Subgroups — C₆₇×#60167 #30739418

Order 17152.

2706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3428	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2952	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2951	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2950	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2949	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2948	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2943	C₂	1	4	yes	maximal, complemented
9	C₄×#135×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2896	C₂	1	2	yes	maximal, complemented
12	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	C₂×#135×C₆₇	C₄	1	16	yes	complemented
15	C₂×#135×C₆₇	C₂²	1	2	yes	—
16	C₂×#135×C₆₇	C₂²	1	2	yes	—
17	C₂×#135×C₆₇	C₂²	1	2	yes	—
18	C₂×#135×C₆₇	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#520×C₆₇	—	2	4	no	—
25	#520×C₆₇	C₂²	1	8	yes	complemented
26	#519×C₆₇	C₂²	1	4	yes	—
27	#518×C₆₇	—	2	4	no	—
28	#518×C₆₇	C₂²	1	4	yes	—
29	C₄×#41×C₆₇	—	2	4	no	—
30	C₄×#41×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#117×C₆₇	—	2	4	no	—
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	#442×C₆₇	C₂²	1	8	yes	complemented
35	#441×C₆₇	C₂²	1	8	yes	complemented
36	#440×C₆₇	C₂²	1	16	yes	complemented
37	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
38	C₄×SD₁₆×C₆₇	C₂²	1	16	yes	complemented
39	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
40	#427×C₆₇	C₂²	1	2	yes	—
41	#427×C₆₇	C₂²	1	2	yes	—
42	#427×C₆₇	C₂²	1	2	yes	complemented
43	#427×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
46	#419×C₆₇	C₂²	1	4	yes	—
47	#418×C₆₇	C₂²	1	4	yes	—
48	#417×C₆₇	C₂²	1	8	yes	complemented
49	#416×C₆₇	C₂²	1	8	yes	complemented
50	#405×C₆₇	C₂²	1	2	yes	—
51	#405×C₆₇	C₂²	1	2	yes	—
52	#405×C₆₇	C₂²	1	2	yes	complemented
53	#405×C₆₇	C₂²	1	2	yes	complemented
54	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
56	C₂×#41×C₆₇	—	2	2	no	—
57	C₂×#41×C₆₇	—	2	2	no	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
60	C₂×#41×C₆₇	C₂³	1	4	yes	—
61	C₂×#41×C₆₇	C₂³	1	4	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	—	2	2	no	—
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	#135×C₆₇	C₂×C₄	1	16	yes	complemented
68	#135×C₆₇	C₂×C₄	1	16	yes	complemented
69	#135×C₆₇	C₂×C₄	1	16	yes	complemented
70	#135×C₆₇	C₂×C₄	1	16	yes	complemented
71	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
72	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
75	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
78	C₂×D₈×C₆₇	C₂³	1	2	yes	—
79	C₂×D₈×C₆₇	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₇	—	2	4	no	—
81	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
84	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
85	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₆₇	—	2	8	no	—
92	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
93	C₄×D₄×C₆₇	—	2	16	no	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	16	no	—
96	C₄×D₄×C₆₇	C₂³	1	16	yes	—
97	#117×C₆₇	—	2	2	no	—
98	#117×C₆₇	—	2	2	no	—
99	#117×C₆₇	—	2	8	no	—
100	#117×C₆₇	—	2	8	no	—
101	#117×C₆₇	—	2	8	no	—
102	#117×C₆₇	D₄	1	2	yes	—
103	#117×C₆₇	D₄	1	2	yes	—
104	#117×C₆₇	C₂³	1	4	yes	—
105	#117×C₆₇	C₂³	1	4	yes	—
106	#117×C₆₇	C₂³	1	2	yes	—
107	#117×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
109	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
113	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
114	C₂×C₄²×C₆₇	—	2	2	no	—
115	C₂×C₄²×C₆₇	—	2	8	no	—
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	#107×C₆₇	C₂³	1	4	yes	—
118	#107×C₆₇	C₂³	1	4	yes	—
119	#106×C₆₇	C₂³	1	4	yes	—
120	#106×C₆₇	C₂³	1	4	yes	—
121	#103×C₆₇	C₂³	1	16	yes	—
122	#102×C₆₇	C₂³	1	16	yes	—
123	#97×C₆₇	C₂³	1	4	yes	—
124	#97×C₆₇	C₂³	1	4	yes	—
125	C₄×C₈×C₆₇	C₂³	1	4	yes	—
126	C₄×C₈×C₆₇	C₂³	1	4	yes	—
127	C₂⁴×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
129	#41×C₆₇	—	2	16	no	—
130	#41×C₆₇	—	2	16	no	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	C₂²×C₄	1	16	yes	—
134	#41×C₆₇	C₂²×C₄	1	16	yes	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	4	no	—
137	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
138	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
139	C₂×D₄×C₆₇	—	2	16	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	2	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
145	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	16	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	16	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	#41	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
161	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
162	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
163	SD₁₆×C₆₇	C₂²×C₄	1	16	yes	—
164	SD₁₆×C₆₇	C₂²×C₄	1	16	yes	—
165	D₈×C₆₇	C₂²×C₄	1	8	yes	—
166	D₈×C₆₇	C₂²×C₄	1	8	yes	—
167	M₁₆×C₆₇	—	2	4	no	—
168	M₁₆×C₆₇	—	2	4	no	—
169	C₂×C₈×C₆₇	—	2	4	no	—
170	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
174	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
176	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
177	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	16	no	—
183	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	—	2	4	no	—
188	C₂²⋊C₄×C₆₇	—	2	16	no	—
189	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
191	C₄²×C₆₇	—	2	16	no	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	—	2	4	no	—
194	C₄²×C₆₇	C₂×D₄	1	4	yes	—
195	C₄²×C₆₇	C₂⁴	1	4	yes	—
196	C₂³×C₆₇	—	2	4	no	—
197	C₂³×C₆₇	—	2	8	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	8	4	no	—
200	C₂³×C₆₇	C₄×D₄	1	4	yes	—
201	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	8	no	—
205	Q₈×C₆₇	—	2	4	no	—
206	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
207	D₄×C₆₇	—	2	16	no	—
208	D₄×C₆₇	—	2	16	no	—
209	D₄×C₆₇	—	2	8	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	C₂³×C₄	1	8	yes	—
212	C₂×C₄×C₆₇	—	2	8	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	16	no	—
215	C₂×C₄×C₆₇	—	2	16	no	—
216	C₂×C₄×C₆₇	—	2	8	no	—
217	C₂×C₄×C₆₇	—	4	4	no	—
218	C₂×C₄×C₆₇	—	8	16	no	—
219	C₂×C₄×C₆₇	—	8	4	no	—
220	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
221	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
223	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
231	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
232	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
233	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
234	C₈×C₆₇	—	2	4	no	—
235	C₈×C₆₇	C₂³×C₄	1	2	yes	—
236	C₈×C₆₇	C₂³×C₄	1	2	yes	—
237	C₈×C₆₇	C₂³×C₄	1	2	yes	—
238	C₈×C₆₇	C₂³×C₄	1	2	yes	—
239	C₂²×C₆₇	—	2	2	no	—
240	C₂²×C₆₇	—	2	2	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	8	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	4	8	no	—
245	C₂²×C₆₇	—	8	8	no	—
246	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
247	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
248	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
249	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₆₇	—	2	8	no	—
251	C₄×C₆₇	—	2	8	no	—
252	C₄×C₆₇	—	4	8	no	—
253	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
254	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
255	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
256	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
257	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
258	#60167	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₆₇	—	2	2	no	—
260	C₂×C₆₇	—	2	2	no	—
261	C₂×C₆₇	—	2	2	no	—
262	C₂×C₆₇	—	4	8	no	—
263	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
264	C₂×C₆₇	#3584	1	2	yes	—
265	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
266	#3428	C₂×C₆₇	1	2	yes	complemented
267	#2952	C₂×C₆₇	1	4	yes	complemented
268	#2951	C₂×C₆₇	1	4	yes	complemented
269	#2950	C₂×C₆₇	1	2	yes	complemented
270	#2949	C₂×C₆₇	1	4	yes	complemented
271	#2948	C₂×C₆₇	1	2	yes	complemented
272	#2943	C₂×C₆₇	1	4	yes	complemented
273	C₄×#135	C₂×C₆₇	1	4	yes	complemented
274	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
275	#2896	C₂×C₆₇	1	2	yes	complemented
276	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
277	C₆₇	#60167	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²×#41	C₂²×C₆₇	1	2	yes	—
279	C₂×#135	C₄×C₆₇	1	16	yes	complemented
280	C₂×#135	C₂²×C₆₇	1	2	yes	—
281	C₂×#135	C₂²×C₆₇	1	2	yes	—
282	C₂×#135	C₂²×C₆₇	1	2	yes	—
283	C₂×#135	C₂²×C₆₇	1	2	yes	—
284	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
285	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
286	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
288	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
289	#520	—	2	4	no	—
290	#520	C₂²×C₆₇	1	8	yes	complemented
291	#519	C₂²×C₆₇	1	4	yes	—
292	#518	—	2	4	no	—
293	#518	C₂²×C₆₇	1	4	yes	—
294	C₄×#41	—	2	4	no	—
295	C₄×#41	C₂²×C₆₇	1	8	yes	complemented
296	C₂×#117	—	2	4	no	—
297	C₂×#117	C₂²×C₆₇	1	2	yes	—
298	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
299	#442	C₂²×C₆₇	1	8	yes	complemented
300	#441	C₂²×C₆₇	1	8	yes	complemented
301	#440	C₂²×C₆₇	1	16	yes	complemented
302	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
303	C₄×SD₁₆	C₂²×C₆₇	1	16	yes	complemented
304	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
305	#427	C₂²×C₆₇	1	2	yes	—
306	#427	C₂²×C₆₇	1	2	yes	—
307	#427	C₂²×C₆₇	1	2	yes	complemented
308	#427	C₂²×C₆₇	1	2	yes	complemented
309	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
310	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
311	#419	C₂²×C₆₇	1	4	yes	—
312	#418	C₂²×C₆₇	1	4	yes	—
313	#417	C₂²×C₆₇	1	8	yes	complemented
314	#416	C₂²×C₆₇	1	8	yes	complemented
315	#405	C₂²×C₆₇	1	2	yes	—
316	#405	C₂²×C₆₇	1	2	yes	—
317	#405	C₂²×C₆₇	1	2	yes	complemented
318	#405	C₂²×C₆₇	1	2	yes	complemented
319	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
320	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	8	no	—
324	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
325	C₂×#41	C₂³×C₆₇	1	4	yes	—
326	C₂×#41	C₂³×C₆₇	1	4	yes	—
327	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
330	C₂³×C₄	—	2	2	no	—
331	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
332	#135	C₂×C₄×C₆₇	1	16	yes	complemented
333	#135	C₂×C₄×C₆₇	1	16	yes	complemented
334	#135	C₂×C₄×C₆₇	1	16	yes	complemented
335	#135	C₂×C₄×C₆₇	1	16	yes	complemented
336	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
337	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
338	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
339	C₂×SD₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
340	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
341	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
342	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
343	C₂×D₈	C₂³×C₆₇	1	2	yes	—
344	C₂×D₈	C₂³×C₆₇	1	2	yes	—
345	C₂×M₁₆	—	2	4	no	—
346	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
347	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
348	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
349	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
350	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₄×Q₈	—	2	8	no	—
357	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
358	C₄×D₄	—	2	16	no	—
359	C₄×D₄	—	2	8	no	—
360	C₄×D₄	—	2	16	no	—
361	C₄×D₄	C₂³×C₆₇	1	16	yes	—
362	#117	—	2	2	no	—
363	#117	—	2	2	no	—
364	#117	—	2	8	no	—
365	#117	—	2	8	no	—
366	#117	—	2	8	no	—
367	#117	D₄×C₆₇	1	2	yes	—
368	#117	D₄×C₆₇	1	2	yes	—
369	#117	C₂³×C₆₇	1	4	yes	—
370	#117	C₂³×C₆₇	1	4	yes	—
371	#117	C₂³×C₆₇	1	2	yes	—
372	#117	C₂³×C₆₇	1	2	yes	—
373	C₂×C₄⋊C₄	—	2	8	no	—
374	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	—	2	8	no	—
378	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
379	C₂×C₄²	—	2	2	no	—
380	C₂×C₄²	—	2	8	no	—
381	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
382	#107	C₂³×C₆₇	1	4	yes	—
383	#107	C₂³×C₆₇	1	4	yes	—
384	#106	C₂³×C₆₇	1	4	yes	—
385	#106	C₂³×C₆₇	1	4	yes	—
386	#103	C₂³×C₆₇	1	16	yes	—
387	#102	C₂³×C₆₇	1	16	yes	—
388	#97	C₂³×C₆₇	1	4	yes	—
389	#97	C₂³×C₆₇	1	4	yes	—
390	C₄×C₈	C₂³×C₆₇	1	4	yes	—
391	C₄×C₈	C₂³×C₆₇	1	4	yes	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
394	#41	—	2	16	no	—
395	#41	—	2	16	no	—
396	#41	—	2	8	no	—
397	#41	—	2	8	no	—
398	#41	C₂²×C₄×C₆₇	1	16	yes	—
399	#41	C₂²×C₄×C₆₇	1	16	yes	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
403	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
404	C₂×D₄	—	2	16	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
410	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	16	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	16	no	—
417	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
418	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	#41×C₆₇	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
426	Q₁₆	C₂²×C₄×C₆₇	1	8	yes	—
427	Q₁₆	C₂²×C₄×C₆₇	1	8	yes	—
428	SD₁₆	C₂²×C₄×C₆₇	1	16	yes	—
429	SD₁₆	C₂²×C₄×C₆₇	1	16	yes	—
430	D₈	C₂²×C₄×C₆₇	1	8	yes	—
431	D₈	C₂²×C₄×C₆₇	1	8	yes	—
432	M₁₆	—	2	4	no	—
433	M₁₆	—	2	4	no	—
434	C₂×C₈	—	2	4	no	—
435	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
436	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
437	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
438	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
439	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
440	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
441	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
442	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
443	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
444	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
445	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
449	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
450	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
456	C₄²	—	2	4	no	—
457	C₄²	—	2	16	no	—
458	C₄²	—	2	4	no	—
459	C₄²	C₂×D₄×C₆₇	1	4	yes	—
460	C₄²	C₂⁴×C₆₇	1	4	yes	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	8	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	8	4	no	—
465	C₂³	C₄×D₄×C₆₇	1	4	yes	—
466	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
467	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
468	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
469	Q₈	—	2	8	no	—
470	Q₈	—	2	4	no	—
471	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
472	D₄	—	2	16	no	—
473	D₄	—	2	16	no	—
474	D₄	—	2	8	no	—
475	D₄	—	2	4	no	—
476	D₄	C₂³×C₄×C₆₇	1	8	yes	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	16	no	—
480	C₂×C₄	—	2	16	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	8	16	no	—
484	C₂×C₄	—	8	4	no	—
485	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
486	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
487	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
488	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
489	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
492	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
494	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
495	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
496	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
497	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
498	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
499	C₈	—	2	4	no	—
500	C₈	C₂³×C₄×C₆₇	1	2	yes	—
501	C₈	C₂³×C₄×C₆₇	1	2	yes	—
502	C₈	C₂³×C₄×C₆₇	1	2	yes	—
503	C₈	C₂³×C₄×C₆₇	1	2	yes	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	8	no	—
510	C₂²	—	8	8	no	—
511	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
512	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
513	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
514	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
515	C₄	—	2	8	no	—
516	C₄	—	2	8	no	—
517	C₄	—	4	8	no	—
518	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
519	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
520	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
521	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
522	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
523	C₂	—	2	2	no	—
524	C₂	—	2	2	no	—
525	C₂	—	2	2	no	—
526	C₂	—	4	8	no	—
527	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
528	C₂	C₆₇×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3428	C₂	semidirect
3. C₆₇×#2952	C₂	semidirect
4. C₆₇×#2951	C₂	semidirect
5. C₆₇×#2950	C₂	semidirect
6. C₆₇×#2949	C₂	semidirect
7. C₆₇×#2948	C₂	semidirect
8. C₆₇×#2943	C₂	semidirect
9. C₄×#135×C₆₇	C₂	semidirect
10. C₂×#427×C₆₇	C₂	semidirect
11. C₆₇×#2896	C₂	semidirect
12. C₂×#405×C₆₇	C₂	semidirect
14. C₂×#135×C₆₇	C₄	semidirect
25. #520×C₆₇	C₂²	semidirect
30. C₄×#41×C₆₇	C₂²	semidirect
34. #442×C₆₇	C₂²	semidirect
35. #441×C₆₇	C₂²	semidirect
36. #440×C₆₇	C₂²	semidirect
37. C₄×Q₁₆×C₆₇	C₂²	semidirect
38. C₄×SD₁₆×C₆₇	C₂²	semidirect
39. C₄×D₈×C₆₇	C₂²	semidirect
42. #427×C₆₇	C₂²	semidirect
43. #427×C₆₇	C₂²	semidirect
44. C₂×#107×C₆₇	C₂²	semidirect
45. C₂×#106×C₆₇	C₂²	semidirect
48. #417×C₆₇	C₂²	semidirect
49. #416×C₆₇	C₂²	semidirect
52. #405×C₆₇	C₂²	semidirect
53. #405×C₆₇	C₂²	semidirect
54. C₂×#97×C₆₇	C₂²	semidirect
55. C₂×C₄×C₈×C₆₇	C₂²	semidirect
59. C₂×#41×C₆₇	C₂×C₄	semidirect
67. #135×C₆₇	C₂×C₄	semidirect
68. #135×C₆₇	C₂×C₄	semidirect
69. #135×C₆₇	C₂×C₄	semidirect
70. #135×C₆₇	C₂×C₄	semidirect
71. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
74. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
77. C₂×D₈×C₆₇	C₂×C₄	semidirect
83. C₂²×C₈×C₆₇	C₂×C₄	semidirect
258. #60167	C₆₇	direct (N × H)
266. #3428	C₂×C₆₇	semidirect
267. #2952	C₂×C₆₇	semidirect
268. #2951	C₂×C₆₇	semidirect
269. #2950	C₂×C₆₇	semidirect
270. #2949	C₂×C₆₇	semidirect
271. #2948	C₂×C₆₇	semidirect
272. #2943	C₂×C₆₇	semidirect
273. C₄×#135	C₂×C₆₇	semidirect
274. C₂×#427	C₂×C₆₇	semidirect
275. #2896	C₂×C₆₇	semidirect
276. C₂×#405	C₂×C₆₇	semidirect
277. C₆₇	#60167	direct (N × H)
279. C₂×#135	C₄×C₆₇	semidirect
290. #520	C₂²×C₆₇	semidirect
295. C₄×#41	C₂²×C₆₇	semidirect
299. #442	C₂²×C₆₇	semidirect
300. #441	C₂²×C₆₇	semidirect
301. #440	C₂²×C₆₇	semidirect
302. C₄×Q₁₆	C₂²×C₆₇	semidirect
303. C₄×SD₁₆	C₂²×C₆₇	semidirect
304. C₄×D₈	C₂²×C₆₇	semidirect
307. #427	C₂²×C₆₇	semidirect
308. #427	C₂²×C₆₇	semidirect
309. C₂×#107	C₂²×C₆₇	semidirect
310. C₂×#106	C₂²×C₆₇	semidirect
313. #417	C₂²×C₆₇	semidirect
314. #416	C₂²×C₆₇	semidirect
317. #405	C₂²×C₆₇	semidirect
318. #405	C₂²×C₆₇	semidirect
319. C₂×#97	C₂²×C₆₇	semidirect
320. C₂×C₄×C₈	C₂²×C₆₇	semidirect
324. C₂×#41	C₂×C₄×C₆₇	semidirect
332. #135	C₂×C₄×C₆₇	semidirect
333. #135	C₂×C₄×C₆₇	semidirect
334. #135	C₂×C₄×C₆₇	semidirect
335. #135	C₂×C₄×C₆₇	semidirect
336. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
339. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
342. C₂×D₈	C₂×C₄×C₆₇	semidirect
348. C₂²×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₆₇×#3428	2
C₆₇×#2952	2
C₆₇×#2951	2
C₆₇×#2950	2
C₆₇×#2949	2
C₆₇×#2948	2
C₆₇×#2943	2
C₄×#135×C₆₇	2
C₂×#427×C₆₇	2
C₆₇×#2896	2
C₂×#405×C₆₇	2
#60167	67

Subgroups — C67×#60167 #30739418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60167 #30739418