Subgroups — #30772323

Order 17152.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573311	C₂	1	1	yes	maximal, characteristic
2	#22561974	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561838	C₂	1	1	yes	maximal, characteristic
4	#22560046	C₂	1	1	yes	maximal, characteristic
5	#22558294	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558291	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	C₂²	1	1	yes	characteristic
9	#16863814	—	2	2	no	—
10	#16863812	—	2	2	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	M₃₂×D₆₇	—	2	2	no	—
14	C₂×#14523971	C₄	1	1	yes	characteristic
15	#16863393	C₄	1	1	yes	characteristic
16	#16863351	C₂²	1	1	yes	characteristic
17	#16863216	—	2	2	no	—
18	#16863181	—	2	2	no	—
19	#16863161	C₄	1	1	yes	characteristic
20	#16863156	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₆₇	—	2	2	no	—
25	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
26	#14524049	—	2	1	no	—
27	#14524049	—	2	1	no	—
28	#14524049	D₄	1	2	yes	—
29	#14524049	Q₈	1	2	yes	—
30	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
31	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
32	C₂×#11761611	—	2	1	no	—
33	C₂×C₈×D₆₇	—	2	1	no	—
34	#14524002	C₂×C₄	1	1	yes	characteristic
35	#14523983	—	2	1	no	—
36	#14523983	—	2	1	no	—
37	#14523983	—	2	1	no	—
38	#14523983	D₄	1	2	yes	—
39	#14523982	—	2	4	no	—
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₁₆×D₆₇	—	2	4	no	—
43	#14523971	C₈	1	2	yes	—
44	#14523971	C₈	1	2	yes	—
45	#14523968	—	2	1	no	—
46	#14523965	—	2	1	no	—
47	#14523953	—	2	1	no	—
48	C₈×Dic₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₆₇	—	2	1	no	—
53	M₃₂×C₆₇	—	2	1	no	—
54	M₃₂×C₆₇	—	2	1	no	—
55	M₃₂×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	#11761628	—	2	1	no	—
61	#11761628	—	2	1	no	—
62	#11761628	C₄⋊C₄	1	2	yes	—
63	#11761628	C₂×C₈	1	2	yes	—
64	C₂×Dic₁₃₄	M₁₆	1	1	yes	characteristic
65	C₂×D₂₆₈	M₁₆	1	1	yes	characteristic
66	C₂×C₂²⋊D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	#11761612	—	2	1	no	—
69	#11761612	—	2	1	no	—
70	#11761612	C₄⋊C₄	1	2	yes	—
71	#11761612	C₂×C₈	1	2	yes	—
72	#11761611	—	2	2	no	—
73	#11761611	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	1	no	—
75	C₂×C₆₇⋊C₈	—	2	1	no	—
76	C₈×D₆₇	—	2	2	no	—
77	C₈×D₆₇	—	2	2	no	—
78	C₄⋊Dic₆₇	—	2	1	no	—
79	C₂²⋊Dic₆₇	—	2	1	no	—
80	C₄×Dic₆₇	—	2	1	no	—
81	C₆₇⋊C₁₆	—	2	2	no	—
82	C₆₇⋊C₁₆	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
97	C₁₆×C₆₇	—	2	2	no	—
98	C₁₆×C₆₇	—	4	1	no	—
99	C₂³×C₆₇	#131	1	1	yes	characteristic
100	C₂²×D₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
110	C₂²⋊D₆₇	—	2	2	no	—
111	C₂²⋊D₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	2	1	no	—
113	C₂×Dic₆₇	—	4	1	no	—
114	D₂₆₈	—	2	1	no	—
115	D₂₆₈	C₄⋊C₈	1	2	yes	—
116	C₄×D₆₇	—	2	2	no	—
117	C₄×D₆₇	—	2	2	no	—
118	Dic₁₃₄	—	2	1	no	—
119	Dic₁₃₄	C₄⋊C₈	1	2	yes	—
120	C₈×C₆₇	—	4	1	no	—
121	C₈×C₆₇	#120	1	1	yes	characteristic
122	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₆₇	#123	1	1	yes	characteristic
125	C₆₇⋊C₈	—	2	2	no	—
126	C₆₇⋊C₈	—	2	2	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	#424	1	1	yes	characteristic
129	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
130	C₂²×C₆₇	#435	1	1	yes	characteristic
131	C₂×D₆₇	—	2	2	no	—
132	C₂×D₆₇	—	8	1	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	4	1	no	—
135	C₄×C₆₇	#385	1	1	yes	characteristic
136	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
138	C₄×C₆₇	#435	1	1	yes	characteristic
139	Dic₆₇	—	2	2	no	—
140	Dic₆₇	—	8	1	no	—
141	#12918	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	#1850	1	1	yes	characteristic
146	D₆₇	—	4	2	no	—
147	#3407	—	67	1	no	—
148	#2153	—	67	1	no	—
149	#2150	D₆₇	1	1	yes	characteristic, complemented
150	#2117	—	67	1	no	—
151	#1764	—	67	1	no	—
152	#1378	—	67	1	no	—
153	#1376	—	67	1	no	—
154	C₆₇	#12918	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	67	1	no	—
156	#504	—	134	2	no	—
157	#504	—	134	2	no	—
158	C₂×M₃₂	—	67	1	no	—
159	C₂×M₃₂	—	67	1	no	—
160	C₂×M₃₂	—	134	2	no	—
161	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
162	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
163	C₂×#104	—	67	1	no	—
164	#416	—	67	1	no	—
165	#405	—	67	1	no	—
166	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#364	—	134	2	no	—
168	C₄⋊C₁₆	—	2	2	no	—
169	#350	—	134	2	no	—
170	#341	—	67	1	no	—
171	#339	—	67	1	no	—
172	C₂×#41	—	67	1	no	—
173	#131	—	67	2	no	—
174	#131	—	67	2	no	—
175	#131	—	134	1	no	—
176	#131	—	134	1	no	—
177	C₂×M₁₆	—	67	1	no	—
178	C₂×M₁₆	—	67	1	no	—
179	C₂×M₁₆	—	134	1	no	—
180	C₂²×C₈	—	134	1	no	—
181	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	#117	—	67	1	no	—
184	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	67	2	no	—
190	M₃₂	—	134	4	no	—
191	M₃₂	—	134	1	no	—
192	M₃₂	—	134	1	no	—
193	M₃₂	—	134	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	134	4	no	—
196	C₂×C₁₆	—	134	1	no	—
197	C₂×C₁₆	—	134	1	no	—
198	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
199	#104	—	67	2	no	—
200	#104	—	67	2	no	—
201	#103	—	134	1	no	—
202	#102	—	134	1	no	—
203	#97	—	134	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	134	1	no	—
207	#41	—	67	2	no	—
208	#41	—	67	2	no	—
209	#41	—	134	1	no	—
210	#41	—	134	1	no	—
211	C₂×Q₈	—	67	1	no	—
212	C₂×D₄	—	67	1	no	—
213	C₂×D₄	—	134	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	134	1	no	—
216	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
217	M₁₆	—	67	2	no	—
218	M₁₆	—	67	2	no	—
219	M₁₆	—	134	1	no	—
220	M₁₆	—	134	1	no	—
221	M₁₆	—	134	2	no	—
222	M₁₆	—	134	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	134	1	no	—
227	C₂×C₈	—	134	1	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	—	134	2	no	—
230	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
231	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
233	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
234	C₂×C₈	#11761628	1	1	yes	characteristic
235	C₂×C₈	#11761629	1	1	yes	characteristic
236	C₄⋊C₄	—	134	1	no	—
237	C₂²⋊C₄	—	134	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	134	1	no	—
240	C₄²	C₈×D₆₇	1	1	yes	characteristic
241	C₄²	C₈×D₆₇	1	1	yes	characteristic
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	268	1	no	—
245	C₁₆	—	134	2	no	—
246	C₂³	—	134	1	no	—
247	C₂³	#14524049	1	1	yes	characteristic
248	Q₈	—	67	2	no	—
249	Q₈	—	134	1	no	—
250	D₄	—	67	2	no	—
251	D₄	—	134	2	no	—
252	D₄	—	134	2	no	—
253	D₄	—	134	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	268	1	no	—
258	C₂×C₄	—	134	1	no	—
259	C₂×C₄	—	134	2	no	—
260	C₂×C₄	—	134	2	no	—
261	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₂×C₄	#14524005	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
266	C₂×C₄	#14524051	1	1	yes	characteristic
267	C₈	—	4	1	no	—
268	C₈	—	134	2	no	—
269	C₈	—	134	2	no	—
270	C₈	#14524011	1	1	yes	characteristic
271	C₈	#14524019	1	1	yes	characteristic
272	C₈	#14524021	1	1	yes	characteristic
273	C₈	#14524028	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	536	1	no	—
276	C₂²	—	134	2	no	—
277	C₂²	#16863421	1	1	yes	characteristic
278	C₂²	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
279	C₂²	#16863466	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	536	1	no	—
283	C₄	—	134	2	no	—
284	C₄	#16863267	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
286	C₄	#16863461	1	1	yes	characteristic
287	C₄	#16863465	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	268	2	no	—
292	C₂	#22560385	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524011
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂×C₆₇	#1850
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#12918
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22561974	C₂	semidirect
5. #22558294	C₂	semidirect
6. #22558291	C₂	semidirect
7. C₆₇×#2150	C₂	semidirect
149. #2150	D₆₇	semidirect
154. C₆₇	#12918	semidirect

Maximal subgroups

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

subgroup	index
#22573311	2
#22561974	2
#22561838	2
#22560046	2
#22558294	2
#22558291	2
C₆₇×#2150	2
#12918	67