Subgroups — #30852925

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572929	C₂	1	2	yes	maximal
2	#22572494	C₂	1	2	yes	maximal, complemented
3	#22559819	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559763	C₂	1	2	yes	maximal, complemented
5	#22559757	C₂	1	2	yes	maximal, complemented
6	#22559468	C₂	1	2	yes	maximal, complemented
7	#22559460	C₂	1	2	yes	maximal, complemented
8	#22559095	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864119	C₂²	1	2	yes	—
11	Q₈×D₂₆₈	C₂²	1	2	yes	—
12	#16863801	C₂²	1	1	yes	characteristic
13	#16863801	C₂²	1	1	yes	characteristic
14	#16863756	—	2	2	no	—
15	#16863747	C₂²	1	2	yes	—
16	#16863747	C₂²	1	2	yes	—
17	#16863739	—	2	2	no	—
18	#16863692	—	2	2	no	—
19	#16863681	C₂²	1	2	yes	—
20	#16863677	C₂²	1	2	yes	—
21	#16863653	—	2	2	no	—
22	#16863612	—	2	2	no	—
23	#16863606	C₂²	1	2	yes	—
24	#16863602	C₂²	1	2	yes	—
25	#16863550	—	2	2	no	—
26	#16863498	—	2	2	no	—
27	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
28	#16863475	C₂²	1	2	yes	—
29	#16863403	—	2	2	no	—
30	#16863149	C₂²	1	1	yes	characteristic
31	#16863143	C₂²	1	2	yes	—
32	#16863141	C₂²	1	2	yes	—
33	#16863125	C₂²	1	2	yes	—
34	#446×C₆₇	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#335×C₆₇	C₂²	1	1	yes	characteristic
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×Q₈×D₆₇	—	2	2	no	—
40	C₂×#11761618	—	2	2	no	—
41	C₂×#11761617	—	2	2	no	—
42	C₂×D₅₃₆	D₄	1	1	yes	characteristic
43	C₂×#11761611	—	2	2	no	—
44	#14524043	—	2	2	no	—
45	#14524043	C₂³	1	2	yes	—
46	#14524043	C₂³	1	2	yes	—
47	#14524038	—	4	2	no	—
48	#14524031	—	2	2	no	—
49	#14524023	—	2	2	no	—
50	#14524023	—	4	2	no	—
51	#14524019	—	2	2	no	—
52	C₄×Dic₁₃₄	—	2	2	no	—
53	C₄×D₂₆₈	—	2	2	no	—
54	C₄×D₂₆₈	C₂³	1	2	yes	—
55	#14524001	—	2	2	no	—
56	#14523977	—	2	2	no	—
57	#14523977	—	4	1	no	—
58	#14523977	—	4	1	no	—
59	#14523977	D₄	1	1	yes	characteristic
60	#14523974	—	2	2	no	—
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
62	#14523968	—	2	2	no	—
63	#14523968	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523968	—	2	2	no	—
66	#14523968	—	2	2	no	—
67	#14523967	—	2	2	no	—
68	#14523966	—	2	2	no	—
69	#14523965	—	2	2	no	—
70	#14523965	D₄	1	2	yes	—
71	(C₂×D₆₇)⋊C₈	—	2	2	no	—
72	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
77	#103×C₆₇	D₄	1	2	yes	—
78	#97×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂×Dic₁₃₄	—	2	2	no	—
84	C₂×Dic₁₃₄	—	4	2	no	—
85	Q₈×D₆₇	—	4	4	no	—
86	Q₈×D₆₇	—	8	2	no	—
87	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
88	C₂×C₄×D₆₇	—	2	2	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	#11761618	—	8	2	no	—
91	#11761617	—	4	4	no	—
92	D₅₃₆	—	4	1	no	—
93	#11761611	—	8	2	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	C₄⋊Dic₆₇	—	2	2	no	—
98	C₄⋊Dic₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	4	2	no	—
100	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
101	Dic₆₇⋊C₄	—	4	2	no	—
102	(C₂×D₆₇)⋊C₄	—	2	2	no	—
103	(C₂×D₆₇)⋊C₄	—	4	2	no	—
104	(C₂×D₆₇)⋊C₄	—	4	2	no	—
105	(C₂×D₆₇)⋊C₄	—	4	2	no	—
106	C₄×Dic₆₇	—	4	2	no	—
107	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
108	Q₁₆×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂²×D₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	D₂₆₈	—	2	2	no	—
131	D₂₆₈	—	2	2	no	—
132	C₄×D₆₇	—	4	2	no	—
133	C₄×D₆₇	—	4	2	no	—
134	C₄×D₆₇	—	8	4	no	—
135	Dic₁₃₄	—	4	2	no	—
136	Dic₁₃₄	—	4	2	no	—
137	Dic₁₃₄	—	4	2	no	—
138	Dic₁₃₄	—	4	2	no	—
139	Dic₁₃₄	—	8	4	no	—
140	C₈×C₆₇	—	2	2	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₆₇⋊C₈	—	4	2	no	—
147	C₂²×C₆₇	D₄²	1	1	yes	characteristic
148	C₂×D₆₇	—	4	2	no	—
149	C₂×D₆₇	—	4	2	no	—
150	C₂×D₆₇	—	4	2	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	#575	1	1	yes	characteristic
155	C₄×C₆₇	#575	1	1	yes	characteristic
156	C₄×C₆₇	#576	1	1	yes	characteristic
157	C₄×C₆₇	#578	1	1	yes	characteristic
158	Dic₆₇	—	4	2	no	—
159	Dic₆₇	—	8	2	no	—
160	Dic₆₇	—	8	2	no	—
161	#20306	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	#3273	1	1	yes	characteristic
163	C₂×C₆₇	#3277	1	1	yes	characteristic
164	C₂×C₆₇	#3293	1	1	yes	characteristic
165	D₆₇	—	8	2	no	—
166	#3367	—	67	2	no	—
167	#3328	—	67	2	no	—
168	#1710	—	67	1	no	—
169	#1700	—	67	2	no	—
170	#1699	—	67	2	no	—
171	#1650	—	67	2	no	—
172	#1649	—	67	2	no	—
173	#1591	D₆₇	1	1	yes	characteristic, complemented
174	#1589	—	67	1	no	—
175	C₆₇	#20306	1	1	yes	characteristic, complemented, Sylow
176	#556	—	67	2	no	—
177	D₄×Q₈	—	67	2	no	—
178	#500	—	67	1	no	—
179	#500	—	67	1	no	—
180	#489	—	134	2	no	—
181	#487	—	67	2	no	—
182	#487	—	67	2	no	—
183	#485	—	134	2	no	—
184	#477	—	134	2	no	—
185	#475	—	67	2	no	—
186	#474	—	67	2	no	—
187	#470	—	134	2	no	—
188	#463	—	134	2	no	—
189	#462	—	67	2	no	—
190	#461	—	67	2	no	—
191	#451	—	134	2	no	—
192	#446	C₂×D₆₇	1	2	yes	—
193	#440	—	134	2	no	—
194	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
195	C₄×D₈	—	67	1	no	—
196	#436	—	67	2	no	—
197	#419	—	134	2	no	—
198	#335	—	67	1	no	—
199	#335	C₂×D₆₇	1	1	yes	characteristic
200	#332	—	67	2	no	—
201	#332	—	67	2	no	—
202	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
203	D₄⋊C₈	—	67	2	no	—
204	C₈²	C₂×D₆₇	1	1	yes	characteristic
205	C₂²×Q₈	—	134	2	no	—
206	C₂×Q₁₆	—	134	2	no	—
207	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic
208	C₂×SD₁₆	—	134	2	no	—
209	C₂×D₈	—	67	1	no	—
210	C₂×M₁₆	—	134	2	no	—
211	C₄⋊Q₈	—	67	2	no	—
212	C₄⋊Q₈	—	67	2	no	—
213	C₄⋊Q₈	—	134	2	no	—
214	#126	—	268	2	no	—
215	#124	—	134	2	no	—
216	C₂²⋊Q₈	—	268	2	no	—
217	C₂²⋊Q₈	—	134	2	no	—
218	C₂²⋊Q₈	—	134	2	no	—
219	C₄×Q₈	—	134	2	no	—
220	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
221	C₄×D₄	—	67	2	no	—
222	C₄×D₄	—	134	2	no	—
223	#117	—	134	2	no	—
224	#107	—	268	1	no	—
225	#107	—	268	1	no	—
226	#107	—	67	1	no	—
227	#107	—	134	2	no	—
228	#107	D₂₆₈	1	1	yes	characteristic
229	#106	—	134	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	67	2	no	—
232	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
233	#103	—	134	2	no	—
234	#103	—	134	2	no	—
235	#103	—	134	2	no	—
236	#103	—	134	2	no	—
237	#103	—	134	2	no	—
238	#103	—	134	2	no	—
239	#103	—	134	2	no	—
240	#103	D₂₆₈	1	2	yes	—
241	#102	—	67	2	no	—
242	#102	—	134	2	no	—
243	C₂²⋊C₈	—	134	2	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	C₂×Q₈	—	268	2	no	—
249	C₂×Q₈	—	268	4	no	—
250	C₂×Q₈	—	536	2	no	—
251	C₂×Q₈	—	134	2	no	—
252	C₂×Q₈	—	134	2	no	—
253	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
254	C₂×D₄	—	67	2	no	—
255	C₂²×C₄	—	268	2	no	—
256	C₂²×C₄	—	134	2	no	—
257	Q₁₆	—	4	1	no	—
258	Q₁₆	—	536	2	no	—
259	SD₁₆	—	268	4	no	—
260	D₈	—	268	1	no	—
261	M₁₆	—	536	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	134	2	no	—
265	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
266	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
267	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	67	2	no	—
273	C₄⋊C₄	—	134	2	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₄⋊C₄	—	134	2	no	—
277	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
278	C₂²⋊C₄	—	268	2	no	—
279	C₂²⋊C₄	—	268	2	no	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₂²⋊C₄	—	134	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	268	2	no	—
284	C₄²	C₂³×D₆₇	1	1	yes	characteristic
285	C₂³	—	134	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	268	2	no	—
289	Q₈	—	268	2	no	—
290	Q₈	—	268	2	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	536	4	no	—
293	D₄	—	134	2	no	—
294	D₄	—	134	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	268	2	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	536	4	no	—
302	C₂×C₄	—	134	2	no	—
303	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
305	C₂×C₄	#14524089	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	268	2	no	—
313	C₂²	—	268	2	no	—
314	C₂²	—	268	2	no	—
315	C₂²	—	268	2	no	—
316	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	268	2	no	—
321	C₄	—	536	2	no	—
322	C₄	—	536	2	no	—
323	C₄	#16864193	1	1	yes	characteristic
324	C₄	#16864196	1	1	yes	characteristic
325	C₄	#16864201	1	1	yes	characteristic
326	C₄	#16864209	1	1	yes	characteristic
327	C₂	—	536	2	no	—
328	C₂	#22571821	1	1	yes	characteristic
329	C₂	#22571865	1	1	yes	characteristic
330	C₂	#22572014	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1591	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×D₅₃₆	D₄
hypercenter	#1591	D₆₇
hypocenter	C₆₇	#20306
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
2. #22572494	C₂	semidirect
3. #22559819	C₂	semidirect
4. #22559763	C₂	semidirect
5. #22559757	C₂	semidirect
6. #22559468	C₂	semidirect
7. #22559460	C₂	semidirect
9. C₆₇×#1591	C₂	semidirect
173. #1591	D₆₇	semidirect
175. C₆₇	#20306	semidirect

Maximal subgroups

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

subgroup	index
#22572929	2
#22572494	2
#22559819	2
#22559763	2
#22559757	2
#22559468	2
#22559460	2
#22559095	2
C₆₇×#1591	2
#20306	67