Subgroups — #30853917

Order 17152.

1126 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	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
3	#22562565	C₂	1	2	yes	maximal
4	#22560108	C₂	1	4	yes	maximal, complemented
5	#106×Dic₆₇	C₂	1	4	yes	maximal, complemented
6	#22559930	C₂	1	2	yes	maximal
7	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	1	yes	characteristic
9	C₂×#14524002	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
11	C₂×#14523975	C₄	1	8	yes	complemented
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×#14523975	C₂²	1	2	yes	—
14	#16863351	C₄	1	8	yes	complemented
15	C₂×#14523953	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₆₇	C₂²	1	8	yes	—
18	C₄×C₄⋊Dic₆₇	—	2	4	no	—
19	C₂²⋊C₄×Dic₆₇	—	2	4	no	—
20	#16863158	C₂²	1	8	yes	—
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	#427×C₆₇	C₄	1	8	yes	complemented
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂³×Dic₆₇	—	2	2	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	C₂×C₄	1	4	yes	—
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	#14524002	C₂×C₄	1	4	yes	—
32	#14524002	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₆₇	—	2	8	no	—
40	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
41	#14523975	C₂×C₄	1	8	yes	—
42	#14523975	C₂×C₄	1	8	yes	—
43	#14523953	C₂×C₄	1	8	yes	complemented
44	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
45	C₆₇⋊#95	—	2	8	no	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	#117×C₆₇	—	2	4	no	—
53	#117×C₆₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	#107×C₆₇	C₂×C₄	1	8	yes	complemented
58	#106×C₆₇	C₂×C₄	1	8	yes	complemented
59	C₂²×Dic₆₇	—	2	4	no	—
60	C₂²×Dic₆₇	—	2	8	no	—
61	C₂²×Dic₆₇	—	8	4	no	—
62	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
63	#11761612	—	2	8	no	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
67	C₄⋊Dic₆₇	—	2	8	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	C₄⋊C₄	1	8	yes	—
70	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
71	C₂²⋊Dic₆₇	C₄⋊C₄	1	8	yes	—
72	C₄×Dic₆₇	—	8	8	no	—
73	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
74	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
86	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
94	C₂²⋊C₄×C₆₇	—	2	8	no	—
95	C₄²×C₆₇	—	2	8	no	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	C₄×D₄	1	2	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	8	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	4	yes	—
107	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
110	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₆₇	—	2	8	no	—
118	C₂×Dic₆₇	—	2	8	no	—
119	C₂×Dic₆₇	—	4	8	no	—
120	C₂×Dic₆₇	—	8	8	no	—
121	C₂×Dic₆₇	C₂×C₄⋊C₄	1	8	yes	—
122	C₈×C₆₇	C₂×C₄²	1	4	yes	—
123	C₈×C₆₇	C₂×C₄²	1	4	yes	—
124	C₆₇⋊C₈	—	2	4	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₆₇	#577	1	4	yes	—
133	C₄×C₆₇	—	4	8	no	—
134	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
139	Dic₆₇	—	2	8	no	—
140	Dic₆₇	—	4	8	no	—
141	#20454	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	4	no	—
143	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₆₇	#2912	1	2	yes	—
145	C₂×C₆₇	#2949	1	2	yes	—
146	C₂×C₆₇	#2952	1	2	yes	—
147	C₂×#427	D₆₇	1	1	yes	characteristic
148	C₂²×#106	—	67	1	no	—
149	C₂×#405	—	67	1	no	—
150	#2274	—	67	2	no	—
151	#1776	—	67	4	no	—
152	C₄×#106	—	67	4	no	—
153	#1736	—	67	2	no	—
154	C₆₇	#20454	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	67	1	no	—
156	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂×#117	—	67	1	no	—
158	C₂×#117	C₂×D₆₇	1	2	yes	—
159	C₂²×C₄⋊C₄	—	67	2	no	—
160	#427	Dic₆₇	1	8	yes	complemented
161	C₂×#107	C₂×D₆₇	1	2	yes	—
162	C₂×#106	—	67	8	no	—
163	C₂×#106	—	67	2	no	—
164	C₂×#106	—	67	2	no	—
165	C₂×#106	C₂×D₆₇	1	2	yes	—
166	#405	—	67	8	no	—
167	C₂×#97	—	67	2	no	—
168	C₂×C₄×C₈	—	67	2	no	—
169	C₄×C₄⋊C₄	—	67	8	no	—
170	C₄×C₄⋊C₄	—	134	4	no	—
171	C₄×C₂²⋊C₄	—	134	4	no	—
172	#340	—	67	8	no	—
173	C₂³×C₄	—	134	2	no	—
174	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×M₁₆	—	67	4	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂²×C₈	—	67	2	no	—
179	C₂²×C₈	C₄×D₆₇	1	4	yes	—
180	C₂²×C₈	C₄×D₆₇	1	4	yes	—
181	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	#117	—	2	4	no	—
185	#117	—	67	4	no	—
186	#117	—	67	4	no	—
187	#117	C₂×Dic₆₇	1	8	yes	—
188	C₂×C₄⋊C₄	—	67	2	no	—
189	C₂×C₄⋊C₄	—	67	8	no	—
190	C₂×C₄⋊C₄	—	67	4	no	—
191	C₂×C₄⋊C₄	—	134	2	no	—
192	C₂×C₄⋊C₄	—	134	4	no	—
193	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	67	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	67	2	no	—
198	C₂×C₄²	—	134	8	no	—
199	#107	C₂×Dic₆₇	1	8	yes	complemented
200	#106	—	67	8	no	—
201	#106	—	67	8	no	—
202	#106	C₂×Dic₆₇	1	8	yes	complemented
203	#97	—	67	8	no	—
204	C₄×C₈	—	67	8	no	—
205	#95	—	134	8	no	—
206	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	536	4	no	—
209	C₂²×C₄	—	67	4	no	—
210	C₂²×C₄	—	134	4	no	—
211	C₂²×C₄	—	134	8	no	—
212	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
215	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
217	C₂²×C₄	#11761627	1	2	yes	—
218	C₂²×C₄	#11761629	1	2	yes	—
219	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
220	M₁₆	—	134	8	no	—
221	C₂×C₈	—	67	4	no	—
222	C₂×C₈	—	134	2	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
225	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
229	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
230	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	67	8	no	—
233	C₄⋊C₄	—	67	8	no	—
234	C₄⋊C₄	—	134	8	no	—
235	C₄⋊C₄	—	134	4	no	—
236	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	67	8	no	—
239	C₄²	—	2	8	no	—
240	C₄²	—	536	8	no	—
241	C₄²	—	67	4	no	—
242	C₄²	—	67	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	C₄⋊C₄×D₆₇	1	2	yes	—
245	C₂³	#14524005	1	2	yes	—
246	C₂³	D₄×Dic₆₇	1	2	yes	—
247	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	268	8	no	—
252	C₂×C₄	—	536	8	no	—
253	C₂×C₄	—	67	8	no	—
254	C₂×C₄	—	134	8	no	—
255	C₂×C₄	—	134	8	no	—
256	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
257	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
258	C₂×C₄	#14524001	1	4	yes	—
259	C₂×C₄	#14524005	1	2	yes	—
260	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
261	C₂×C₄	#14524008	1	4	yes	—
262	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
268	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
269	C₈	—	134	4	no	—
270	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
271	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄⋊C₄×Dic₆₇	1	4	yes	—
276	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
277	C₂²	C₂×#14524005	1	1	yes	characteristic
278	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
279	C₂²	#16864203	1	4	yes	—
280	C₄	—	4	8	no	—
281	C₄	—	268	8	no	—
282	C₄	—	134	8	no	—
283	C₄	C₄⋊C₄×Dic₆₇	1	4	yes	—
284	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
285	C₄	C₂×#14524001	1	1	yes	characteristic
286	C₄	C₂×#14524008	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
290	C₂	#22568801	1	2	yes	—
291	C₂	#22569067	1	2	yes	—
292	C₂	#22569093	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#20454
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #22560108	C₂	semidirect
5. #106×Dic₆₇	C₂	semidirect
11. C₂×#14523975	C₄	semidirect
14. #16863351	C₄	semidirect
23. #427×C₆₇	C₄	semidirect
43. #14523953	C₂×C₄	semidirect
44. C₈×Dic₆₇	C₂×C₄	semidirect
57. #107×C₆₇	C₂×C₄	semidirect
58. #106×C₆₇	C₂×C₄	semidirect
85. C₂×C₈×C₆₇	C₄²	semidirect
154. C₆₇	#20454	semidirect
160. #427	Dic₆₇	semidirect
199. #107	C₂×Dic₆₇	semidirect
202. #106	C₂×Dic₆₇	semidirect
224. C₂×C₈	C₄×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523975	2
C₂×#16863351	2
#22562565	2
#22560108	2
#106×Dic₆₇	2
#22559930	2
C₂×#427×C₆₇	2
#20454	67