Subgroups — #31010126

Order 17152.

702 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	C₂×#16863837	C₂	1	2	yes	maximal
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22562256	C₂	1	4	yes	maximal, complemented
4	#22562251	C₂	1	4	yes	maximal, complemented
5	#22561965	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561929	C₂	1	2	yes	maximal, complemented
7	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524065	C₂²	1	1	yes	characteristic
9	C₄×#11761628	—	2	1	no	—
10	#16863837	—	2	8	no	—
11	C₂×#14523989	C₂²	1	4	yes	—
12	C₂×#14523986	C₂²	1	4	yes	—
13	C₂×#14523983	C₂²	1	2	yes	—
14	C₄×Dic₂₆₈	C₂²	1	2	yes	—
15	C₄×#11761617	—	2	2	no	—
16	C₄×D₅₃₆	C₂²	1	2	yes	—
17	#16863434	C₂²	1	1	yes	characteristic
18	#16863393	—	2	1	no	—
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
20	#16863200	C₂²	1	8	yes	—
21	C₂²×Q₁₆×C₆₇	—	2	1	no	—
22	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
24	#494×C₆₇	—	2	2	no	—
25	#494×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	—	2	1	no	—
28	#14524065	—	2	2	no	—
29	#14524065	D₄	1	4	yes	—
30	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
31	C₂×#11761617	—	2	1	no	—
32	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
33	C₄×Dic₁₃₄	—	2	2	no	—
34	C₄×D₂₆₈	—	2	2	no	—
35	C₄×C₂²⋊D₆₇	—	4	2	no	—
36	#14524002	—	2	1	no	—
37	#14524000	—	4	1	no	—
38	C₄²×D₆₇	—	4	1	no	—
39	#14523989	—	2	8	no	—
40	#14523986	—	2	8	no	—
41	#14523983	—	2	4	no	—
42	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
43	#14523977	C₂³	1	2	yes	—
44	#14523975	—	2	1	no	—
45	#14523968	—	2	2	no	—
46	#14523965	—	2	2	no	—
47	C₂²×Q₈×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	4	no	—
49	C₂×Q₁₆×C₆₇	—	2	2	no	—
50	C₂×Q₁₆×C₆₇	—	2	4	no	—
51	C₂×Q₁₆×C₆₇	—	2	4	no	—
52	C₂×Q₁₆×C₆₇	—	4	4	no	—
53	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	—	2	4	no	—
57	C₄⋊Q₈×C₆₇	—	4	1	no	—
58	C₄⋊Q₈×C₆₇	—	4	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	D₄	1	2	yes	—
62	C₄×C₈×C₆₇	C₂³	1	2	yes	—
63	#11761628	—	2	4	no	—
64	#11761628	—	4	2	no	—
65	C₂×Dic₁₃₄	—	2	1	no	—
66	C₂×D₂₆₈	—	2	1	no	—
67	C₂×C₂²⋊D₆₇	—	4	1	no	—
68	C₂×C₄×D₆₇	—	4	1	no	—
69	C₂×C₄×D₆₇	—	4	2	no	—
70	Dic₂₆₈	—	2	1	no	—
71	Dic₂₆₈	C₂×D₄	1	2	yes	—
72	#11761617	—	2	4	no	—
73	D₅₃₆	—	2	1	no	—
74	D₅₃₆	C₂×D₄	1	2	yes	—
75	C₄⋊Dic₆₇	—	2	2	no	—
76	Dic₆₇⋊C₄	—	4	2	no	—
77	C₂²⋊Dic₆₇	—	4	1	no	—
78	(C₂×D₆₇)⋊C₄	—	4	2	no	—
79	C₄×Dic₆₇	—	4	1	no	—
80	C₄×Dic₆₇	—	4	2	no	—
81	C₆₇⋊C₁₆	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	4	4	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	D₈	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₆₇	—	2	4	no	—
90	Q₁₆×C₆₇	—	2	4	no	—
91	Q₁₆×C₆₇	—	4	8	no	—
92	Q₁₆×C₆₇	—	4	4	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	#41	1	2	yes	—
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₇	—	4	4	no	—
100	C₄⋊C₄×C₆₇	—	4	4	no	—
101	C₄²×C₆₇	D₈	1	1	yes	characteristic
102	C₄²×C₆₇	D₈	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	2	yes	—
104	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₆₇	—	4	1	no	—
106	Q₈×C₆₇	—	4	4	no	—
107	Q₈×C₆₇	—	4	4	no	—
108	Q₈×C₆₇	—	8	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	8	4	no	—
112	C₂×C₄×C₆₇	#121	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₆₇	—	4	4	no	—
119	C₂×Dic₆₇	—	4	1	no	—
120	C₂×Dic₆₇	—	4	2	no	—
121	D₂₆₈	—	2	2	no	—
122	D₂₆₈	—	4	1	no	—
123	C₄×D₆₇	—	4	4	no	—
124	C₄×D₆₇	—	8	4	no	—
125	Dic₁₃₄	—	2	2	no	—
126	Dic₁₃₄	—	4	1	no	—
127	C₈×C₆₇	—	2	2	no	—
128	C₈×C₆₇	#121	1	2	yes	—
129	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	#459	1	2	yes	—
133	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₆₇	—	4	2	no	—
135	C₂×D₆₇	—	8	2	no	—
136	C₄×C₆₇	—	2	2	no	—
137	C₄×C₆₇	—	8	4	no	—
138	C₄×C₆₇	#459	1	2	yes	—
139	C₄×C₆₇	#510	1	4	yes	complemented
140	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
141	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	Dic₆₇	—	4	2	no	—
143	Dic₆₇	—	8	2	no	—
144	#33583	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#510	1	2	yes	—
148	D₆₇	—	8	2	no	—
149	C₂×#510	—	67	2	no	—
150	C₂×#494	D₆₇	1	1	yes	characteristic, complemented
151	C₄×#135	—	67	1	no	—
152	#2209	—	67	4	no	—
153	#2208	—	67	4	no	—
154	#2150	—	67	1	no	—
155	#2141	—	67	2	no	—
156	C₆₇	#33583	1	1	yes	characteristic, complemented, Sylow
157	C₂×#135	—	67	1	no	—
158	C₂²×Q₁₆	—	2	1	no	—
159	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
160	C₂×C₄⋊Q₈	—	2	1	no	—
161	C₄×#41	—	134	1	no	—
162	#510	—	134	8	no	—
163	C₂×Q₃₂	—	67	4	no	—
164	C₂×SD₃₂	—	67	4	no	—
165	C₂×M₃₂	—	67	2	no	—
166	#494	—	2	2	no	—
167	#494	C₂×D₆₇	1	4	yes	—
168	C₄×Q₁₆	—	67	2	no	—
169	C₄×SD₁₆	—	134	2	no	—
170	C₄×D₈	—	67	2	no	—
171	#427	—	67	1	no	—
172	#416	—	134	1	no	—
173	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
174	C₄⋊C₁₆	—	67	4	no	—
175	#358	—	67	8	no	—
176	C₂×#41	—	134	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#135	—	67	4	no	—
179	#135	—	134	2	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	4	no	—
184	C₂×Q₁₆	—	4	4	no	—
185	C₂×Q₁₆	—	67	1	no	—
186	C₂×Q₁₆	C₂²×D₆₇	1	4	yes	—
187	C₂×SD₁₆	—	134	1	no	—
188	C₂×D₈	—	67	1	no	—
189	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	C₄⋊Q₈	—	4	1	no	—
194	C₄×Q₈	—	134	2	no	—
195	C₄×D₄	—	268	2	no	—
196	C₄×D₄	—	134	2	no	—
197	#117	—	268	1	no	—
198	#117	—	134	1	no	—
199	C₂×C₄⋊C₄	—	4	2	no	—
200	C₂×C₄²	—	268	1	no	—
201	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
202	Q₃₂	—	134	8	no	—
203	SD₃₂	—	134	8	no	—
204	M₃₂	—	134	4	no	—
205	C₂×C₁₆	—	67	4	no	—
206	#107	—	67	2	no	—
207	#106	—	134	1	no	—
208	#103	—	134	2	no	—
209	#102	—	134	2	no	—
210	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
211	C₄×C₈	C₂²×D₆₇	1	2	yes	—
212	#41	—	268	2	no	—
213	#41	—	134	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	134	1	no	—
219	C₂×D₄	—	268	1	no	—
220	C₂×D₄	—	134	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	268	1	no	—
223	C₂²×C₄	—	268	2	no	—
224	C₂²×C₄	#11761613	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	4	8	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	67	2	no	—
231	Q₁₆	—	134	1	no	—
232	SD₁₆	—	134	4	no	—
233	D₈	—	67	2	no	—
234	D₈	—	134	1	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	D₄×D₆₇	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
239	C₂×C₈	#11761627	1	2	yes	—
240	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	268	2	no	—
244	C₄⋊C₄	—	134	2	no	—
245	C₂²⋊C₄	—	268	1	no	—
246	C₂²⋊C₄	—	268	2	no	—
247	C₄²	—	268	1	no	—
248	C₄²	—	268	2	no	—
249	C₄²	#11761613	1	1	yes	characteristic
250	C₄²	#11761613	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
252	C₁₆	—	134	4	no	—
253	C₂³	—	268	1	no	—
254	C₂³	C₂×#11761613	1	1	yes	characteristic
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	268	1	no	—
259	Q₈	—	134	2	no	—
260	D₄	—	268	4	no	—
261	D₄	—	268	1	no	—
262	D₄	—	134	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	268	1	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	268	2	no	—
269	C₂×C₄	—	536	4	no	—
270	C₂×C₄	#14524014	1	2	yes	—
271	C₂×C₄	C₂×#11761613	1	2	yes	—
272	C₂×C₄	C₂×#11761613	1	2	yes	—
273	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
274	C₂×C₄	#14524069	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	#14524014	1	2	yes	—
278	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
279	C₈	C₂×#11761627	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	268	2	no	—
282	C₂²	—	536	2	no	—
283	C₂²	#16863588	1	2	yes	—
284	C₂²	C₂²×#11761613	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	8	4	no	—
287	C₄	—	268	2	no	—
288	C₄	—	536	2	no	—
289	C₄	#16863588	1	2	yes	—
290	C₄	#16863837	1	4	yes	complemented
291	C₄	C₂×#14524014	1	1	yes	characteristic
292	C₄	C₂×#14524069	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	536	2	no	—
295	C₂	C₂×#16863588	1	1	yes	characteristic
296	C₂	C₂×#16863837	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#14524065	C₂²
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#33583
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #22562256	C₂	semidirect
4. #22562251	C₂	semidirect
5. #22561965	C₂	semidirect
6. #22561929	C₂	semidirect
7. C₂×#494×C₆₇	C₂	semidirect
139. C₄×C₆₇	#510	semidirect
150. C₂×#494	D₆₇	semidirect
156. C₆₇	#33583	semidirect
290. C₄	#16863837	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863837	2
C₄×#14524065	2
#22562256	2
#22562251	2
#22561965	2
#22561929	2
C₂×#494×C₆₇	2
#33583	67