Subgroups — #30856206

Order 17152.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560358	C₂	1	2	yes	maximal, complemented
2	#22560333	C₂	1	2	yes	maximal, complemented
3	#22560313	C₂	1	2	yes	maximal
4	#22560259	C₂	1	1	yes	maximal, characteristic
5	#22560255	C₂	1	1	yes	maximal, characteristic
6	#22560093	C₂	1	2	yes	maximal, complemented
7	#22560009	C₂	1	2	yes	maximal
8	#22559964	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523979	C₂²	1	2	yes	—
11	#16863434	C₂²	1	2	yes	—
12	#16863380	C₂²	1	2	yes	—
13	#16863370	C₂²	1	2	yes	—
14	#16863351	C₂²	1	4	yes	—
15	#16863167	C₂²	1	4	yes	—
16	#16863156	C₂²	1	4	yes	—
17	C₆₇⋊#338	C₂²	1	2	yes	—
18	#16863153	C₂²	1	2	yes	—
19	#16863152	C₂²	1	4	yes	—
20	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	C₄	1	2	yes	complemented
24	#405×C₆₇	C₄	1	2	yes	complemented
25	#405×C₆₇	C₂²	1	1	yes	characteristic
26	#405×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₆₇	C₄	1	2	yes	complemented
28	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₇	C₄	1	2	yes	complemented
30	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#11761612	C₂³	1	4	yes	—
32	#14524002	C₂³	1	4	yes	—
33	#14523979	—	2	2	no	—
34	#14523979	—	2	2	no	—
35	#14523977	—	2	2	no	—
36	#14523975	—	2	2	no	—
37	#14523962	—	2	2	no	—
38	#14523960	—	2	2	no	—
39	#14523958	—	2	4	no	—
40	#14523953	—	2	4	no	—
41	C₈×Dic₆₇	—	2	4	no	—
42	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
43	#131×C₆₇	—	2	4	no	—
44	#131×C₆₇	—	2	2	no	—
45	#131×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	—	2	1	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	2	1	no	—
57	#117×C₆₇	C₂×C₄	1	2	yes	—
58	#117×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
62	#97×C₆₇	C₂×C₄	1	2	yes	—
63	#97×C₆₇	C₂×C₄	1	2	yes	—
64	#97×C₆₇	C₂×C₄	1	2	yes	—
65	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	#11761612	—	2	4	no	—
72	#11761612	—	2	4	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₂²⋊Dic₆₇	—	2	4	no	—
76	C₄×Dic₆₇	—	2	4	no	—
77	#41×C₆₇	—	2	2	no	—
78	#41×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
80	C₂×D₄×C₆₇	#41	1	2	yes	—
81	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₆₇	—	2	2	no	—
87	M₁₆×C₆₇	—	2	2	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	M₁₆×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	#41	1	4	yes	—
100	C₂×C₈×C₆₇	#41	1	2	yes	—
101	C₂×C₈×C₆₇	#41	1	2	yes	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂²⋊C₄×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂³×C₆₇	#117	1	2	yes	—
112	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	2	no	—
114	D₄×C₆₇	—	2	4	no	—
115	D₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	8	1	no	—
122	C₂×C₄×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	#117	1	2	yes	—
124	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	—	2	2	no	—
131	C₈×C₆₇	#117	1	1	yes	characteristic
132	C₈×C₆₇	#117	1	1	yes	characteristic
133	C₈×C₆₇	#117	1	2	yes	—
134	C₈×C₆₇	#117	1	2	yes	—
135	C₈×C₆₇	#117	1	2	yes	—
136	C₆₇⋊C₈	—	4	4	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	4	1	no	—
139	C₂²×C₆₇	—	4	2	no	—
140	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
141	C₂²×C₆₇	#529	1	2	yes	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	—	4	1	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
148	C₄×C₆₇	#529	1	2	yes	—
149	Dic₆₇	—	8	4	no	—
150	#20788	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	1	no	—
153	C₂×C₆₇	—	4	2	no	—
154	C₂×C₆₇	#2283	1	1	yes	characteristic
155	#2875	D₆₇	1	1	yes	characteristic
156	#1841	—	67	2	no	—
157	#1833	—	67	2	no	—
158	#1827	—	67	2	no	—
159	#1813	—	67	1	no	—
160	#1812	—	67	1	no	—
161	#1773	—	67	2	no	—
162	#1757	—	67	2	no	—
163	#1747	—	67	2	no	—
164	C₆₇	#20788	1	1	yes	characteristic, complemented, Sylow
165	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
166	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
167	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#108	—	67	2	no	—
169	#427	—	67	2	no	—
170	#413	—	67	2	no	—
171	#410	—	67	2	no	—
172	#405	—	67	4	no	—
173	#405	Dic₆₇	1	2	yes	complemented
174	#405	Dic₆₇	1	2	yes	complemented
175	#405	C₂×D₆₇	1	1	yes	characteristic
176	#405	C₂×D₆₇	1	1	yes	characteristic
177	C₄×M₁₆	Dic₆₇	1	2	yes	complemented
178	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
179	C₂×#97	Dic₆₇	1	2	yes	complemented
180	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
181	#344	—	67	4	no	—
182	#339	—	67	4	no	—
183	#338	—	67	2	no	—
184	H(3,ℤ/4)	—	67	2	no	—
185	H(3,ℤ/4)	—	67	4	no	—
186	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
187	#131	—	2	4	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	—	67	4	no	—
191	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
192	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
193	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
194	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
196	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
197	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄×Q₈	—	2	1	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	1	no	—
202	#117	—	67	4	no	—
203	#117	C₂×Dic₆₇	1	2	yes	—
204	#117	C₂²×D₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
206	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
207	#108	—	134	2	no	—
208	#108	—	134	2	no	—
209	#107	—	134	2	no	—
210	#106	—	134	2	no	—
211	#101	—	134	2	no	—
212	#100	—	134	2	no	—
213	#99	—	134	4	no	—
214	#97	—	134	4	no	—
215	#97	C₂×Dic₆₇	1	1	yes	characteristic
216	#97	C₂×Dic₆₇	1	2	yes	—
217	#97	C₂×Dic₆₇	1	2	yes	—
218	#97	C₂×Dic₆₇	1	2	yes	—
219	#97	C₂×Dic₆₇	1	1	yes	characteristic
220	#97	C₂×Dic₆₇	1	1	yes	characteristic
221	#97	C₂×Dic₆₇	1	1	yes	characteristic
222	C₄×C₈	—	134	4	no	—
223	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
224	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
225	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#11761628	1	1	yes	characteristic
229	C₂×D₄	#11761628	1	2	yes	—
230	C₂×D₄	#11761628	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	134	4	no	—
240	M₁₆	—	134	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	134	4	no	—
245	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
246	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
247	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
251	C₂×C₈	#11761628	1	4	yes	—
252	C₂×C₈	#11761628	1	2	yes	—
253	C₂×C₈	#11761628	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	134	4	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	134	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	134	4	no	—
263	C₄²	C₂²×Dic₆₇	1	2	yes	—
264	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂³	#14524002	1	2	yes	—
267	C₂³	C₂×#11761628	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	268	4	no	—
279	C₂×C₄	#14524002	1	2	yes	—
280	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂×#11761628	1	2	yes	—
282	C₂×C₄	C₂×#11761628	1	2	yes	—
283	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	268	4	no	—
287	C₈	#14524002	1	1	yes	characteristic
288	C₈	#14524002	1	1	yes	characteristic
289	C₈	#14524002	1	2	yes	—
290	C₈	#14524002	1	2	yes	—
291	C₈	#14524002	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#14524002	1	1	yes	characteristic
296	C₂²	#16863930	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	536	4	no	—
302	C₄	C₂×#14524002	1	1	yes	characteristic
303	C₄	C₂×#14524002	1	1	yes	characteristic
304	C₄	#16863930	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#22562607	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524002
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2875	C₂
socle	C₂×C₆₇	#2283
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2875	D₆₇
hypocenter	C₆₇	#20788
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2875

Complements

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

N	H ≅ G/N	type
1. #22560358	C₂	semidirect
2. #22560333	C₂	semidirect
6. #22560093	C₂	semidirect
8. #22559964	C₂	semidirect
23. #405×C₆₇	C₄	semidirect
24. #405×C₆₇	C₄	semidirect
27. C₄×M₁₆×C₆₇	C₄	semidirect
29. C₂×#97×C₆₇	C₄	semidirect
164. C₆₇	#20788	semidirect
173. #405	Dic₆₇	semidirect
174. #405	Dic₆₇	semidirect
177. C₄×M₁₆	Dic₆₇	semidirect
179. C₂×#97	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22560358	2
#22560333	2
#22560313	2
#22560259	2
#22560255	2
#22560093	2
#22560009	2
#22559964	2
C₆₇×#2875	2
#20788	67