Subgroups — #30839085

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571362	C₂	1	2	yes	maximal
2	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
3	#22569143	C₂	1	1	yes	maximal, characteristic
4	#22559738	C₂	1	4	yes	maximal, complemented
5	#22559614	C₂	1	4	yes	maximal, complemented
6	#22558889	C₂	1	2	yes	maximal
7	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₂²	1	1	yes	characteristic
9	C₂×#14524043	C₂²	1	2	yes	—
10	#16863801	—	2	1	no	—
11	#16863801	—	2	1	no	—
12	#16863801	—	2	2	no	—
13	#16863801	C₂²	1	2	yes	—
14	#16863801	C₂²	1	2	yes	—
15	#16863747	—	2	2	no	—
16	#16863732	—	2	4	no	—
17	#16863692	C₂²	1	8	yes	—
18	#16863526	—	2	1	no	—
19	#16863526	C₂²	1	2	yes	—
20	C₈×Dic₁₃₄	—	2	1	no	—
21	C₈×Dic₁₃₄	C₂²	1	2	yes	—
22	C₂×#14523977	—	2	1	no	—
23	C₂×#14523977	—	2	1	no	—
24	#16863422	—	2	1	no	—
25	#16863420	C₂²	1	1	yes	characteristic
26	#16863351	—	2	1	no	—
27	#16863144	C₂²	1	8	yes	—
28	#431×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#335×C₆₇	C₂²	1	2	yes	—
31	#334×C₆₇	C₂²	1	2	yes	—
32	C₂²×Dic₁₃₄	—	2	2	no	—
33	C₂×#11761612	—	2	1	no	—
34	#14524043	—	2	2	no	—
35	#14524043	—	2	2	no	—
36	#14524043	—	2	2	no	—
37	#14524043	D₄	1	1	yes	characteristic
38	#14524043	D₄	1	1	yes	characteristic
39	#14524043	C₂³	1	4	yes	—
40	#14524035	—	2	1	no	—
41	#14524022	—	2	1	no	—
42	#14524022	D₄	1	2	yes	—
43	C₄×Dic₁₃₄	—	2	1	no	—
44	C₄×Dic₁₃₄	C₂³	1	2	yes	—
45	#14524002	—	2	1	no	—
46	C₂×C₄⋊Dic₆₇	—	2	2	no	—
47	C₂×C₄⋊Dic₆₇	—	2	2	no	—
48	#14523977	—	2	8	no	—
49	#14523977	—	2	2	no	—
50	#14523977	—	2	2	no	—
51	#14523977	—	2	2	no	—
52	#14523977	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
55	Dic₆₇⋊C₈	—	2	2	no	—
56	Dic₆₇⋊C₈	—	2	2	no	—
57	#14523968	—	2	8	no	—
58	#14523953	—	2	2	no	—
59	C₈×Dic₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
64	C₂²⋊C₈×C₆₇	—	2	2	no	—
65	C₄×C₈×C₆₇	—	2	2	no	—
66	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₃₄	—	2	1	no	—
71	C₂×Dic₁₃₄	—	2	4	no	—
72	C₂×Dic₁₃₄	—	4	2	no	—
73	C₂×Dic₁₃₄	—	8	4	no	—
74	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
75	C₂²×Dic₆₇	—	4	2	no	—
76	#11761612	—	4	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₄⋊Dic₆₇	—	2	1	no	—
79	C₄⋊Dic₆₇	—	2	1	no	—
80	C₄⋊Dic₆₇	—	2	1	no	—
81	C₄⋊Dic₆₇	—	2	4	no	—
82	C₄⋊Dic₆₇	—	2	4	no	—
83	C₄⋊Dic₆₇	—	4	2	no	—
84	C₄⋊Dic₆₇	—	4	2	no	—
85	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
86	Dic₆₇⋊C₄	—	2	2	no	—
87	Dic₆₇⋊C₄	—	2	2	no	—
88	C₂²⋊Dic₆₇	—	2	2	no	—
89	C₄×Dic₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	#41	1	1	yes	characteristic
103	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₆₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	#137	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₆₇	—	2	2	no	—
116	C₂×Dic₆₇	—	2	2	no	—
117	C₂×Dic₆₇	—	4	4	no	—
118	C₂×Dic₆₇	—	8	4	no	—
119	Dic₁₃₄	—	2	4	no	—
120	Dic₁₃₄	—	4	2	no	—
121	Dic₁₃₄	—	8	4	no	—
122	Dic₁₃₄	—	8	4	no	—
123	C₈×C₆₇	—	2	1	no	—
124	C₈×C₆₇	—	2	1	no	—
125	C₈×C₆₇	—	4	4	no	—
126	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
127	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	#462	1	2	yes	—
132	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	#462	1	2	yes	—
136	C₄×C₆₇	#472	1	2	yes	—
137	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
139	C₄×C₆₇	#578	1	2	yes	—
140	Dic₆₇	—	4	2	no	—
141	Dic₆₇	—	4	2	no	—
142	Dic₆₇	—	8	4	no	—
143	#19209	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	2	no	—
145	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
146	C₂×C₆₇	#3059	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
148	#3230	—	67	2	no	—
149	C₂×#500	—	67	1	no	—
150	#2960	—	67	1	no	—
151	#1696	—	67	4	no	—
152	#1676	—	67	4	no	—
153	#1564	D₆₇	1	1	yes	characteristic
154	#1548	—	67	2	no	—
155	C₆₇	#19209	1	1	yes	characteristic, complemented, Sylow
156	#533	—	67	1	no	—
157	C₂×C₄⋊Q₈	—	67	2	no	—
158	#500	—	67	2	no	—
159	#500	—	67	2	no	—
160	#500	—	134	2	no	—
161	#500	—	134	1	no	—
162	#500	—	134	1	no	—
163	#487	—	134	2	no	—
164	#484	—	134	4	no	—
165	#477	—	67	8	no	—
166	#446	—	67	2	no	—
167	#446	—	134	1	no	—
168	C₈×Q₈	—	67	2	no	—
169	C₈×Q₈	—	134	1	no	—
170	#431	C₂×D₆₇	1	2	yes	—
171	C₂×#107	—	134	1	no	—
172	C₂×#107	—	134	1	no	—
173	#424	—	134	1	no	—
174	#423	—	67	1	no	—
175	#405	—	134	1	no	—
176	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
177	#335	C₂×D₆₇	1	2	yes	—
178	#334	C₂×D₆₇	1	2	yes	—
179	#333	—	67	8	no	—
180	C₂²×Q₈	—	134	2	no	—
181	C₂×M₁₆	—	134	1	no	—
182	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
183	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
184	C₄⋊Q₈	—	67	1	no	—
185	C₄⋊Q₈	—	67	1	no	—
186	C₄⋊Q₈	—	67	4	no	—
187	C₄⋊Q₈	—	134	2	no	—
188	C₄⋊Q₈	—	134	2	no	—
189	C₄⋊Q₈	—	134	2	no	—
190	#125	—	134	1	no	—
191	C₂²⋊Q₈	—	67	2	no	—
192	C₂²⋊Q₈	—	134	1	no	—
193	C₄×Q₈	—	67	2	no	—
194	C₄×Q₈	—	134	1	no	—
195	#117	—	134	1	no	—
196	C₂×C₄⋊C₄	—	134	2	no	—
197	C₂×C₄⋊C₄	—	134	2	no	—
198	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	#107	—	134	8	no	—
200	#107	—	134	2	no	—
201	#107	—	134	2	no	—
202	#107	—	134	2	no	—
203	#107	—	134	2	no	—
204	C₄⋊C₈	—	67	2	no	—
205	C₄⋊C₈	—	134	2	no	—
206	C₄⋊C₈	—	134	2	no	—
207	C₄⋊C₈	—	134	1	no	—
208	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
209	#103	—	134	8	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	#97	—	134	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	134	2	no	—
214	C₄×C₈	D₂₆₈	1	1	yes	characteristic
215	C₄×C₈	D₂₆₈	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	C₂×Q₈	—	268	2	no	—
219	C₂×Q₈	—	536	4	no	—
220	C₂×Q₈	—	67	2	no	—
221	C₂×Q₈	—	134	4	no	—
222	C₂×Q₈	—	134	1	no	—
223	C₂²×C₄	—	268	2	no	—
224	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
226	C₂²×C₄	#11761629	1	1	yes	characteristic
227	M₁₆	—	268	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	134	2	no	—
231	C₂×C₈	Dic₂₆₈	1	4	yes	—
232	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
233	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
234	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
236	C₄⋊C₄	—	268	2	no	—
237	C₄⋊C₄	—	268	2	no	—
238	C₄⋊C₄	—	67	2	no	—
239	C₄⋊C₄	—	134	4	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₄⋊C₄	—	134	1	no	—
242	C₄⋊C₄	—	134	1	no	—
243	C₄⋊C₄	—	134	2	no	—
244	C₄⋊C₄	—	134	1	no	—
245	C₄⋊C₄	—	134	2	no	—
246	C₂²⋊C₄	—	134	2	no	—
247	C₄²	—	134	2	no	—
248	C₄²	D₄×D₆₇	1	1	yes	characteristic
249	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
250	C₄²	#11761629	1	1	yes	characteristic
251	C₄²	C₂³×D₆₇	1	1	yes	characteristic
252	C₂³	#14524018	1	1	yes	characteristic
253	Q₈	—	268	2	no	—
254	Q₈	—	536	4	no	—
255	Q₈	—	536	4	no	—
256	Q₈	—	134	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	536	4	no	—
261	C₂×C₄	—	134	2	no	—
262	C₂×C₄	—	134	2	no	—
263	C₂×C₄	#14524018	1	1	yes	characteristic
264	C₂×C₄	#14524018	1	1	yes	characteristic
265	C₂×C₄	#14524018	1	1	yes	characteristic
266	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
267	C₂×C₄	#14524076	1	2	yes	—
268	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	4	no	—
274	C₈	—	268	2	no	—
275	C₈	C₂×Dic₂₆₈	1	2	yes	—
276	C₈	C₂×Dic₂₆₈	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	#16863607	1	2	yes	—
280	C₂²	C₂×#14524018	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	268	2	no	—
284	C₄	—	268	2	no	—
285	C₄	—	536	4	no	—
286	C₄	#16863607	1	2	yes	—
287	C₄	#16863667	1	2	yes	—
288	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
289	C₄	C₂×#14524076	1	1	yes	characteristic
290	C₄	#16864210	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	C₂×#16863607	1	1	yes	characteristic
293	C₂	#22569865	1	1	yes	characteristic
294	C₂	C₂×#16863667	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂³×C₆₇	#121
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#19209
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #22559738	C₂	semidirect
5. #22559614	C₂	semidirect
155. C₆₇	#19209	semidirect

Maximal subgroups

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

subgroup	index
#22571362	2
C₂×#16863801	2
#22569143	2
#22559738	2
#22559614	2
#22558889	2
C₆₇×#1564	2
#19209	67