Subgroups — C₆₇×#60126 #30739377

Order 17152.

2042 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2925	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2924	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2923	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2922	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2879	C₂	1	1	yes	maximal, characteristic
9	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#520×C₆₇	C₄	1	4	yes	—
18	#520×C₆₇	C₄	1	4	yes	—
19	#519×C₆₇	C₄	1	2	yes	—
20	#518×C₆₇	C₄	1	2	yes	—
21	#518×C₆₇	C₄	1	4	yes	—
22	C₄×#41×C₆₇	C₂²	1	4	yes	—
23	C₄×#41×C₆₇	C₂²	1	4	yes	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₂²	1	4	yes	—
27	#433×C₆₇	C₂²	1	16	yes	complemented
28	#433×C₆₇	C₂²	1	16	yes	complemented
29	#432×C₆₇	C₂²	1	16	yes	complemented
30	#431×C₆₇	C₂²	1	16	yes	complemented
31	#424×C₆₇	C₂²	1	4	yes	—
32	#424×C₆₇	C₂²	1	4	yes	—
33	#424×C₆₇	C₂²	1	2	yes	complemented
34	#424×C₆₇	C₂²	1	2	yes	complemented
35	#424×C₆₇	C₂²	1	4	yes	complemented
36	#423×C₆₇	C₂²	1	4	yes	complemented
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
38	#408×C₆₇	C₂²	1	2	yes	—
39	#408×C₆₇	C₂²	1	2	yes	—
40	#408×C₆₇	C₂²	1	4	yes	—
41	#408×C₆₇	C₂²	1	4	yes	—
42	#408×C₆₇	C₂²	1	4	yes	—
43	#407×C₆₇	C₂²	1	4	yes	—
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
45	#405×C₆₇	C₂²	1	4	yes	—
46	#405×C₆₇	C₂²	1	4	yes	—
47	#405×C₆₇	C₂²	1	4	yes	complemented
48	#405×C₆₇	C₂²	1	4	yes	complemented
49	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
50	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
51	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	—	2	4	no	—
54	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×#41×C₆₇	C₂³	1	2	yes	—
57	C₂×#41×C₆₇	C₂³	1	2	yes	—
58	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
81	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
82	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	—
83	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
84	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	—
85	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
86	#117×C₆₇	C₂×C₄	1	8	yes	—
87	#117×C₆₇	C₂×C₄	1	2	yes	—
88	#117×C₆₇	C₂×C₄	1	2	yes	—
89	#117×C₆₇	C₂×C₄	1	8	yes	—
90	#117×C₆₇	C₂×C₄	1	2	yes	—
91	#117×C₆₇	C₂×C₄	1	2	yes	—
92	#117×C₆₇	C₂³	1	2	yes	—
93	#117×C₆₇	C₂³	1	2	yes	—
94	#117×C₆₇	C₂³	1	2	yes	—
95	#117×C₆₇	C₂³	1	2	yes	—
96	#117×C₆₇	C₂³	1	8	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
106	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
107	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
108	C₄⋊C₈×C₆₇	C₂³	1	16	yes	—
109	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
111	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
112	#97×C₆₇	C₂³	1	8	yes	—
113	#97×C₆₇	C₂³	1	8	yes	—
114	C₄×C₈×C₆₇	C₂³	1	8	yes	—
115	C₄×C₈×C₆₇	C₂³	1	8	yes	—
116	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	#41×C₆₇	—	2	8	no	—
119	#41×C₆₇	—	2	8	no	—
120	#41×C₆₇	—	2	4	no	—
121	#41×C₆₇	—	2	4	no	—
122	#41×C₆₇	—	2	4	no	—
123	#41×C₆₇	—	2	4	no	—
124	C₂×Q₈×C₆₇	—	2	4	no	—
125	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂×D₄×C₆₇	—	2	8	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
138	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	#41	1	4	yes	—
144	C₂²×C₄×C₆₇	#41	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₆₇	—	2	4	no	—
150	M₁₆×C₆₇	—	2	4	no	—
151	M₁₆×C₆₇	—	2	4	no	—
152	M₁₆×C₆₇	—	2	4	no	—
153	M₁₆×C₆₇	—	2	4	no	—
154	M₁₆×C₆₇	—	2	4	no	—
155	C₂×C₈×C₆₇	—	2	4	no	—
156	C₂×C₈×C₆₇	—	2	4	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	4	no	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	—	2	4	no	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
164	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
165	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
166	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
170	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
174	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
175	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
177	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
178	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
179	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
180	C₄²×C₆₇	C₂⁴	1	4	yes	—
181	C₄²×C₆₇	C₂⁴	1	4	yes	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	—	4	4	no	—
185	C₂³×C₆₇	#117	1	4	yes	—
186	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
187	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
190	Q₈×C₆₇	—	2	4	no	—
191	Q₈×C₆₇	—	2	4	no	—
192	D₄×C₆₇	—	2	8	no	—
193	D₄×C₆₇	—	2	8	no	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	8	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	4	4	no	—
201	C₂×C₄×C₆₇	—	4	4	no	—
202	C₂×C₄×C₆₇	—	4	16	no	—
203	C₂×C₄×C₆₇	#117	1	4	yes	—
204	C₂×C₄×C₆₇	#117	1	4	yes	—
205	C₂×C₄×C₆₇	#117	1	4	yes	—
206	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
207	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
209	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
213	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
217	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
218	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
219	C₈×C₆₇	—	2	2	no	—
220	C₈×C₆₇	—	2	2	no	—
221	C₈×C₆₇	—	2	4	no	—
222	C₈×C₆₇	—	2	4	no	—
223	C₈×C₆₇	—	2	4	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	2	8	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	—	8	4	no	—
231	C₂²×C₆₇	C₂×#117	1	2	yes	—
232	C₂²×C₆₇	C₂×#117	1	2	yes	—
233	C₂²×C₆₇	C₂×#117	1	2	yes	—
234	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	—	2	8	no	—
237	C₄×C₆₇	—	2	8	no	—
238	C₄×C₆₇	C₂×#117	1	2	yes	—
239	C₄×C₆₇	C₂×#117	1	2	yes	—
240	C₄×C₆₇	C₂×#117	1	2	yes	—
241	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
242	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
243	#60126	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	2	2	no	—
245	C₂×C₆₇	—	2	2	no	—
246	C₂×C₆₇	—	2	2	no	—
247	C₂×C₆₇	—	4	4	no	—
248	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
249	C₂×C₆₇	#3573	1	2	yes	—
250	#3428	C₂×C₆₇	1	1	yes	characteristic
251	#2925	C₂×C₆₇	1	8	yes	complemented
252	#2924	C₂×C₆₇	1	8	yes	complemented
253	#2923	C₂×C₆₇	1	4	yes	complemented
254	#2922	C₂×C₆₇	1	4	yes	complemented
255	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₂×#424	C₂×C₆₇	1	1	yes	characteristic, complemented
257	#2879	C₂×C₆₇	1	1	yes	characteristic
258	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
259	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₆₇	#60126	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
263	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
265	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
266	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
267	#520	C₄×C₆₇	1	4	yes	—
268	#520	C₄×C₆₇	1	4	yes	—
269	#519	C₄×C₆₇	1	2	yes	—
270	#518	C₄×C₆₇	1	2	yes	—
271	#518	C₄×C₆₇	1	4	yes	—
272	C₄×#41	C₂²×C₆₇	1	4	yes	—
273	C₄×#41	C₂²×C₆₇	1	4	yes	—
274	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#117	C₂²×C₆₇	1	4	yes	—
277	#433	C₂²×C₆₇	1	16	yes	complemented
278	#433	C₂²×C₆₇	1	16	yes	complemented
279	#432	C₂²×C₆₇	1	16	yes	complemented
280	#431	C₂²×C₆₇	1	16	yes	complemented
281	#424	C₂²×C₆₇	1	4	yes	—
282	#424	C₂²×C₆₇	1	4	yes	—
283	#424	C₂²×C₆₇	1	2	yes	complemented
284	#424	C₂²×C₆₇	1	2	yes	complemented
285	#424	C₂²×C₆₇	1	4	yes	complemented
286	#423	C₂²×C₆₇	1	4	yes	complemented
287	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
288	#408	C₂²×C₆₇	1	2	yes	—
289	#408	C₂²×C₆₇	1	2	yes	—
290	#408	C₂²×C₆₇	1	4	yes	—
291	#408	C₂²×C₆₇	1	4	yes	—
292	#408	C₂²×C₆₇	1	4	yes	—
293	#407	C₂²×C₆₇	1	4	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
295	#405	C₂²×C₆₇	1	4	yes	—
296	#405	C₂²×C₆₇	1	4	yes	—
297	#405	C₂²×C₆₇	1	4	yes	complemented
298	#405	C₂²×C₆₇	1	4	yes	complemented
299	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
300	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
301	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
302	C₂×#41	—	2	4	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
306	C₂×#41	C₂³×C₆₇	1	2	yes	—
307	C₂×#41	C₂³×C₆₇	1	2	yes	—
308	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
310	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
312	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×M₁₆	—	2	4	no	—
314	C₂×M₁₆	—	2	4	no	—
315	C₂×M₁₆	—	2	4	no	—
316	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
317	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
318	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
319	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
320	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
321	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
322	C₂²×C₈	—	2	4	no	—
323	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
324	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
326	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
327	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
328	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
329	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
330	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
331	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
332	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	—
333	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
334	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	—
335	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
336	#117	C₂×C₄×C₆₇	1	8	yes	—
337	#117	C₂×C₄×C₆₇	1	2	yes	—
338	#117	C₂×C₄×C₆₇	1	2	yes	—
339	#117	C₂×C₄×C₆₇	1	8	yes	—
340	#117	C₂×C₄×C₆₇	1	2	yes	—
341	#117	C₂×C₄×C₆₇	1	2	yes	—
342	#117	C₂³×C₆₇	1	2	yes	—
343	#117	C₂³×C₆₇	1	2	yes	—
344	#117	C₂³×C₆₇	1	2	yes	—
345	#117	C₂³×C₆₇	1	2	yes	—
346	#117	C₂³×C₆₇	1	8	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
348	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
349	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
353	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
354	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
356	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
357	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
358	C₄⋊C₈	C₂³×C₆₇	1	16	yes	—
359	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
360	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
361	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
362	#97	C₂³×C₆₇	1	8	yes	—
363	#97	C₂³×C₆₇	1	8	yes	—
364	C₄×C₈	C₂³×C₆₇	1	8	yes	—
365	C₄×C₈	C₂³×C₆₇	1	8	yes	—
366	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
367	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
368	#41	—	2	8	no	—
369	#41	—	2	8	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	2	4	no	—
373	#41	—	2	4	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
376	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
380	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
381	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
386	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
387	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
388	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
389	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
390	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	#41×C₆₇	1	4	yes	—
394	C₂²×C₄	#41×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	4	no	—
401	M₁₆	—	2	4	no	—
402	M₁₆	—	2	4	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	2	4	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	4	no	—
410	C₂×C₈	—	2	4	no	—
411	C₂×C₈	—	2	4	no	—
412	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
413	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
414	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
415	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
416	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
417	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
418	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
419	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
420	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
421	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
422	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
425	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
426	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
427	C₄²	C₂²×C₄×C₆₇	1	16	yes	—
428	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
429	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
430	C₄²	C₂⁴×C₆₇	1	4	yes	—
431	C₄²	C₂⁴×C₆₇	1	4	yes	—
432	C₂³	—	2	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	4	no	—
435	C₂³	#117×C₆₇	1	4	yes	—
436	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
437	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
438	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
439	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
440	Q₈	—	2	4	no	—
441	Q₈	—	2	4	no	—
442	D₄	—	2	8	no	—
443	D₄	—	2	8	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	16	no	—
453	C₂×C₄	#117×C₆₇	1	4	yes	—
454	C₂×C₄	#117×C₆₇	1	4	yes	—
455	C₂×C₄	#117×C₆₇	1	4	yes	—
456	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
457	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
458	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
459	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
463	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
464	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
465	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
466	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
467	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
468	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	4	no	—
472	C₈	—	2	4	no	—
473	C₈	—	2	4	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	8	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	4	no	—
481	C₂²	C₂×#117×C₆₇	1	2	yes	—
482	C₂²	C₂×#117×C₆₇	1	2	yes	—
483	C₂²	C₂×#117×C₆₇	1	2	yes	—
484	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
485	C₄	—	2	8	no	—
486	C₄	—	2	8	no	—
487	C₄	—	2	4	no	—
488	C₄	C₂×#117×C₆₇	1	2	yes	—
489	C₄	C₂×#117×C₆₇	1	2	yes	—
490	C₄	C₂×#117×C₆₇	1	2	yes	—
491	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
492	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
493	C₂	—	2	2	no	—
494	C₂	—	2	2	no	—
495	C₂	—	2	2	no	—
496	C₂	—	4	4	no	—
497	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
498	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	C₂²×#41×C₆₇	C₂²
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
2. C₆₇×#2925	C₂	semidirect
3. C₆₇×#2924	C₂	semidirect
4. C₆₇×#2923	C₂	semidirect
5. C₆₇×#2922	C₂	semidirect
6. C₆₇×#2905	C₂	semidirect
7. C₂×#424×C₆₇	C₂	semidirect
10. C₆₇×#2873	C₂	semidirect
11. C₂×#405×C₆₇	C₂	semidirect
27. #433×C₆₇	C₂²	semidirect
28. #433×C₆₇	C₂²	semidirect
29. #432×C₆₇	C₂²	semidirect
30. #431×C₆₇	C₂²	semidirect
33. #424×C₆₇	C₂²	semidirect
34. #424×C₆₇	C₂²	semidirect
35. #424×C₆₇	C₂²	semidirect
36. #423×C₆₇	C₂²	semidirect
37. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
47. #405×C₆₇	C₂²	semidirect
48. #405×C₆₇	C₂²	semidirect
49. C₄×M₁₆×C₆₇	C₂²	semidirect
50. C₂×#97×C₆₇	C₂²	semidirect
51. C₂×C₄×C₈×C₆₇	C₂²	semidirect
243. #60126	C₆₇	direct (N × H)
251. #2925	C₂×C₆₇	semidirect
252. #2924	C₂×C₆₇	semidirect
253. #2923	C₂×C₆₇	semidirect
254. #2922	C₂×C₆₇	semidirect
255. #2905	C₂×C₆₇	semidirect
256. C₂×#424	C₂×C₆₇	semidirect
259. #2873	C₂×C₆₇	semidirect
260. C₂×#405	C₂×C₆₇	semidirect
261. C₆₇	#60126	direct (N × H)
277. #433	C₂²×C₆₇	semidirect
278. #433	C₂²×C₆₇	semidirect
279. #432	C₂²×C₆₇	semidirect
280. #431	C₂²×C₆₇	semidirect
283. #424	C₂²×C₆₇	semidirect
284. #424	C₂²×C₆₇	semidirect
285. #424	C₂²×C₆₇	semidirect
286. #423	C₂²×C₆₇	semidirect
287. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
297. #405	C₂²×C₆₇	semidirect
298. #405	C₂²×C₆₇	semidirect
299. C₄×M₁₆	C₂²×C₆₇	semidirect
300. C₂×#97	C₂²×C₆₇	semidirect
301. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3428	2
C₆₇×#2925	2
C₆₇×#2924	2
C₆₇×#2923	2
C₆₇×#2922	2
C₆₇×#2905	2
C₂×#424×C₆₇	2
C₆₇×#2879	2
C₂×#408×C₆₇	2
C₆₇×#2873	2
C₂×#405×C₆₇	2
#60126	67

Subgroups — C67×#60126 #30739377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60126 #30739377