Subgroups — C₆₇×#60618 #30739869

Order 17152.

2170 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3142	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3141	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3140	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3139	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3138	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3135	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3131	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2896	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	#533×C₆₇	—	2	2	no	—
15	#533×C₆₇	—	2	2	no	—
16	#533×C₆₇	C₂²	1	2	yes	—
17	#533×C₆₇	C₂²	1	2	yes	—
18	#533×C₆₇	C₂²	1	2	yes	complemented
19	#533×C₆₇	C₂²	1	2	yes	complemented
20	#532×C₆₇	—	2	2	no	—
21	#532×C₆₇	—	2	2	no	—
22	#532×C₆₇	C₂²	1	2	yes	—
23	#532×C₆₇	C₂²	1	2	yes	—
24	#532×C₆₇	C₂²	1	2	yes	—
25	#532×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#124×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
32	C₂×#121×C₆₇	—	2	2	no	—
33	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
34	C₂×C₄×D₄×C₆₇	—	2	2	no	—
35	C₂×#117×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
37	#491×C₆₇	C₂²	1	8	yes	complemented
38	#490×C₆₇	C₂²	1	8	yes	complemented
39	#489×C₆₇	C₂²	1	16	yes	complemented
40	#488×C₆₇	C₂²	1	16	yes	complemented
41	#487×C₆₇	C₂²	1	8	yes	complemented
42	#486×C₆₇	C₂²	1	8	yes	complemented
43	#419×C₆₇	C₂²	1	8	yes	—
44	#418×C₆₇	C₂²	1	8	yes	—
45	#417×C₆₇	C₂²	1	16	yes	complemented
46	#416×C₆₇	C₂²	1	16	yes	complemented
47	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
48	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
49	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
50	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
51	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
52	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	2	2	no	—
55	C₂×#41×C₆₇	—	2	16	no	—
56	C₂×#41×C₆₇	C₂³	1	4	yes	—
57	C₂×#41×C₆₇	C₂³	1	4	yes	—
58	C₂²×Q₈×C₆₇	—	2	2	no	—
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₇	—	2	2	no	—
61	C₂²×D₄×C₆₇	—	2	2	no	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	D₄	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	—	2	8	no	—
67	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
70	C₄⋊Q₈×C₆₇	—	2	2	no	—
71	C₄⋊Q₈×C₆₇	—	2	2	no	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	#127×C₆₇	—	2	2	no	—
78	#127×C₆₇	C₂³	1	2	yes	—
79	#127×C₆₇	C₂³	1	2	yes	—
80	#125×C₆₇	—	2	4	no	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#125×C₆₇	C₂³	1	4	yes	—
83	#124×C₆₇	—	2	4	no	—
84	#124×C₆₇	C₂³	1	4	yes	—
85	#124×C₆₇	C₂³	1	4	yes	—
86	C₂²⋊Q₈×C₆₇	—	2	4	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	4	no	—
89	C₂²⋊Q₈×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	4	no	—
91	#121×C₆₇	—	2	4	no	—
92	#121×C₆₇	—	2	4	no	—
93	#121×C₆₇	—	2	4	no	—
94	C₄×Q₈×C₆₇	—	2	4	no	—
95	C₄×Q₈×C₆₇	—	2	4	no	—
96	C₄×Q₈×C₆₇	—	2	4	no	—
97	C₄×Q₈×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	2	4	no	—
99	C₄×D₄×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	—	2	4	no	—
101	C₄×D₄×C₆₇	—	2	4	no	—
102	#117×C₆₇	—	2	2	no	—
103	#117×C₆₇	—	2	2	no	—
104	#117×C₆₇	C₂³	1	4	yes	—
105	#117×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
112	C₂×C₄²×C₆₇	—	2	2	no	—
113	C₂×C₄²×C₆₇	D₄	1	2	yes	—
114	C₂×C₄²×C₆₇	D₄	1	2	yes	—
115	C₂×C₄²×C₆₇	D₄	1	2	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	#103×C₆₇	C₂³	1	32	yes	—
123	#102×C₆₇	C₂³	1	32	yes	—
124	#97×C₆₇	C₂³	1	4	yes	—
125	#97×C₆₇	C₂³	1	4	yes	—
126	C₄×C₈×C₆₇	C₂³	1	4	yes	—
127	C₄×C₈×C₆₇	C₂³	1	4	yes	—
128	C₂⁴×C₆₇	—	2	2	no	—
129	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
130	#41×C₆₇	—	2	16	no	—
131	#41×C₆₇	—	2	16	no	—
132	#41×C₆₇	—	4	8	no	—
133	#41×C₆₇	—	4	8	no	—
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	8	no	—
138	C₂×Q₈×C₆₇	—	4	8	no	—
139	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	2	no	—
145	C₂×D₄×C₆₇	—	2	8	no	—
146	C₂×D₄×C₆₇	—	4	8	no	—
147	C₂×D₄×C₆₇	—	4	8	no	—
148	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	4	4	no	—
154	C₂²×C₄×C₆₇	—	4	8	no	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	#41	1	8	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₆₇	—	2	8	no	—
169	M₁₆×C₆₇	—	2	8	no	—
170	C₂×C₈×C₆₇	—	2	8	no	—
171	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	8	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	4	no	—
177	C₄⋊C₄×C₆₇	—	2	4	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₂²⋊C₄×C₆₇	—	2	8	no	—
183	C₂²⋊C₄×C₆₇	—	2	8	no	—
184	C₄²×C₆₇	—	2	4	no	—
185	C₄²×C₆₇	—	2	4	no	—
186	C₄²×C₆₇	C₂×D₄	1	2	yes	—
187	C₄²×C₆₇	C₂×D₄	1	2	yes	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₄²×C₆₇	C₂×D₄	1	2	yes	—
190	C₄²×C₆₇	C₂×D₄	1	2	yes	—
191	C₄²×C₆₇	C₂×D₄	1	2	yes	—
192	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₂³×C₆₇	—	2	4	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	8	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	#124	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	—	4	8	no	—
206	Q₈×C₆₇	—	4	4	no	—
207	Q₈×C₆₇	—	4	8	no	—
208	D₄×C₆₇	—	2	8	no	—
209	D₄×C₆₇	—	4	8	no	—
210	D₄×C₆₇	—	4	8	no	—
211	D₄×C₆₇	—	4	8	no	—
212	D₄×C₆₇	—	4	4	no	—
213	D₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	8	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	—	4	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	8	no	—
223	C₂×C₄×C₆₇	—	4	4	no	—
224	C₂×C₄×C₆₇	—	4	8	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	#124	1	4	yes	—
227	C₂×C₄×C₆₇	#124	1	4	yes	—
228	C₂×C₄×C₆₇	#124	1	4	yes	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
237	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
238	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
239	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
240	C₈×C₆₇	—	2	8	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	2	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	8	no	—
247	C₂²×C₆₇	—	8	8	no	—
248	C₂²×C₆₇	C₂×#124	1	2	yes	—
249	C₂²×C₆₇	C₂×#124	1	2	yes	—
250	C₂²×C₆₇	C₂×#124	1	2	yes	—
251	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	2	2	no	—
254	C₄×C₆₇	—	2	2	no	—
255	C₄×C₆₇	—	4	4	no	—
256	C₄×C₆₇	—	4	8	no	—
257	C₄×C₆₇	C₂×#124	1	2	yes	—
258	C₄×C₆₇	C₂×#124	1	2	yes	—
259	C₄×C₆₇	C₂×#124	1	2	yes	—
260	C₄×C₆₇	#575	1	4	yes	—
261	C₄×C₆₇	C₂²×#41	1	2	yes	—
262	#60618	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	2	2	no	—
264	C₂×C₆₇	—	2	2	no	—
265	C₂×C₆₇	—	2	2	no	—
266	C₂×C₆₇	—	8	4	no	—
267	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
268	C₂×C₆₇	C₂×#575	1	2	yes	—
269	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
271	#3142	C₂×C₆₇	1	4	yes	complemented
272	#3141	C₂×C₆₇	1	4	yes	complemented
273	#3140	C₂×C₆₇	1	4	yes	complemented
274	#3139	C₂×C₆₇	1	2	yes	complemented
275	#3138	C₂×C₆₇	1	2	yes	complemented
276	#3135	C₂×C₆₇	1	4	yes	complemented
277	#3131	C₂×C₆₇	1	4	yes	complemented
278	#2896	C₂×C₆₇	1	4	yes	complemented
279	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₆₇	#60618	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²×#41	C₂²×C₆₇	1	2	yes	—
282	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
283	#533	—	2	2	no	—
284	#533	—	2	2	no	—
285	#533	C₂²×C₆₇	1	2	yes	—
286	#533	C₂²×C₆₇	1	2	yes	—
287	#533	C₂²×C₆₇	1	2	yes	complemented
288	#533	C₂²×C₆₇	1	2	yes	complemented
289	#532	—	2	2	no	—
290	#532	—	2	2	no	—
291	#532	C₂²×C₆₇	1	2	yes	—
292	#532	C₂²×C₆₇	1	2	yes	—
293	#532	C₂²×C₆₇	1	2	yes	—
294	#532	C₂²×C₆₇	1	2	yes	—
295	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
297	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
299	C₂×#124	C₂²×C₆₇	1	2	yes	—
300	C₂×C₂²⋊Q₈	—	2	2	no	—
301	C₂×#121	—	2	2	no	—
302	C₂×C₄×Q₈	—	2	2	no	—
303	C₂×C₄×D₄	—	2	2	no	—
304	C₂×#117	C₂²×C₆₇	1	2	yes	—
305	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
306	#491	C₂²×C₆₇	1	8	yes	complemented
307	#490	C₂²×C₆₇	1	8	yes	complemented
308	#489	C₂²×C₆₇	1	16	yes	complemented
309	#488	C₂²×C₆₇	1	16	yes	complemented
310	#487	C₂²×C₆₇	1	8	yes	complemented
311	#486	C₂²×C₆₇	1	8	yes	complemented
312	#419	C₂²×C₆₇	1	8	yes	—
313	#418	C₂²×C₆₇	1	8	yes	—
314	#417	C₂²×C₆₇	1	16	yes	complemented
315	#416	C₂²×C₆₇	1	16	yes	complemented
316	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
317	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
318	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
319	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
320	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
321	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	16	no	—
325	C₂×#41	C₂³×C₆₇	1	4	yes	—
326	C₂×#41	C₂³×C₆₇	1	4	yes	—
327	C₂²×Q₈	—	2	2	no	—
328	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
332	C₂³×C₄	—	2	2	no	—
333	C₂³×C₄	D₄×C₆₇	1	2	yes	—
334	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
335	C₂×M₁₆	—	2	8	no	—
336	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
337	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
338	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
343	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
344	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
345	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
346	#127	—	2	2	no	—
347	#127	C₂³×C₆₇	1	2	yes	—
348	#127	C₂³×C₆₇	1	2	yes	—
349	#125	—	2	4	no	—
350	#125	C₂³×C₆₇	1	4	yes	—
351	#125	C₂³×C₆₇	1	4	yes	—
352	#124	—	2	4	no	—
353	#124	C₂³×C₆₇	1	4	yes	—
354	#124	C₂³×C₆₇	1	4	yes	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	#121	—	2	4	no	—
360	#121	—	2	4	no	—
361	#121	—	2	4	no	—
362	#121	—	2	4	no	—
363	C₄×Q₈	—	2	4	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	4	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	#117	—	2	2	no	—
372	#117	—	2	2	no	—
373	#117	C₂³×C₆₇	1	4	yes	—
374	#117	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	4	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₂²⋊C₄	—	2	4	no	—
381	C₂×C₄²	—	2	2	no	—
382	C₂×C₄²	D₄×C₆₇	1	2	yes	—
383	C₂×C₄²	D₄×C₆₇	1	2	yes	—
384	C₂×C₄²	D₄×C₆₇	1	2	yes	—
385	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	#103	C₂³×C₆₇	1	32	yes	—
392	#102	C₂³×C₆₇	1	32	yes	—
393	#97	C₂³×C₆₇	1	4	yes	—
394	#97	C₂³×C₆₇	1	4	yes	—
395	C₄×C₈	C₂³×C₆₇	1	4	yes	—
396	C₄×C₈	C₂³×C₆₇	1	4	yes	—
397	C₂⁴	—	2	2	no	—
398	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
399	#41	—	2	16	no	—
400	#41	—	2	16	no	—
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	—	4	8	no	—
408	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	4	8	no	—
416	C₂×D₄	—	4	8	no	—
417	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	#41×C₆₇	1	8	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
437	M₁₆	—	2	8	no	—
438	M₁₆	—	2	8	no	—
439	C₂×C₈	—	2	8	no	—
440	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	8	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₆₇	1	2	yes	—
456	C₄²	C₂×D₄×C₆₇	1	2	yes	—
457	C₄²	C₂×D₄×C₆₇	1	2	yes	—
458	C₄²	C₂×D₄×C₆₇	1	2	yes	—
459	C₄²	C₂×D₄×C₆₇	1	2	yes	—
460	C₄²	C₂×D₄×C₆₇	1	2	yes	—
461	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
462	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
463	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₂³	—	2	4	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	8	no	—
468	C₂³	—	4	4	no	—
469	C₂³	#124×C₆₇	1	4	yes	—
470	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
471	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
472	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
473	Q₈	—	2	8	no	—
474	Q₈	—	4	8	no	—
475	Q₈	—	4	4	no	—
476	Q₈	—	4	8	no	—
477	D₄	—	2	8	no	—
478	D₄	—	4	8	no	—
479	D₄	—	4	8	no	—
480	D₄	—	4	8	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	8	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	#124×C₆₇	1	4	yes	—
496	C₂×C₄	#124×C₆₇	1	4	yes	—
497	C₂×C₄	#124×C₆₇	1	4	yes	—
498	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
501	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
505	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
506	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
507	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
508	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
509	C₈	—	2	8	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	8	8	no	—
517	C₂²	C₂×#124×C₆₇	1	2	yes	—
518	C₂²	C₂×#124×C₆₇	1	2	yes	—
519	C₂²	C₂×#124×C₆₇	1	2	yes	—
520	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	8	no	—
525	C₄	—	4	4	no	—
526	C₄	C₂×#124×C₆₇	1	2	yes	—
527	C₄	C₂×#124×C₆₇	1	2	yes	—
528	C₄	C₂×#124×C₆₇	1	2	yes	—
529	C₄	#575×C₆₇	1	4	yes	—
530	C₄	C₂²×#41×C₆₇	1	2	yes	—
531	C₂	—	2	2	no	—
532	C₂	—	2	2	no	—
533	C₂	—	2	2	no	—
534	C₂	—	8	4	no	—
535	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
536	C₂	C₂×#575×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#533×C₆₇	C₂	semidirect
3. C₆₇×#3142	C₂	semidirect
4. C₆₇×#3141	C₂	semidirect
5. C₆₇×#3140	C₂	semidirect
6. C₆₇×#3139	C₂	semidirect
7. C₆₇×#3138	C₂	semidirect
8. C₆₇×#3135	C₂	semidirect
9. C₆₇×#3131	C₂	semidirect
10. C₆₇×#2896	C₂	semidirect
11. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
18. #533×C₆₇	C₂²	semidirect
19. #533×C₆₇	C₂²	semidirect
27. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
29. C₂×#125×C₆₇	C₂²	semidirect
37. #491×C₆₇	C₂²	semidirect
38. #490×C₆₇	C₂²	semidirect
39. #489×C₆₇	C₂²	semidirect
40. #488×C₆₇	C₂²	semidirect
41. #487×C₆₇	C₂²	semidirect
42. #486×C₆₇	C₂²	semidirect
45. #417×C₆₇	C₂²	semidirect
46. #416×C₆₇	C₂²	semidirect
49. C₄×M₁₆×C₆₇	C₂²	semidirect
50. C₄×M₁₆×C₆₇	C₂²	semidirect
51. C₂×#97×C₆₇	C₂²	semidirect
52. C₂×C₄×C₈×C₆₇	C₂²	semidirect
262. #60618	C₆₇	direct (N × H)
269. C₂×#533	C₂×C₆₇	semidirect
271. #3142	C₂×C₆₇	semidirect
272. #3141	C₂×C₆₇	semidirect
273. #3140	C₂×C₆₇	semidirect
274. #3139	C₂×C₆₇	semidirect
275. #3138	C₂×C₆₇	semidirect
276. #3135	C₂×C₆₇	semidirect
277. #3131	C₂×C₆₇	semidirect
278. #2896	C₂×C₆₇	semidirect
279. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
280. C₆₇	#60618	direct (N × H)
287. #533	C₂²×C₆₇	semidirect
288. #533	C₂²×C₆₇	semidirect
296. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
298. C₂×#125	C₂²×C₆₇	semidirect
306. #491	C₂²×C₆₇	semidirect
307. #490	C₂²×C₆₇	semidirect
308. #489	C₂²×C₆₇	semidirect
309. #488	C₂²×C₆₇	semidirect
310. #487	C₂²×C₆₇	semidirect
311. #486	C₂²×C₆₇	semidirect
314. #417	C₂²×C₆₇	semidirect
315. #416	C₂²×C₆₇	semidirect
318. C₄×M₁₆	C₂²×C₆₇	semidirect
319. C₄×M₁₆	C₂²×C₆₇	semidirect
320. C₂×#97	C₂²×C₆₇	semidirect
321. 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
C₂×#533×C₆₇	2
C₂×#532×C₆₇	2
C₆₇×#3142	2
C₆₇×#3141	2
C₆₇×#3140	2
C₆₇×#3139	2
C₆₇×#3138	2
C₆₇×#3135	2
C₆₇×#3131	2
C₆₇×#2896	2
C₂×C₄×M₁₆×C₆₇	2
#60618	67

Subgroups — C67×#60618 #30739869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60618 #30739869