Subgroups — #31009200

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523986	C₂	1	1	yes	maximal, characteristic
2	#22569062	C₂	1	1	yes	maximal, characteristic
3	#22562071	C₂	1	4	yes	maximal, complemented
4	C₄×#14523986	C₂	1	4	yes	maximal, complemented
5	#22561991	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561929	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561924	C₂	1	1	yes	maximal, characteristic
8	#22561806	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
11	#16863870	—	2	1	no	—
12	C₂×#14523986	C₄	1	8	yes	complemented
13	C₂×#14523986	C₂²	1	2	yes	—
14	C₂×#14523986	C₂²	1	2	yes	—
15	C₂×#14523983	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16863509	—	2	2	no	—
18	C₄×D₅₃₆	C₂²	1	4	yes	—
19	#16863434	C₂²	1	1	yes	characteristic
20	#16863402	—	2	1	no	—
21	#16863223	C₂²	1	2	yes	complemented
22	#16863221	C₂²	1	2	yes	complemented
23	#16863200	C₂²	1	4	yes	complemented
24	#16863196	C₂²	1	4	yes	—
25	#16863172	C₂²	1	2	yes	complemented
26	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
27	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	#519×C₆₇	—	2	1	no	—
29	#441×C₆₇	—	2	2	no	—
30	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
31	#427×C₆₇	C₂²	1	1	yes	characteristic
32	#419×C₆₇	—	2	1	no	—
33	#405×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×D₂₆₈	—	2	1	no	—
35	C₂×D₅₃₆	—	2	1	no	—
36	C₂×D₅₃₆	—	2	2	no	—
37	C₂×D₅₃₆	C₂×C₄	1	4	yes	—
38	C₂×D₅₃₆	C₂³	1	2	yes	—
39	C₄×D₂₆₈	—	2	4	no	—
40	#14524005	—	4	2	no	—
41	#14524002	—	2	1	no	—
42	C₂²⋊C₄×D₆₇	—	4	2	no	—
43	#14523986	C₂×C₄	1	8	yes	complemented
44	#14523986	C₂×C₄	1	8	yes	complemented
45	#14523983	—	2	2	no	—
46	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
50	#14523977	C₂³	1	2	yes	—
51	#14523975	—	2	1	no	—
52	#14523965	—	2	4	no	—
53	C₂²×Q₈×C₆₇	—	2	1	no	—
54	C₂×Q₁₆×C₆₇	—	2	1	no	—
55	C₂×Q₁₆×C₆₇	—	2	2	no	—
56	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
57	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
58	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₆₇	—	2	4	no	—
61	C₄×Q₈×C₆₇	—	4	2	no	—
62	#117×C₆₇	—	2	1	no	—
63	#117×C₆₇	—	4	2	no	—
64	#117×C₆₇	D₄	1	1	yes	characteristic
65	#107×C₆₇	C₂³	1	2	yes	—
66	#106×C₆₇	—	2	1	no	—
67	#103×C₆₇	—	2	4	no	—
68	#97×C₆₇	D₄	1	2	yes	—
69	C₄×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂³×D₆₇	—	4	1	no	—
71	C₂×D₂₆₈	—	2	4	no	—
72	C₂×D₂₆₈	—	2	2	no	—
73	C₂×D₂₆₈	—	4	1	no	—
74	C₂×D₂₆₈	—	4	2	no	—
75	C₂×C₄×D₆₇	—	4	4	no	—
76	D₅₃₆	—	2	4	no	—
77	D₅₃₆	—	2	2	no	—
78	D₅₃₆	C₂²×C₄	1	4	yes	—
79	C₄⋊Dic₆₇	—	2	2	no	—
80	C₂²⋊Dic₆₇	—	4	1	no	—
81	(C₂×D₆₇)⋊C₄	—	4	4	no	—
82	C₄×Dic₆₇	—	4	1	no	—
83	C₆₇⋊C₁₆	—	2	2	no	—
84	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
85	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	4	1	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₆₇	—	2	4	no	—
93	Q₁₆×C₆₇	—	2	2	no	—
94	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
95	M₁₆×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	#41	1	2	yes	—
100	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	4	4	no	—
103	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
104	C₂²⋊C₄×C₆₇	—	4	1	no	—
105	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
106	C₄²×C₆₇	—	4	1	no	—
107	C₄²×C₆₇	—	4	4	no	—
108	C₄²×C₆₇	C₂×D₄	1	2	yes	—
109	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₆₇	—	4	2	no	—
111	C₂²×D₆₇	—	4	4	no	—
112	C₂²×D₆₇	—	16	2	no	—
113	Q₈×C₆₇	—	2	4	no	—
114	Q₈×C₆₇	—	4	2	no	—
115	Q₈×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#135	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₆₇	—	4	2	no	—
125	D₂₆₈	—	2	4	no	—
126	D₂₆₈	—	4	2	no	—
127	D₂₆₈	—	4	4	no	—
128	C₄×D₆₇	—	16	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₄×D₄	1	2	yes	—
131	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	C₄×D₈	1	2	yes	—
135	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₆₇	—	4	4	no	—
137	C₂×D₆₇	—	8	4	no	—
138	C₂×D₆₇	—	16	4	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	—	4	4	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	C₄×D₈	1	2	yes	—
143	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
145	Dic₆₇	—	8	2	no	—
146	#33457	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
149	C₂×C₆₇	#3416	1	2	yes	—
150	D₆₇	—	8	4	no	—
151	C₂²×SD₃₂	—	67	1	no	—
152	#2950	D₆₇	1	1	yes	characteristic, complemented
153	#2948	—	67	1	no	—
154	#2175	—	67	4	no	—
155	C₄×SD₃₂	—	67	4	no	—
156	#2158	—	67	1	no	—
157	#2141	—	67	1	no	—
158	#2140	—	67	1	no	—
159	#2108	—	67	1	no	—
160	C₆₇	#33457	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
162	C₂²×D₈	—	67	1	no	—
163	#519	—	2	1	no	—
164	#518	—	134	1	no	—
165	C₂×SD₃₂	—	67	8	no	—
166	C₂×SD₃₂	—	67	2	no	—
167	C₂×SD₃₂	—	67	2	no	—
168	C₂×M₃₂	—	67	1	no	—
169	C₂²×C₁₆	—	67	1	no	—
170	#442	—	134	2	no	—
171	#441	—	2	2	no	—
172	C₄×Q₁₆	C₂×D₆₇	1	4	yes	complemented
173	C₄×D₈	—	67	4	no	—
174	#427	—	67	1	no	—
175	#427	C₂×D₆₇	1	1	yes	characteristic
176	#419	—	2	1	no	—
177	#418	—	134	1	no	—
178	#405	C₂×D₆₇	1	1	yes	characteristic
179	#367	—	67	2	no	—
180	#366	—	67	2	no	—
181	#358	—	67	4	no	—
182	#357	—	67	4	no	—
183	#346	—	67	2	no	—
184	C₄×C₁₆	—	67	2	no	—
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	134	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	C₄×D₆₇	1	4	yes	complemented
190	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
191	C₂×D₈	—	67	4	no	—
192	C₂×D₈	—	67	2	no	—
193	C₂×D₈	—	134	1	no	—
194	C₂×D₈	—	134	2	no	—
195	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	4	2	no	—
199	C₄×D₄	—	268	2	no	—
200	C₄×D₄	—	134	4	no	—
201	#117	—	2	1	no	—
202	#117	—	4	2	no	—
203	#117	—	134	1	no	—
204	#117	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	268	2	no	—
206	SD₃₂	—	67	8	no	—
207	SD₃₂	—	67	8	no	—
208	M₃₂	—	134	2	no	—
209	C₂×C₁₆	—	67	4	no	—
210	C₂×C₁₆	—	67	1	no	—
211	C₂×C₁₆	—	67	1	no	—
212	C₂×C₁₆	—	67	2	no	—
213	#107	—	67	2	no	—
214	#107	C₂²×D₆₇	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	134	1	no	—
217	#103	—	2	4	no	—
218	#102	—	134	4	no	—
219	#97	C₂²⋊D₆₇	1	2	yes	—
220	C₄×C₈	C₂²×D₆₇	1	2	yes	—
221	C₂⁴	—	268	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	268	1	no	—
227	C₂×D₄	—	268	2	no	—
228	C₂×D₄	—	134	4	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	268	4	no	—
232	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
236	D₈	—	67	4	no	—
237	D₈	—	134	4	no	—
238	D₈	—	134	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂×C₈	#11761628	1	2	yes	—
244	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	134	2	no	—
248	C₄⋊C₄	#11761613	1	2	yes	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	268	1	no	—
251	C₂²⋊C₄	—	268	4	no	—
252	C₂²⋊C₄	#11761613	1	2	yes	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	4	no	—
255	C₄²	—	268	1	no	—
256	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
257	C₁₆	—	67	2	no	—
258	C₁₆	—	67	2	no	—
259	C₁₆	—	134	2	no	—
260	C₂³	—	268	2	no	—
261	C₂³	—	268	4	no	—
262	C₂³	—	1072	2	no	—
263	C₂³	C₂×#11761613	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	268	2	no	—
268	D₄	—	268	4	no	—
269	D₄	—	134	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	268	2	no	—
274	C₂×C₄	—	1072	4	no	—
275	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
276	C₂×C₄	C₂×#11761613	1	4	yes	—
277	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
278	C₂×C₄	#14524063	1	2	yes	—
279	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
282	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
283	C₈	C₂×#11761628	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	268	4	no	—
286	C₂²	—	536	4	no	—
287	C₂²	—	1072	4	no	—
288	C₂²	C₄×#11761613	1	2	yes	—
289	C₂²	C₂²×#11761613	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	2	no	—
293	C₄	—	536	2	no	—
294	C₄	C₄×#11761613	1	2	yes	—
295	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
296	C₄	C₂×#14524063	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	536	4	no	—
299	C₂	C₂×C₄×#11761613	1	1	yes	characteristic
300	C₂	#22573353	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2950	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₅₃₆	C₂²
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2950	D₆₇
hypocenter	C₆₇	#33457
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #22562071	C₂	semidirect
4. C₄×#14523986	C₂	semidirect
5. #22561991	C₂	semidirect
6. #22561929	C₂	semidirect
8. #22561806	C₂	semidirect
9. C₆₇×#2950	C₂	semidirect
12. C₂×#14523986	C₄	semidirect
21. #16863223	C₂²	semidirect
22. #16863221	C₂²	semidirect
23. #16863200	C₂²	semidirect
25. #16863172	C₂²	semidirect
26. C₄×C₆₇⋊C₁₆	C₂²	semidirect
30. C₄×Q₁₆×C₆₇	C₂²	semidirect
43. #14523986	C₂×C₄	semidirect
44. #14523986	C₂×C₄	semidirect
46. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
56. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
152. #2950	D₆₇	semidirect
160. C₆₇	#33457	semidirect
172. C₄×Q₁₆	C₂×D₆₇	semidirect
189. C₂×Q₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523986	2
#22569062	2
#22562071	2
C₄×#14523986	2
#22561991	2
#22561929	2
#22561924	2
#22561806	2
C₆₇×#2950	2
#33457	67