	no	—
220	C₂³×C₆₇	—	4	4	no	—
221	C₂³×C₆₇	C₄×D₄	1	4	yes	—
222	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
223	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
224	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
225	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
226	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
227	Q₈×C₆₇	—	2	4	no	—
228	Q₈×C₆₇	—	2	4	no	—
229	Q₈×C₆₇	—	2	4	no	—
230	Q₈×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	2	8	no	—
232	D₄×C₆₇	—	2	8	no	—
233	D₄×C₆₇	—	2	8	no	—
234	D₄×C₆₇	—	2	8	no	—
235	D₄×C₆₇	—	2	4	no	—
236	D₄×C₆₇	—	2	4	no	—
237	D₄×C₆₇	—	2	4	no	—
238	D₄×C₆₇	—	2	4	no	—
239	D₄×C₆₇	—	4	16	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	8	no	—
243	C₂×C₄×C₆₇	—	2	8	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	4	4	no	—
249	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
250	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
251	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
252	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
253	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
254	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
255	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
262	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
263	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
264	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
265	C₈×C₆₇	—	2	2	no	—
266	C₈×C₆₇	—	2	2	no	—
267	C₈×C₆₇	—	2	4	no	—
268	C₈×C₆₇	—	2	4	no	—
269	C₈×C₆₇	C₂²×D₄	1	8	yes	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	2	8	no	—
276	C₂²×C₆₇	—	2	8	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	4	8	no	—
279	C₂²×C₆₇	—	8	4	no	—
280	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
281	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
282	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
283	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
284	C₄×C₆₇	—	2	4	no	—
285	C₄×C₆₇	—	2	4	no	—
286	C₄×C₆₇	—	2	8	no	—
287	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
288	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
289	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
290	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
291	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
292	#60142	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
293	C₂×C₆₇	—	2	2	no	—
294	C₂×C₆₇	—	2	2	no	—
295	C₂×C₆₇	—	2	2	no	—
296	C₂×C₆₇	—	4	4	no	—
297	C₂×C₆₇	—	4	4	no	—
298	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
299	C₂×C₆₇	#3573	1	2	yes	—
300	C₂×#568	C₂×C₆₇	1	1	yes	characteristic, complemented
301	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3449	C₂×C₆₇	1	1	yes	characteristic
303	#2935	C₂×C₆₇	1	8	yes	complemented
304	#2933	C₂×C₆₇	1	8	yes	complemented
305	#2930	C₂×C₆₇	1	4	yes	complemented
306	#2929	C₂×C₆₇	1	4	yes	complemented
307	C₂×#424	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#2879	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#60142	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
313	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
314	#568	—	2	8	no	—
315	C₂×#131	—	2	8	no	—
316	C₂×#131	C₂²×C₆₇	1	4	yes	—
317	C₂×#131	C₂²×C₆₇	1	4	yes	—
318	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
319	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
320	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
323	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
324	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	complemented
325	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
327	#537	C₄×C₆₇	1	4	yes	—
328	#536	C₄×C₆₇	1	2	yes	—
329	#536	C₄×C₆₇	1	2	yes	—
330	#535	C₄×C₆₇	1	2	yes	—
331	#535	C₄×C₆₇	1	2	yes	—
332	#534	C₄×C₆₇	1	4	yes	—
333	#532	C₂²×C₆₇	1	4	yes	—
334	#525	C₂²×C₆₇	1	4	yes	—
335	#525	C₂²×C₆₇	1	4	yes	—
336	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
337	#436	C₂²×C₆₇	1	16	yes	complemented
338	#435	C₂²×C₆₇	1	16	yes	complemented
339	#435	C₂²×C₆₇	1	16	yes	complemented
340	C₈×D₄	C₂²×C₆₇	1	16	yes	complemented
341	#424	C₂²×C₆₇	1	2	yes	complemented
342	#424	C₂²×C₆₇	1	2	yes	complemented
343	#424	C₂²×C₆₇	1	4	yes	complemented
344	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
345	#408	C₂²×C₆₇	1	2	yes	—
346	#408	C₂²×C₆₇	1	2	yes	—
347	#408	C₂²×C₆₇	1	4	yes	—
348	#408	C₂²×C₆₇	1	4	yes	complemented
349	#408	C₂²×C₆₇	1	4	yes	complemented
350	#407	C₂²×C₆₇	1	4	yes	complemented
351	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
352	#405	C₂²×C₆₇	1	4	yes	complemented
353	#405	C₂²×C₆₇	1	4	yes	complemented
354	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
355	C₂×#41	—	2	4	no	—
356	C₂×#41	—	2	4	no	—
357	C₂×#41	—	2	4	no	—
358	C₂×#41	—	2	4	no	—
359	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
360	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
361	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
362	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
363	C₂×#41	C₂³×C₆₇	1	2	yes	—
364	C₂×#41	C₂³×C₆₇	1	2	yes	—
365	C₂×#41	C₂³×C₆₇	1	2	yes	—
366	C₂×#41	C₂³×C₆₇	1	2	yes	—
367	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
368	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
369	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
370	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
371	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
372	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
374	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
375	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
376	#131	—	2	16	no	—
377	#131	—	2	16	no	—
378	#131	—	2	16	no	—
379	#131	—	2	16	no	—
380	C₂×M₁₆	—	2	4	no	—
381	C₂×M₁₆	—	2	4	no	—
382	C₂×M₁₆	—	2	16	no	—
383	C₂×M₁₆	—	2	8	no	—
384	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
385	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
386	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
387	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
388	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
389	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
390	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
391	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
392	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
393	C₂²×C₈	—	2	4	no	—
394	C₂²×C₈	—	2	8	no	—
395	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
398	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
399	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
400	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
401	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
402	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
403	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
404	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
405	#127	C₂×C₄×C₆₇	1	4	yes	—
406	#124	C₂×C₄×C₆₇	1	4	yes	—
407	#124	C₂×C₄×C₆₇	1	4	yes	—
408	#123	C₂×C₄×C₆₇	1	8	yes	—
409	#123	C₂×C₄×C₆₇	1	8	yes	—
410	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
411	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
412	#121	C₂×C₄×C₆₇	1	8	yes	—
413	#121	C₂×C₄×C₆₇	1	8	yes	—
414	#121	C₂×C₄×C₆₇	1	16	yes	—
415	#120	C₂×C₄×C₆₇	1	8	yes	—
416	#120	C₂×C₄×C₆₇	1	8	yes	—
417	C₄×D₄	C₂³×C₆₇	1	16	yes	—
418	#117	C₂×C₄×C₆₇	1	2	yes	—
419	#117	C₂×C₄×C₆₇	1	2	yes	—
420	#117	C₂³×C₆₇	1	2	yes	—
421	#117	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
424	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
425	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
426	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
427	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
428	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
429	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
430	#97	C₂³×C₆₇	1	8	yes	—
431	C₄×C₈	C₂³×C₆₇	1	8	yes	—
432	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
433	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
434	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	#41	—	2	8	no	—
438	#41	—	2	8	no	—
439	#41	—	2	4	no	—
440	#41	—	2	4	no	—
441	#41	—	2	4	no	—
442	#41	—	2	4	no	—
443	#41	—	2	4	no	—
444	#41	—	2	4	no	—
445	#41	—	2	4	no	—
446	#41	—	2	4	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
450	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
451	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
452	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
458	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
459	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
460	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
461	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
462	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
463	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
464	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
465	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
471	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
472	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
474	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
475	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
479	C₂²×C₄	#41×C₆₇	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
485	M₁₆	—	2	4	no	—
486	M₁₆	—	2	4	no	—
487	M₁₆	—	2	4	no	—
488	M₁₆	—	2	4	no	—
489	M₁₆	—	2	16	no	—
490	M₁₆	—	2	16	no	—
491	M₁₆	C₂×D₄×C₆₇	1	8	yes	—
492	M₁₆	C₂×D₄×C₆₇	1	8	yes	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	2	4	no	—
496	C₂×C₈	—	2	4	no	—
497	C₂×C₈	—	2	4	no	—
498	C₂×C₈	—	2	4	no	—
499	C₂×C₈	—	2	16	no	—
500	C₂×C₈	—	2	16	no	—
501	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
502	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
503	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
504	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
505	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
506	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
507	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
508	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
509	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
510	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
511	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
512	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
513	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
514	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
515	C₄²	C₂⁴×C₆₇	1	4	yes	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	4	no	—
520	C₂³	—	4	4	no	—
521	C₂³	C₄×D₄×C₆₇	1	4	yes	—
522	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
523	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
524	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
525	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
526	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
527	Q₈	—	2	4	no	—
528	Q₈	—	2	4	no	—
529	Q₈	—	2	4	no	—
530	Q₈	—	2	4	no	—
531	D₄	—	2	8	no	—
532	D₄	—	2	8	no	—
533	D₄	—	2	8	no	—
534	D₄	—	2	8	no	—
535	D₄	—	2	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	2	4	no	—
539	D₄	—	4	16	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
550	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
551	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
552	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
553	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
554	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
555	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
556	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
557	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
561	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
562	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
563	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
564	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	2	4	no	—
568	C₈	—	2	4	no	—
569	C₈	C₂²×D₄×C₆₇	1	8	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	8	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	8	no	—
579	C₂²	—	8	4	no	—
580	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
581	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
582	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
583	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
584	C₄	—	2	8	no	—
585	C₄	—	2	4	no	—
586	C₄	—	2	4	no	—
587	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
588	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
589	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
590	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
591	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
592	C₂	—	2	2	no	—
593	C₂	—	2	2	no	—
594	C₂	—	2	2	no	—
595	C₂	—	4	4	no	—
596	C₂	—	4	4	no	—
597	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
598	C₂	C₆₇×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	C₆₇×#3449	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#568×C₆₇	C₂	semidirect
2. C₂²×#131×C₆₇	C₂	semidirect
4. C₆₇×#2935	C₂	semidirect
5. C₆₇×#2933	C₂	semidirect
6. C₆₇×#2930	C₂	semidirect
7. C₆₇×#2929	C₂	semidirect
8. C₂×#424×C₆₇	C₂	semidirect
9. C₆₇×#2879	C₂	semidirect
10. C₂×#408×C₆₇	C₂	semidirect
11. C₆₇×#2873	C₂	semidirect
18. C₂×#131×C₆₇	C₂²	semidirect
19. C₂×#131×C₆₇	C₂²	semidirect
23. C₂²×M₁₆×C₆₇	C₂²	semidirect
24. C₂²×M₁₆×C₆₇	C₂²	semidirect
26. C₂³×C₈×C₆₇	C₂²	semidirect
37. #436×C₆₇	C₂²	semidirect
38. #435×C₆₇	C₂²	semidirect
39. #435×C₆₇	C₂²	semidirect
40. C₈×D₄×C₆₇	C₂²	semidirect
41. #424×C₆₇	C₂²	semidirect
42. #424×C₆₇	C₂²	semidirect
43. #424×C₆₇	C₂²	semidirect
44. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
48. #408×C₆₇	C₂²	semidirect
49. #408×C₆₇	C₂²	semidirect
50. #407×C₆₇	C₂²	semidirect
51. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
52. #405×C₆₇	C₂²	semidirect
53. #405×C₆₇	C₂²	semidirect
54. C₄×M₁₆×C₆₇	C₂²	semidirect
84. C₂×M₁₆×C₆₇	D₄	semidirect
292. #60142	C₆₇	direct (N × H)
300. C₂×#568	C₂×C₆₇	semidirect
301. C₂²×#131	C₂×C₆₇	semidirect
303. #2935	C₂×C₆₇	semidirect
304. #2933	C₂×C₆₇	semidirect
305. #2930	C₂×C₆₇	semidirect
306. #2929	C₂×C₆₇	semidirect
307. C₂×#424	C₂×C₆₇	semidirect
308. #2879	C₂×C₆₇	semidirect
309. C₂×#408	C₂×C₆₇	semidirect
310. #2873	C₂×C₆₇	semidirect
311. C₆₇	#60142	direct (N × H)
318. C₂×#131	C₂²×C₆₇	semidirect
319. C₂×#131	C₂²×C₆₇	semidirect
323. C₂²×M₁₆	C₂²×C₆₇	semidirect
324. C₂²×M₁₆	C₂²×C₆₇	semidirect
326. C₂³×C₈	C₂²×C₆₇	semidirect
337. #436	C₂²×C₆₇	semidirect
338. #435	C₂²×C₆₇	semidirect
339. #435	C₂²×C₆₇	semidirect
340. C₈×D₄	C₂²×C₆₇	semidirect
341. #424	C₂²×C₆₇	semidirect
342. #424	C₂²×C₆₇	semidirect
343. #424	C₂²×C₆₇	semidirect
344. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
348. #408	C₂²×C₆₇	semidirect
349. #408	C₂²×C₆₇	semidirect
350. #407	C₂²×C₆₇	semidirect
351. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
352. #405	C₂²×C₆₇	semidirect
353. #405	C₂²×C₆₇	semidirect
354. C₄×M₁₆	C₂²×C₆₇	semidirect
384. C₂×M₁₆	D₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#568×C₆₇	2
C₂²×#131×C₆₇	2
C₆₇×#3449	2
C₆₇×#2935	2
C₆₇×#2933	2
C₆₇×#2930	2
C₆₇×#2929	2
C₂×#424×C₆₇	2
C₆₇×#2879	2
C₂×#408×C₆₇	2
C₆₇×#2873	2
#60142	67

Subgroups — C67×#60142 #30739393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60142 #30739393