Subgroups — #30825690

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569324	C₂	1	2	yes	maximal, complemented
2	C₄×#14524062	C₂	1	1	yes	maximal, characteristic
3	#22559123	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559119	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559117	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559102	C₂	1	2	yes	maximal, complemented
7	#22559097	C₂	1	2	yes	maximal, complemented
8	#334×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558593	C₂	1	2	yes	maximal, complemented
10	#22558494	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524062	C₄	1	1	yes	characteristic
13	C₄×#11761627	C₂²	1	2	yes	—
14	#16863526	C₂²	1	2	yes	—
15	#16863523	C₂²	1	2	yes	—
16	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
17	#16863490	C₄	1	1	yes	characteristic
18	#16863487	C₄	1	2	yes	—
19	C₄×#11761616	C₂²	1	2	yes	—
20	D₈×Dic₆₇	C₄	1	1	yes	characteristic
21	#16863474	C₂²	1	2	yes	—
22	#16863470	—	2	2	no	—
23	#16863465	—	2	2	no	—
24	#16863450	—	2	2	no	—
25	#16863447	C₂²	1	1	yes	characteristic
26	#16863447	C₂²	1	1	yes	characteristic
27	#16863433	C₄	1	1	yes	characteristic
28	#16863419	—	2	2	no	—
29	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
30	#16863389	C₄	1	2	yes	—
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863147	C₂²	1	1	yes	characteristic
33	#16863146	C₂²	1	2	yes	—
34	#16863128	C₂²	1	2	yes	—
35	#16863126	C₂²	1	2	yes	—
36	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
37	#16863117	C₂²	1	1	yes	characteristic
38	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
41	#436×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761627	C₂×C₄	1	2	yes	—
46	#14524062	—	2	4	no	—
47	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
48	C₂×#11761616	C₂×C₄	1	2	yes	—
49	C₂×#11761612	—	2	2	no	—
50	C₂×C₈×D₆₇	—	2	2	no	—
51	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₃₄	C₂³	1	2	yes	—
54	#14524008	C₂×C₄	1	2	yes	—
55	D₄×Dic₆₇	C₂×C₄	1	2	yes	—
56	C₄×C₂²⋊D₆₇	—	2	2	no	—
57	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
58	#14524001	C₂×C₄	1	2	yes	—
59	#14523999	—	2	2	no	—
60	C₂×C₄×Dic₆₇	—	2	2	no	—
61	C₄²×D₆₇	C₂³	1	1	yes	characteristic
62	#14523977	C₂×C₄	1	1	yes	characteristic
63	#14523974	C₂×C₄	1	2	yes	—
64	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₆₇⋊C₈	—	2	2	no	—
66	Dic₆₇⋊C₈	—	2	2	no	—
67	Dic₆₇⋊C₈	—	2	1	no	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	Dic₆₇⋊C₈	D₄	1	2	yes	—
70	#14523968	C₂×C₄	1	2	yes	—
71	#14523966	C₂×C₄	1	2	yes	—
72	#14523964	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
74	(C₂×D₆₇)⋊C₈	—	2	1	no	—
75	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
76	#14523953	—	2	2	no	—
77	#14523953	—	2	2	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×M₁₆×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	C₂³	1	2	yes	—
86	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₆₇	—	2	2	no	—
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	#11761627	—	2	4	no	—
94	#11761627	—	2	4	no	—
95	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₆₇	—	2	2	no	—
97	C₂²×Dic₆₇	—	2	2	no	—
98	C₂²×Dic₆₇	—	2	2	no	—
99	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
102	Dic₂₆₈	—	2	2	no	—
103	#11761616	—	2	4	no	—
104	#11761612	—	8	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
110	C₈×D₆₇	—	2	2	no	—
111	C₈×D₆₇	—	8	2	no	—
112	C₈×D₆₇	M₁₆	1	4	yes	complemented
113	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
114	Dic₆₇⋊C₄	—	2	2	no	—
115	Dic₆₇⋊C₄	—	2	2	no	—
116	C₂²⋊Dic₆₇	—	2	2	no	—
117	(C₂×D₆₇)⋊C₄	—	2	2	no	—
118	C₄×Dic₆₇	—	2	2	no	—
119	C₄×Dic₆₇	—	2	2	no	—
120	C₄×Dic₆₇	—	2	2	no	—
121	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₆₇	#41	1	1	yes	characteristic
124	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	D₈×C₆₇	—	2	2	no	—
127	M₁₆×C₆₇	—	8	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²⋊C₄×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₆₇	—	2	2	no	—
137	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
138	D₄×C₆₇	—	2	2	no	—
139	D₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₆₇	—	4	4	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	4	4	no	—
150	C₂×Dic₆₇	—	8	2	no	—
151	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
154	C₄×D₆₇	—	2	2	no	—
155	C₄×D₆₇	—	4	1	no	—
156	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
157	C₄×D₆₇	#131	1	2	yes	—
158	Dic₁₃₄	—	2	2	no	—
159	Dic₁₃₄	—	2	2	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	2	no	—
162	C₈×C₆₇	—	4	1	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
165	C₆₇⋊C₈	—	2	1	no	—
166	C₆₇⋊C₈	—	2	2	no	—
167	C₆₇⋊C₈	—	4	1	no	—
168	C₆₇⋊C₈	—	4	2	no	—
169	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₆₇	—	2	1	no	—
175	C₂×D₆₇	—	2	1	no	—
176	C₂×D₆₇	#436	1	2	yes	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
181	C₄×C₆₇	#577	1	2	yes	—
182	Dic₆₇	—	2	1	no	—
183	Dic₆₇	—	2	1	no	—
184	Dic₆₇	—	4	2	no	—
185	Dic₆₇	—	4	2	no	—
186	Dic₆₇	#436	1	2	yes	—
187	#17899	—	67	1	no	maximal, Sylow
188	C₂×C₆₇	—	8	2	no	—
189	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
190	C₂×C₆₇	#2952	1	1	yes	characteristic
191	C₂×C₆₇	#2957	1	1	yes	characteristic
192	D₆₇	—	2	2	no	—
193	#2989	—	67	2	no	—
194	C₄×#135	—	67	1	no	—
195	#1594	—	67	1	no	—
196	#1594	D₆₇	1	1	yes	characteristic, complemented
197	#1593	—	67	1	no	—
198	#1593	—	67	1	no	—
199	#1590	—	67	2	no	—
200	#1590	—	67	2	no	—
201	C₂×#334	—	67	1	no	—
202	#1477	—	67	2	no	—
203	#1450	—	67	1	no	—
204	C₆₇	#17899	1	1	yes	characteristic, complemented, Sylow
205	C₂×#135	—	67	1	no	—
206	C₄×#41	—	67	2	no	—
207	#446	—	67	2	no	—
208	#446	—	67	2	no	—
209	C₄×Q₁₆	—	67	1	no	—
210	C₄×Q₁₆	—	67	1	no	—
211	C₄×SD₁₆	—	67	2	no	—
212	C₄×SD₁₆	—	67	2	no	—
213	C₄×D₈	—	67	1	no	—
214	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
215	#436	—	67	2	no	—
216	#436	C₂×D₆₇	1	2	yes	—
217	#435	—	134	2	no	—
218	#435	—	134	2	no	—
219	#432	—	134	2	no	—
220	#431	—	67	1	no	—
221	#431	—	67	1	no	—
222	#427	—	67	1	no	—
223	#423	—	134	2	no	—
224	C₂×C₄⋊C₈	—	67	2	no	—
225	#416	—	67	2	no	—
226	C₂×C₄×C₈	—	67	1	no	—
227	#334	—	67	2	no	—
228	#334	—	67	1	no	—
229	#334	C₂×D₆₇	1	1	yes	characteristic
230	Q₈⋊C₈	—	67	2	no	—
231	Q₈⋊C₈	—	67	2	no	—
232	D₄⋊C₈	—	67	2	no	—
233	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
234	#322	—	67	1	no	—
235	#322	C₂×D₆₇	1	1	yes	characteristic
236	C₈²	—	67	1	no	—
237	C₈²	—	67	1	no	—
238	C₂×#41	—	67	2	no	—
239	#135	—	134	4	no	—
240	C₂×Q₁₆	—	67	1	no	—
241	C₂×SD₁₆	—	67	2	no	—
242	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	134	2	no	—
245	C₂²×C₈	—	67	1	no	—
246	C₂²×C₈	—	67	1	no	—
247	C₂²×C₈	—	134	2	no	—
248	C₄×Q₈	—	67	2	no	—
249	C₄×Q₈	—	67	2	no	—
250	C₄×D₄	—	67	2	no	—
251	C₄×D₄	—	134	2	no	—
252	C₄×D₄	—	134	2	no	—
253	C₄×D₄	C₂²×D₆₇	1	2	yes	—
254	#117	—	67	2	no	—
255	#117	—	134	2	no	—
256	C₂×C₄²	—	67	1	no	—
257	C₂×C₄²	—	134	2	no	—
258	#107	—	67	1	no	—
259	#107	C₄×D₆₇	1	1	yes	characteristic
260	#106	—	67	2	no	—
261	C₄⋊C₈	—	67	2	no	—
262	C₄⋊C₈	—	67	2	no	—
263	C₄⋊C₈	—	134	2	no	—
264	C₄⋊C₈	—	134	1	no	—
265	C₄⋊C₈	—	134	2	no	—
266	C₄⋊C₈	—	134	2	no	—
267	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
268	#103	—	67	2	no	—
269	#103	—	67	2	no	—
270	#102	—	67	2	no	—
271	#102	C₄×D₆₇	1	2	yes	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	67	2	no	—
274	C₂²⋊C₈	—	134	1	no	—
275	C₂²⋊C₈	—	134	2	no	—
276	#97	—	134	2	no	—
277	#97	—	134	2	no	—
278	C₄×C₈	—	67	1	no	—
279	C₄×C₈	—	67	1	no	—
280	C₄×C₈	—	67	1	no	—
281	C₄×C₈	—	67	1	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
286	#41	—	134	4	no	—
287	#41	—	134	4	no	—
288	C₂×Q₈	—	67	2	no	—
289	C₂×D₄	—	134	2	no	—
290	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	67	1	no	—
293	C₂²×C₄	—	67	1	no	—
294	C₂²×C₄	—	67	1	no	—
295	C₂²×C₄	—	134	2	no	—
296	C₂²×C₄	—	134	2	no	—
297	Q₁₆	—	134	2	no	—
298	SD₁₆	—	134	4	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	8	2	no	—
301	M₁₆	—	536	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	536	2	no	—
305	C₂×C₈	—	67	1	no	—
306	C₂×C₈	—	67	4	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	67	2	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	1	no	—
312	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₈	D₄×D₆₇	1	2	yes	—
315	C₄⋊C₄	—	67	2	no	—
316	C₄⋊C₄	—	134	2	no	—
317	C₄⋊C₄	—	134	2	no	—
318	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	134	2	no	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₄²	—	67	1	no	—
323	C₄²	—	67	1	no	—
324	C₄²	—	67	1	no	—
325	C₄²	—	134	2	no	—
326	C₄²	—	134	2	no	—
327	C₄²	—	134	2	no	—
328	C₄²	C₂³×D₆₇	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	—	67	1	no	—
331	Q₈	—	134	2	no	—
332	Q₈	—	134	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	2	no	—
335	D₄	—	268	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	268	1	no	—
340	C₂×C₄	—	268	4	no	—
341	C₂×C₄	—	536	2	no	—
342	C₂×C₄	—	67	1	no	—
343	C₂×C₄	—	67	2	no	—
344	C₂×C₄	—	67	2	no	—
345	C₂×C₄	—	67	1	no	—
346	C₂×C₄	—	67	1	no	—
347	C₂×C₄	—	134	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	—	134	2	no	—
350	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
352	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	268	1	no	—
358	C₈	—	268	2	no	—
359	C₈	—	67	2	no	—
360	C₈	—	134	1	no	—
361	C₈	—	134	2	no	—
362	C₈	M₁₆×D₆₇	1	2	yes	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	67	2	no	—
367	C₂²	—	134	1	no	—
368	C₂²	—	134	1	no	—
369	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	268	2	no	—
373	C₄	—	268	2	no	—
374	C₄	—	67	2	no	—
375	C₄	—	134	1	no	—
376	C₄	—	134	1	no	—
377	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
378	C₄	#131×D₆₇	1	1	yes	characteristic
379	C₄	#16864206	1	2	yes	—
380	C₂	—	8	2	no	—
381	C₂	—	134	2	no	—
382	C₂	#436×D₆₇	1	1	yes	characteristic
383	C₂	#22569091	1	1	yes	characteristic
384	C₂	#22569123	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1594	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524062	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1594	D₆₇
hypocenter	C₆₇	#17899
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
1. #22569324	C₂	semidirect
3. #22559123	C₂	semidirect
4. #22559119	C₂	semidirect
5. #22559117	C₂	semidirect
6. #22559102	C₂	semidirect
7. #22559097	C₂	semidirect
8. #334×D₆₇	C₂	semidirect
9. #22558593	C₂	semidirect
10. #22558494	C₂	semidirect
11. C₆₇×#1594	C₂	semidirect
112. C₈×D₆₇	M₁₆	semidirect
196. #1594	D₆₇	semidirect
204. C₆₇	#17899	semidirect

Maximal subgroups

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

subgroup	index
#22569324	2
C₄×#14524062	2
#22559123	2
#22559119	2
#22559117	2
#22559102	2
#22559097	2
#334×D₆₇	2
#22558593	2
#22558494	2
C₆₇×#1594	2
#17899	67