Subgroups — #30853856

Order 17152.

1042 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	C₂×#16863445	C₂	1	1	yes	maximal, characteristic
2	#22568799	C₂	1	1	yes	maximal, characteristic
3	#22568627	C₂	1	2	yes	maximal
4	#22560093	C₂	1	4	yes	maximal, complemented
5	#22560088	C₂	1	4	yes	maximal, complemented
6	C₂×#16863156	C₂	1	2	yes	maximal
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂×#14524002	C₂²	1	2	yes	—
10	#16863445	C₄	1	8	yes	complemented
11	C₂×#14523979	C₂²	1	4	yes	—
12	#16863438	C₄	1	8	yes	complemented
13	#16863434	C₂²	1	4	yes	—
14	#16863354	—	2	4	no	—
15	#16863351	C₂²	1	8	yes	—
16	M₁₆×Dic₆₇	—	2	4	no	—
17	#16863156	C₂²	1	4	yes	—
18	#16863156	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₆₇	C₄	1	8	yes	complemented
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	4	no	—
27	C₂×#11761612	C₂×C₄	1	8	yes	—
28	C₂×#11761612	C₂³	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	#14524002	—	2	2	no	—
32	#14524002	C₂×C₄	1	4	yes	—
33	#14524002	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	#14523979	C₂×C₄	1	16	yes	complemented
38	#14523977	C₂×C₄	1	8	yes	complemented
39	#14523975	C₂×C₄	1	8	yes	complemented
40	#14523953	—	2	8	no	—
41	C₈×Dic₆₇	—	2	8	no	—
42	C₂³×C₄×C₆₇	—	2	1	no	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
49	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#97×C₆₇	C₂×C₄	1	8	yes	complemented
58	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
59	C₂²×Dic₆₇	—	2	4	no	—
60	#11761612	—	2	8	no	—
61	#11761612	—	2	4	no	—
62	#11761612	C₂²×C₄	1	8	yes	—
63	C₂×C₆₇⋊C₈	—	2	8	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₄⋊Dic₆₇	—	2	4	no	—
66	C₄⋊Dic₆₇	C₂²×C₄	1	8	yes	—
67	C₂²⋊Dic₆₇	—	2	4	no	—
68	C₂²⋊Dic₆₇	—	2	4	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	—	2	4	no	—
71	C₂⁴×C₆₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₆₇	C₄²	1	16	yes	—
85	M₁₆×C₆₇	C₄⋊C₄	1	16	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂³×C₆₇	—	4	2	no	—
98	C₂³×C₆₇	#117	1	2	yes	—
99	C₂³×C₆₇	C₄×D₄	1	2	yes	—
100	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
101	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	8	no	—
106	C₂×C₄×C₆₇	—	4	2	no	—
107	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
108	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₆₇	#117	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₆₇	—	2	8	no	—
121	C₂×Dic₆₇	—	8	4	no	—
122	C₈×C₆₇	C₂×C₄²	1	8	yes	—
123	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
124	C₆₇⋊C₈	—	2	8	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
130	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	—	4	2	no	—
135	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
136	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	Dic₆₇	—	4	8	no	—
141	#20445	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	4	no	—
145	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₆₇	#2956	1	2	yes	—
147	C₂×#430	—	67	1	no	—
148	#2911	—	67	1	no	—
149	#2873	—	67	2	no	—
150	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
151	#1773	—	67	4	no	—
152	#1772	—	67	4	no	—
153	C₂×#339	—	67	2	no	—
154	C₆₇	#20445	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	67	2	no	—
156	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
158	C₂×#117	—	67	2	no	—
159	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
160	#430	—	67	8	no	—
161	C₂×#108	—	67	4	no	—
162	#428	—	67	8	no	—
163	#427	—	67	4	no	—
164	#405	—	67	8	no	—
165	#405	—	134	4	no	—
166	C₄×M₁₆	—	134	4	no	—
167	C₄×M₁₆	Dic₆₇	1	8	yes	complemented
168	C₂×#97	C₂×D₆₇	1	2	yes	—
169	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
170	#339	—	67	4	no	—
171	#339	—	67	4	no	—
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×M₁₆	—	67	8	no	—
175	C₂×M₁₆	—	67	4	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	4	no	—
178	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
179	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
180	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
181	C₂×M₁₆	D₂₆₈	1	4	yes	—
182	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
183	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
184	C₂²×C₈	—	134	2	no	—
185	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
186	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
187	#117	—	67	4	no	—
188	#117	—	67	4	no	—
189	#117	—	134	2	no	—
190	#117	—	134	2	no	—
191	C₂×C₄⋊C₄	—	67	4	no	—
192	C₂×C₂²⋊C₄	—	134	2	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	134	2	no	—
196	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
197	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
199	#108	—	67	16	no	—
200	#107	—	67	8	no	—
201	#106	—	67	8	no	—
202	#97	—	134	8	no	—
203	#97	C₂×Dic₆₇	1	8	yes	complemented
204	C₄×C₈	—	134	8	no	—
205	C₄×C₈	C₂×Dic₆₇	1	8	yes	complemented
206	C₂⁴	#11761628	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	134	4	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	C₂×D₂₆₈	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
217	C₂²×C₄	#11761628	1	1	yes	characteristic
218	C₂²×C₄	#11761628	1	2	yes	—
219	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
220	M₁₆	—	67	8	no	—
221	M₁₆	—	134	8	no	—
222	M₁₆	—	134	4	no	—
223	M₁₆	C₄×Dic₆₇	1	16	yes	—
224	M₁₆	C₄⋊Dic₆₇	1	16	yes	—
225	C₂×C₈	—	134	8	no	—
226	C₂×C₈	—	134	4	no	—
227	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
229	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
230	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
231	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
232	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
233	C₄⋊C₄	—	67	8	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₂²⋊C₄	—	134	4	no	—
236	C₂²⋊C₄	—	134	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	134	4	no	—
241	C₄²	—	134	4	no	—
242	C₄²	C₂²×Dic₆₇	1	2	yes	—
243	C₄²	C₂²×Dic₆₇	1	2	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	#14524002	1	2	yes	—
246	C₂³	C₄×D₂₆₈	1	2	yes	—
247	C₂³	C₄×Dic₁₃₄	1	2	yes	—
248	C₂³	C₂×#11761628	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	536	4	no	—
255	C₂×C₄	—	134	8	no	—
256	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
258	C₂×C₄	#14524002	1	2	yes	—
259	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
260	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
261	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
262	C₂×C₄	C₄×Dic₁₃₄	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	C₂²×D₂₆₈	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
268	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
269	C₈	—	134	8	no	—
270	C₈	C₂×C₄×Dic₆₇	1	8	yes	—
271	C₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
277	C₂²	C₂×#14524002	1	1	yes	characteristic
278	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
279	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	4	2	no	—
282	C₄	—	268	8	no	—
283	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
284	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
285	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
286	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
287	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	4	no	—
290	C₂	—	2	1	no	—
291	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
292	C₂	#22569120	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#20445
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22560093	C₂	semidirect
5. #22560088	C₂	semidirect
10. #16863445	C₄	semidirect
12. #16863438	C₄	semidirect
22. C₄×M₁₆×C₆₇	C₄	semidirect
37. #14523979	C₂×C₄	semidirect
38. #14523977	C₂×C₄	semidirect
39. #14523975	C₂×C₄	semidirect
57. #97×C₆₇	C₂×C₄	semidirect
58. C₄×C₈×C₆₇	C₂×C₄	semidirect
154. C₆₇	#20445	semidirect
167. C₄×M₁₆	Dic₆₇	semidirect
203. #97	C₂×Dic₆₇	semidirect
205. C₄×C₈	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863445	2
#22568799	2
#22568627	2
#22560093	2
#22560088	2
C₂×#16863156	2
C₂×C₄×M₁₆×C₆₇	2
#20445	67