Subgroups — #30843763

Order 17152.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571326	C₂	1	2	yes	maximal, complemented
2	#22570786	C₂	1	1	yes	maximal, characteristic
3	#500×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22570663	C₂	1	2	yes	maximal, complemented
5	#22559882	C₂	1	2	yes	maximal, complemented
6	#22559878	C₂	1	2	yes	maximal, complemented
7	#22559827	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559826	C₂	1	1	yes	maximal, characteristic, complemented
9	#22559798	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524072	—	2	4	no	—
13	#16863944	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₆₇	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863801	C₂²	1	1	yes	characteristic
17	#16863801	C₂²	1	2	yes	—
18	#16863800	—	2	2	no	—
19	#16863799	C₂²	1	2	yes	—
20	#16863799	C₂²	1	2	yes	—
21	#16863787	—	2	4	no	—
22	#16863776	C₂²	1	2	yes	—
23	#16863775	C₂²	1	2	yes	—
24	#16863768	C₂²	1	2	yes	—
25	#16863742	C₂²	1	2	yes	—
26	#16863742	C₂²	1	2	yes	—
27	#16863708	—	2	4	no	—
28	#16863674	—	2	4	no	—
29	#16863629	—	2	2	no	—
30	D₈×Dic₆₇	—	2	2	no	—
31	#16863447	C₂²	1	1	yes	characteristic
32	#107×D₆₇	—	2	2	no	—
33	C₄×#11761611	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
35	#16863118	C₂²	1	1	yes	characteristic
36	#16863117	C₂²	1	1	yes	characteristic
37	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₆₇	C₂²	1	1	yes	characteristic
39	#493×C₆₇	C₂²	1	1	yes	characteristic
40	#492×C₆₇	C₂²	1	2	yes	—
41	#486×C₆₇	C₂²	1	2	yes	—
42	C₈²×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#11761627	—	2	2	no	—
44	C₂×Q₈×D₆₇	—	2	2	no	—
45	#14524072	—	4	16	no	—
46	C₂×Dic₂₆₈	—	2	4	no	—
47	C₂×#11761618	—	2	4	no	—
48	C₂×#11761616	—	2	4	no	—
49	C₂×#11761611	D₄	1	2	yes	—
50	C₂×C₈×D₆₇	D₄	1	2	yes	—
51	#14524043	C₂³	1	2	yes	—
52	#14524043	C₂³	1	2	yes	—
53	#14524042	—	2	2	no	—
54	#14524042	—	2	2	no	—
55	#14524041	—	2	2	no	—
56	#14524030	—	2	2	no	—
57	#14524014	—	2	2	no	—
58	#14524014	—	2	2	no	—
59	D₄×Dic₆₇	—	2	2	no	—
60	D₄×Dic₆₇	—	2	2	no	—
61	C₄⋊C₄×D₆₇	—	2	2	no	—
62	C₄⋊C₄×D₆₇	—	2	2	no	—
63	C₄²×D₆₇	C₂³	1	1	yes	characteristic
64	#14523977	—	2	2	no	—
65	#14523977	—	2	2	no	—
66	#14523977	—	2	4	no	—
67	#14523976	—	2	4	no	—
68	#14523976	—	2	4	no	—
69	Dic₆₇⋊C₈	D₄	1	2	yes	—
70	#14523964	—	2	4	no	—
71	#14523964	—	2	4	no	—
72	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
73	#14523953	D₄	1	2	yes	—
74	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₆₇	D₄	1	2	yes	—
78	C₂×SD₁₆×C₆₇	—	2	4	no	—
79	C₂×D₈×C₆₇	—	2	2	no	—
80	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
81	#127×C₆₇	C₂³	1	2	yes	—
82	#107×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	4	no	—
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#11761627	—	4	8	no	—
88	C₂×Dic₁₃₄	—	2	2	no	—
89	C₂×Dic₁₃₄	—	2	2	no	—
90	C₂×Dic₁₃₄	—	4	2	no	—
91	Q₈×D₆₇	—	4	8	no	—
92	C₂×C₂²⋊D₆₇	—	4	2	no	—
93	C₂²×Dic₆₇	—	4	2	no	—
94	C₂×C₄×D₆₇	—	4	2	no	—
95	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
98	Dic₂₆₈	—	4	8	no	—
99	#11761618	—	4	8	no	—
100	#11761616	—	4	8	no	—
101	#11761611	D₈	1	8	yes	complemented
102	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
104	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
105	C₈×D₆₇	—	2	4	no	—
106	C₄⋊Dic₆₇	—	2	2	no	—
107	C₄⋊Dic₆₇	—	2	2	no	—
108	C₄⋊Dic₆₇	—	2	2	no	—
109	C₄⋊Dic₆₇	—	2	2	no	—
110	Dic₆₇⋊C₄	—	4	2	no	—
111	Dic₆₇⋊C₄	—	4	2	no	—
112	C₂²⋊Dic₆₇	—	4	2	no	—
113	C₂²⋊Dic₆₇	—	4	2	no	—
114	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	SD₁₆×C₆₇	—	4	8	no	—
122	D₈×C₆₇	—	4	4	no	—
123	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂³×C₆₇	—	4	2	no	—
130	C₂²×D₆₇	#127	1	1	yes	characteristic
131	Q₈×C₆₇	—	4	4	no	—
132	D₄×C₆₇	—	4	4	no	—
133	D₄×C₆₇	—	4	4	no	—
134	D₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₆₇	—	8	4	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	—	4	2	no	—
142	C₂×Dic₆₇	—	8	4	no	—
143	C₂×Dic₆₇	#127	1	1	yes	characteristic
144	C₂×Dic₆₇	#127	1	1	yes	characteristic
145	C₂×Dic₆₇	#127	1	1	yes	characteristic
146	C₄×D₆₇	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	8	4	no	—
149	C₄×D₆₇	C₂×D₈	1	4	yes	—
150	Dic₁₃₄	—	4	4	no	—
151	Dic₁₃₄	—	4	4	no	—
152	Dic₁₃₄	—	8	4	no	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	C₂×D₈	1	4	yes	—
155	C₈×C₆₇	#137	1	4	yes	complemented
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	2	2	no	—
158	C₆₇⋊C₈	C₂×D₈	1	4	yes	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
163	C₂×D₆₇	—	2	1	no	—
164	C₂×D₆₇	—	2	1	no	—
165	C₂×D₆₇	#493	1	2	yes	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	C₂²×D₈	1	2	yes	—
168	C₄×C₆₇	C₂×#137	1	2	yes	—
169	C₄×C₆₇	#578	1	2	yes	—
170	Dic₆₇	—	2	1	no	—
171	Dic₆₇	—	2	1	no	—
172	Dic₆₇	—	8	2	no	—
173	Dic₆₇	—	8	2	no	—
174	Dic₆₇	#493	1	2	yes	—
175	#19630	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	16	2	no	—
177	C₂×C₆₇	C₂×#493	1	1	yes	characteristic
178	C₂×C₆₇	#3152	1	1	yes	characteristic
179	C₂×C₆₇	#3156	1	1	yes	characteristic
180	D₆₇	—	2	2	no	—
181	#3226	—	67	2	no	—
182	#3169	—	67	1	no	—
183	C₂×#500	—	67	1	no	—
184	#3152	—	67	2	no	—
185	#1722	—	67	2	no	—
186	#1722	—	67	2	no	—
187	#1713	—	67	1	no	—
188	#1713	—	67	1	no	—
189	#1706	—	67	1	no	—
190	#1706	D₆₇	1	1	yes	characteristic, complemented
191	C₈×M₁₆	—	67	1	no	—
192	C₆₇	#19630	1	1	yes	characteristic, complemented, Sylow
193	C₂×#137	—	134	4	no	—
194	#532	—	67	2	no	—
195	C₂×C₄⋊Q₈	—	67	2	no	—
196	#500	—	67	2	no	—
197	#500	—	67	2	no	—
198	#500	—	67	1	no	—
199	#500	—	67	1	no	—
200	#500	—	67	2	no	—
201	#500	—	134	2	no	—
202	#500	C₂×D₆₇	1	1	yes	characteristic
203	#497	—	134	4	no	—
204	#494	—	67	2	no	—
205	#494	—	67	2	no	—
206	#493	C₂×D₆₇	1	1	yes	characteristic
207	#492	—	67	2	no	—
208	#492	C₂×D₆₇	1	2	yes	—
209	#486	—	67	2	no	—
210	#486	—	67	2	no	—
211	#486	C₂×D₆₇	1	2	yes	—
212	#480	—	134	4	no	—
213	#474	—	134	4	no	—
214	#466	—	134	2	no	—
215	C₄×D₈	—	134	2	no	—
216	#431	—	67	1	no	—
217	C₂×#107	—	134	2	no	—
218	C₄×M₁₆	—	67	1	no	—
219	C₂×C₄×C₈	—	67	1	no	—
220	#322	—	67	1	no	—
221	#322	—	67	1	no	—
222	C₈²	—	67	1	no	—
223	C₈²	C₂×D₆₇	1	1	yes	characteristic
224	C₂×#41	—	134	2	no	—
225	C₂²×Q₈	—	134	2	no	—
226	#137	—	268	16	no	—
227	C₂×Q₁₆	—	134	4	no	—
228	C₂×Q₁₆	—	134	4	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×SD₁₆	—	134	4	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×M₁₆	—	67	2	no	—
233	C₂²×C₈	—	67	2	no	—
234	C₄⋊Q₈	—	67	2	no	—
235	C₄⋊Q₈	—	67	2	no	—
236	C₄⋊Q₈	—	134	2	no	—
237	C₄⋊Q₈	—	134	2	no	—
238	C₄⋊Q₈	—	134	2	no	—
239	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
240	#127	C₂²×D₆₇	1	2	yes	—
241	#124	—	134	2	no	—
242	#121	—	134	2	no	—
243	#121	—	134	2	no	—
244	C₄×D₄	—	134	2	no	—
245	C₄×D₄	—	134	2	no	—
246	C₂×C₄⋊C₄	—	134	2	no	—
247	C₂×C₄⋊C₄	—	134	2	no	—
248	C₂×C₄²	—	67	1	no	—
249	#107	—	2	2	no	—
250	#107	—	134	4	no	—
251	#107	—	134	4	no	—
252	#107	—	134	2	no	—
253	#107	—	134	2	no	—
254	#107	—	134	4	no	—
255	C₄⋊C₈	—	67	2	no	—
256	#102	—	2	4	no	—
257	#102	—	134	4	no	—
258	#102	—	134	4	no	—
259	C₂²⋊C₈	—	67	2	no	—
260	#97	—	67	2	no	—
261	C₄×C₈	—	67	2	no	—
262	C₄×C₈	—	67	1	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	1	no	—
265	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
268	#41	—	268	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	268	2	no	—
271	C₂×Q₈	—	268	8	no	—
272	C₂×Q₈	—	134	2	no	—
273	C₂×Q₈	—	134	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	268	2	no	—
278	C₂²×C₄	—	268	2	no	—
279	C₂²×C₄	—	268	2	no	—
280	C₂²×C₄	—	67	1	no	—
281	C₂²×C₄	—	67	1	no	—
282	C₂²×C₄	—	67	1	no	—
283	Q₁₆	—	268	8	no	—
284	Q₁₆	—	268	8	no	—
285	SD₁₆	—	4	8	no	—
286	SD₁₆	—	268	8	no	—
287	D₈	—	4	4	no	—
288	M₁₆	—	67	8	no	—
289	C₂×C₈	—	67	2	no	—
290	C₂×C₈	—	67	2	no	—
291	C₂×C₈	—	67	2	no	—
292	C₂×C₈	—	134	4	no	—
293	C₂×C₈	D₄×D₆₇	1	2	yes	—
294	C₂×C₈	D₄×D₆₇	1	2	yes	—
295	C₂×C₈	D₄×D₆₇	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	268	2	no	—
299	C₄⋊C₄	—	268	2	no	—
300	C₄⋊C₄	—	134	2	no	—
301	C₄⋊C₄	—	134	2	no	—
302	C₄⋊C₄	—	134	2	no	—
303	C₄⋊C₄	—	134	2	no	—
304	C₂²⋊C₄	—	268	2	no	—
305	C₂²⋊C₄	—	268	2	no	—
306	C₄²	—	67	1	no	—
307	C₄²	—	67	1	no	—
308	C₄²	—	67	1	no	—
309	C₄²	C₂³×D₆₇	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	67	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	268	4	no	—
314	Q₈	—	268	4	no	—
315	Q₈	—	536	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	4	no	—
319	D₄	—	536	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	268	2	no	—
322	C₂×C₄	—	268	2	no	—
323	C₂×C₄	—	536	4	no	—
324	C₂×C₄	—	536	4	no	—
325	C₂×C₄	—	67	1	no	—
326	C₂×C₄	—	67	4	no	—
327	C₂×C₄	—	67	1	no	—
328	C₂×C₄	—	67	1	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	67	4	no	—
336	C₈	—	134	2	no	—
337	C₈	—	134	2	no	—
338	C₈	D₈×D₆₇	1	4	yes	—
339	C₈	#14524072	1	4	yes	complemented
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	67	2	no	—
344	C₂²	—	134	1	no	—
345	C₂²	—	134	1	no	—
346	C₂²	#127×D₆₇	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	536	2	no	—
349	C₄	—	536	2	no	—
350	C₄	—	67	2	no	—
351	C₄	—	134	1	no	—
352	C₄	—	134	1	no	—
353	C₄	C₂×D₈×D₆₇	1	2	yes	—
354	C₄	C₂×#14524072	1	2	yes	—
355	C₄	#16864210	1	2	yes	—
356	C₂	—	16	2	no	—
357	C₂	—	134	2	no	—
358	C₂	#493×D₆₇	1	1	yes	characteristic
359	C₂	#22570663	1	1	yes	characteristic
360	C₂	#22570699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₆₇
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1706	C₂
socle	C₂²×C₆₇	C₂×#127
prime radical	#22570786	C₂
hypercenter	#1706	D₆₇
hypocenter	C₆₇	#19630
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #22571326	C₂	semidirect
3. #500×D₆₇	C₂	semidirect
4. #22570663	C₂	semidirect
5. #22559882	C₂	semidirect
6. #22559878	C₂	semidirect
7. #22559827	C₂	semidirect
8. #22559826	C₂	semidirect
9. #22559798	C₂	semidirect
10. C₈×#11761611	C₂	semidirect
11. C₆₇×#1706	C₂	semidirect
101. #11761611	D₈	semidirect
155. C₈×C₆₇	#137	semidirect
190. #1706	D₆₇	semidirect
192. C₆₇	#19630	semidirect
339. C₈	#14524072	semidirect

Maximal subgroups

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

subgroup	index
#22571326	2
#22570786	2
#500×D₆₇	2
#22570663	2
#22559882	2
#22559878	2
#22559827	2
#22559826	2
#22559798	2
C₈×#11761611	2
C₆₇×#1706	2
#19630	67