Subgroups — #30772309

Order 17152.

370 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#22561824	C₂	1	1	yes	maximal, characteristic
4	#22560010	C₂	1	1	yes	maximal, characteristic
5	#22558299	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558298	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2153	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	M₃₂×D₆₇	—	2	2	no	—
13	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#16863378	C₄	1	1	yes	characteristic
15	#16863369	C₄	1	1	yes	characteristic
16	#16863351	C₂²	1	1	yes	characteristic
17	#16863216	—	2	2	no	—
18	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
19	#16863168	C₄	1	1	yes	characteristic
20	#16863166	C₄	1	1	yes	characteristic
21	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	C₂²	1	1	yes	characteristic
24	#364×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	—	2	1	no	—
29	#14524049	—	2	1	no	—
30	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
31	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
32	C₂×#11761611	Q₈	1	2	yes	—
33	C₂×C₈×D₆₇	D₄	1	2	yes	—
34	#14524002	C₂×C₄	1	1	yes	characteristic
35	#14523983	—	2	1	no	—
36	#14523983	D₄	1	2	yes	—
37	#14523982	—	2	4	no	—
38	C₂×C₆₇⋊C₁₆	—	2	1	no	—
39	C₂×C₆₇⋊C₁₆	—	2	1	no	—
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₁₆×D₆₇	—	2	4	no	—
43	#14523961	C₈	1	2	yes	—
44	#14523959	C₈	1	2	yes	—
45	#14523957	—	2	2	no	—
46	#14523953	—	2	1	no	—
47	C₈×Dic₆₇	—	2	1	no	—
48	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₆₇	—	2	1	no	—
52	M₃₂×C₆₇	—	2	1	no	—
53	M₃₂×C₆₇	—	2	1	no	—
54	M₃₂×C₆₇	D₄	1	2	yes	—
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	#97×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	#11761628	—	2	1	no	—
60	#11761628	—	2	1	no	—
61	#11761628	—	2	1	no	—
62	#11761628	—	2	1	no	—
63	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
64	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₂²⋊D₆₇	M₁₆	1	2	yes	—
66	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
67	#11761612	—	2	1	no	—
68	#11761612	—	2	1	no	—
69	#11761612	—	2	1	no	—
70	#11761612	C₂×C₈	1	2	yes	—
71	#11761611	—	2	2	no	—
72	#11761611	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
75	C₈×D₆₇	—	2	2	no	—
76	C₈×D₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	—	2	1	no	—
78	C₂²⋊Dic₆₇	—	2	1	no	—
79	C₄×Dic₆₇	—	2	1	no	—
80	C₆₇⋊C₁₆	—	2	2	no	—
81	C₆₇⋊C₁₆	—	4	1	no	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₆₇	—	2	1	no	—
84	M₁₆×C₆₇	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₁₆×C₆₇	—	2	2	no	—
96	C₁₆×C₆₇	—	4	1	no	—
97	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₆₇	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
106	C₂²⋊D₆₇	—	2	2	no	—
107	C₂²⋊D₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	4	1	no	—
109	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
110	D₂₆₈	—	2	1	no	—
111	D₂₆₈	—	2	1	no	—
112	C₄×D₆₇	—	2	2	no	—
113	C₄×D₆₇	—	2	2	no	—
114	Dic₁₃₄	—	2	1	no	—
115	Dic₁₃₄	—	2	1	no	—
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	#120	1	1	yes	characteristic
118	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₆₇	#123	1	1	yes	characteristic
121	C₆₇⋊C₈	—	2	2	no	—
122	C₆₇⋊C₈	—	2	2	no	—
123	C₂²×C₆₇	—	4	1	no	—
124	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₆₇	#435	1	1	yes	characteristic
126	C₂²×C₆₇	#435	1	1	yes	characteristic
127	C₂×D₆₇	—	2	2	no	—
128	C₂×D₆₇	—	4	2	no	—
129	C₄×C₆₇	—	8	1	no	—
130	C₄×C₆₇	#385	1	1	yes	characteristic
131	C₄×C₆₇	#424	1	1	yes	characteristic
132	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
133	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
134	Dic₆₇	—	2	2	no	—
135	Dic₆₇	—	8	1	no	—
136	#12916	—	67	1	no	maximal, Sylow
137	C₂×C₆₇	—	2	1	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	1	no	—
140	C₂×C₆₇	#1850	1	1	yes	characteristic
141	D₆₇	—	4	2	no	—
142	#3407	—	67	1	no	—
143	#2153	—	67	1	no	—
144	#2153	D₆₇	1	1	yes	characteristic, complemented
145	#2113	—	67	1	no	—
146	#1757	—	67	1	no	—
147	#1379	—	67	1	no	—
148	#1379	—	67	1	no	—
149	C₆₇	#12916	1	1	yes	characteristic, complemented, Sylow
150	C₂×#131	—	67	1	no	—
151	#504	—	134	2	no	—
152	#504	—	134	2	no	—
153	C₂×M₃₂	—	67	1	no	—
154	C₂×M₃₂	—	134	2	no	—
155	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
157	C₂²×C₁₆	—	67	1	no	—
158	#413	—	67	1	no	—
159	#410	—	67	1	no	—
160	#405	—	67	1	no	—
161	#405	C₂×D₆₇	1	1	yes	characteristic
162	#364	—	2	2	no	—
163	#364	—	134	2	no	—
164	C₂²⋊C₁₆	—	134	2	no	—
165	#344	—	67	1	no	—
166	#344	—	67	1	no	—
167	C₂×#41	—	67	1	no	—
168	#131	—	134	1	no	—
169	#131	—	134	1	no	—
170	#131	—	134	1	no	—
171	#131	—	134	1	no	—
172	C₂×M₁₆	—	67	1	no	—
173	C₂×M₁₆	—	67	2	no	—
174	C₂×M₁₆	—	67	1	no	—
175	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
176	C₂²×C₈	—	67	2	no	—
177	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
178	#117	—	67	1	no	—
179	#117	C₄×D₆₇	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	67	2	no	—
184	M₃₂	—	134	4	no	—
185	M₃₂	—	134	1	no	—
186	M₃₂	C₂²⋊D₆₇	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	134	4	no	—
190	C₂×C₁₆	—	134	1	no	—
191	C₂×C₁₆	—	134	1	no	—
192	C₂×C₁₆	—	134	1	no	—
193	C₂×C₁₆	—	134	1	no	—
194	#101	—	67	2	no	—
195	#100	—	67	2	no	—
196	#99	—	134	2	no	—
197	#97	—	2	1	no	—
198	#97	—	134	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	134	1	no	—
201	#41	—	134	1	no	—
202	#41	—	134	1	no	—
203	#41	—	134	1	no	—
204	#41	—	134	1	no	—
205	C₂×Q₈	—	67	1	no	—
206	C₂×D₄	—	67	2	no	—
207	C₂×D₄	—	67	1	no	—
208	C₂²×C₄	—	67	2	no	—
209	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	67	2	no	—
212	M₁₆	—	134	1	no	—
213	M₁₆	—	134	1	no	—
214	M₁₆	—	134	1	no	—
215	M₁₆	—	134	2	no	—
216	M₁₆	—	134	2	no	—
217	M₁₆	C₈×D₆₇	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	67	2	no	—
220	C₂×C₈	—	134	1	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
224	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
227	C₂×C₈	#11761628	1	1	yes	characteristic
228	C₂×C₈	#11761629	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	134	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	134	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	134	1	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	268	1	no	—
238	C₁₆	—	134	2	no	—
239	C₂³	—	67	2	no	—
240	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
241	Q₈	—	134	1	no	—
242	Q₈	—	134	1	no	—
243	D₄	—	134	2	no	—
244	D₄	—	134	2	no	—
245	D₄	—	134	1	no	—
246	D₄	—	134	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	268	1	no	—
249	C₂×C₄	—	67	2	no	—
250	C₂×C₄	—	134	2	no	—
251	C₂×C₄	—	134	2	no	—
252	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
254	C₂×C₄	#14524005	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
256	C₂×C₄	#14524049	1	1	yes	characteristic
257	C₂×C₄	#14524051	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	134	2	no	—
260	C₈	—	134	2	no	—
261	C₈	#14524011	1	1	yes	characteristic
262	C₈	#14524019	1	1	yes	characteristic
263	C₈	#14524021	1	1	yes	characteristic
264	C₈	#14524028	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	268	2	no	—
267	C₂²	—	134	2	no	—
268	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
269	C₂²	#16863465	1	1	yes	characteristic
270	C₂²	#16863466	1	1	yes	characteristic
271	C₄	—	8	1	no	—
272	C₄	—	536	1	no	—
273	C₄	—	134	2	no	—
274	C₄	#16863267	1	1	yes	characteristic
275	C₄	#16863421	1	1	yes	characteristic
276	C₄	C₈×C₂²⋊D₆₇	1	1	yes	characteristic
277	C₄	#16863461	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	268	2	no	—
282	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×#2153	C₂
socle	C₂×C₆₇	#1850
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2153	D₆₇
hypocenter	C₆₇	#12916
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
2. #22561974	C₂	semidirect
5. #22558299	C₂	semidirect
6. #22558298	C₂	semidirect
7. C₆₇×#2153	C₂	semidirect
144. #2153	D₆₇	semidirect
149. C₆₇	#12916	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
#22561824	2
#22560010	2
#22558299	2
#22558298	2
C₆₇×#2153	2
#12916	67