Subgroups — #30749130

Order 17152.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558816	C₂	1	2	yes	maximal
4	#22558080	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic
6	#16863453	C₂²	1	2	yes	—
7	C₂×#14523979	—	2	1	no	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	#16863161	—	2	1	no	—
10	#16863149	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863138	—	2	2	no	—
13	#16863136	—	2	2	no	—
14	#529×C₆₇	C₂²	1	1	yes	characteristic
15	#446×C₆₇	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
17	#436×C₆₇	C₄	1	1	yes	characteristic
18	#435×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	—	2	1	no	—
20	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
21	#433×C₆₇	—	2	1	no	—
22	#432×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	—	2	2	no	—
26	#14523979	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
29	#14523954	—	2	2	no	—
30	#14523954	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	1	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	#126×C₆₇	—	2	1	no	—
37	#125×C₆₇	D₄	1	1	yes	characteristic
38	#124×C₆₇	D₄	1	1	yes	characteristic
39	#123×C₆₇	—	2	1	no	—
40	C₂²⋊Q₈×C₆₇	Q₈	1	1	yes	characteristic
41	#121×C₆₇	Q₈	1	1	yes	characteristic
42	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₆₇	—	2	1	no	—
44	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	#117×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	#97×C₆₇	—	2	1	no	—
55	#97×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	#11761612	—	2	4	no	—
62	#11761612	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂×D₄×C₆₇	—	2	1	no	—
67	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	M₁₆×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	1	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	4	1	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₇	#41	1	1	yes	characteristic
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
100	Q₈×C₆₇	—	4	1	no	—
101	D₄×C₆₇	—	4	1	no	—
102	D₄×C₆₇	—	4	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	—	4	1	no	—
110	C₂×C₄×C₆₇	—	4	1	no	—
111	C₂×C₄×C₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₆₇	—	2	1	no	—
119	C₈×C₆₇	—	2	1	no	—
120	C₈×C₆₇	—	2	1	no	—
121	C₈×C₆₇	—	4	1	no	—
122	C₈×C₆₇	—	4	1	no	—
123	C₈×C₆₇	#104	1	2	yes	—
124	C₆₇⋊C₈	—	4	4	no	—
125	C₆₇⋊C₈	—	8	2	no	—
126	C₂²×C₆₇	—	2	1	no	—
127	C₂²×C₆₇	—	2	1	no	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	4	1	no	—
130	C₂²×C₆₇	—	4	1	no	—
131	C₂²×C₆₇	—	4	1	no	—
132	C₂²×C₆₇	#380	1	1	yes	characteristic
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	4	1	no	—
139	C₄×C₆₇	—	4	1	no	—
140	C₄×C₆₇	—	4	1	no	—
141	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
142	C₄×C₆₇	#421	1	1	yes	characteristic
143	#8693	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	4	1	no	—
145	C₂×C₆₇	—	8	1	no	—
146	C₂×C₆₇	#1806	1	1	yes	characteristic
147	C₂×C₆₇	#1813	1	1	yes	characteristic
148	C₂×C₆₇	#1814	1	1	yes	characteristic
149	#2966	D₆₇	1	1	yes	characteristic
150	C₄×#108	—	67	1	no	—
151	#1568	—	67	1	no	—
152	#1529	—	67	2	no	—
153	#1291	—	67	2	no	—
154	C₆₇	#8693	1	1	yes	characteristic, complemented, Sylow
155	#529	C₂×D₆₇	1	1	yes	characteristic
156	#446	Dic₆₇	1	1	yes	characteristic
157	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
158	#436	Dic₆₇	1	1	yes	characteristic
159	#435	—	2	1	no	—
160	C₈×D₄	—	2	1	no	—
161	C₈×D₄	Dic₆₇	1	1	yes	characteristic
162	#433	—	2	1	no	—
163	#432	—	67	2	no	—
164	#432	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#108	—	134	1	no	—
166	#405	—	2	1	no	—
167	C₄×M₁₆	—	67	2	no	—
168	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
169	#341	—	134	1	no	—
170	#335	—	134	1	no	—
171	#334	—	134	1	no	—
172	#330	—	134	2	no	—
173	#329	—	134	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
179	#126	—	2	1	no	—
180	#125	D₂₆₈	1	1	yes	characteristic
181	#124	D₂₆₈	1	1	yes	characteristic
182	#123	—	2	1	no	—
183	C₂²⋊Q₈	Dic₁₃₄	1	1	yes	characteristic
184	#121	Dic₁₃₄	1	1	yes	characteristic
185	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
190	#108	—	134	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	134	2	no	—
194	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
195	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	134	2	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	#97	—	134	2	no	—
203	#97	—	134	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	134	2	no	—
206	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
216	C₂²×C₄	#11761628	1	1	yes	characteristic
217	M₁₆	—	4	2	no	—
218	M₁₆	—	268	2	no	—
219	M₁₆	—	134	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
236	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
244	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
246	C₄²	#11761628	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	#14524022	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
263	C₂×C₄	#14524002	1	1	yes	characteristic
264	C₂×C₄	#14524017	1	1	yes	characteristic
265	C₂×C₄	#14524017	1	1	yes	characteristic
266	C₂×C₄	#14524022	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	268	4	no	—
273	C₈	—	536	2	no	—
274	C₈	#14523970	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	#16863256	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₂×#14523970	1	1	yes	characteristic
291	C₄	#16863413	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	8	1	no	—
294	C₂	#22560229	1	1	yes	characteristic
295	C₂	#22560259	1	1	yes	characteristic
296	C₂	#22560263	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2966	C₂
socle	C₂²×C₆₇	#380
prime radical	#121×C₆₇	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2966	D₆₇
hypocenter	C₆₇	#8693
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#14523979	C₂	semidirect
2. #22558972	C₂	semidirect
4. #22558080	C₂	semidirect
154. C₆₇	#8693	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523979	2
#22558972	2
#22558816	2
#22558080	2
C₆₇×#2966	2
#8693	67