Subgroups — #46845513

Order 34688.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774028	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#31773975	C₂	1	2	yes	maximal, complemented
4	#31773975	C₂	1	1	yes	maximal, characteristic, complemented
5	#31773969	C₂	1	2	yes	maximal, complemented
6	D₄×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#31773958	C₂	1	2	yes	maximal, complemented
8	#31773950	C₂	1	1	yes	maximal, characteristic, complemented
9	#31773924	C₂	1	2	yes	maximal, complemented
10	#31773921	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₂₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#16864638	C₂²	1	2	yes	—
13	C₂×#16864638	C₂²	1	2	yes	—
14	C₂²×C₂₇₁⋊C₈	C₂²	1	2	yes	—
15	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
19	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
23	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
24	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
26	#22576071	C₂²	1	2	yes	—
27	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	—
28	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
30	#126×C₂₇₁	C₄	1	2	yes	—
31	#125×C₂₇₁	C₄	1	1	yes	characteristic
32	#124×C₂₇₁	C₄	1	1	yes	characteristic
33	#123×C₂₇₁	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₂₇₁	C₄	1	1	yes	characteristic
35	#121×C₂₇₁	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₂₇₁	C₂²	1	2	yes	—
39	#117×C₂₇₁	C₂²	1	2	yes	—
40	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
41	#16864638	—	4	2	no	—
42	#16864638	—	4	2	no	—
43	C₂×C₂₇₁⋊C₈	—	4	2	no	—
44	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
46	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
47	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂×D₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄×C₂₇₁	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
68	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
71	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
72	C₄²×C₂₇₁	C₂³	1	2	yes	—
73	C₂₇₁⋊C₈	—	2	1	no	—
74	C₂₇₁⋊C₈	—	2	2	no	—
75	C₂₇₁⋊C₈	—	2	2	no	—
76	C₂₇₁⋊C₈	—	2	1	no	—
77	C₂₇₁⋊C₈	—	2	1	no	—
78	C₂₇₁⋊C₈	—	2	1	no	—
79	C₂³×C₂₇₁	C₂²×C₄	1	2	yes	—
80	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₂₇₁	—	2	2	no	—
82	D₄×C₂₇₁	—	2	2	no	—
83	D₄×C₂₇₁	—	4	2	no	—
84	C₂×C₄×C₂₇₁	—	2	2	no	—
85	C₂×C₄×C₂₇₁	—	2	2	no	—
86	C₂×C₄×C₂₇₁	—	2	1	no	—
87	C₂×C₄×C₂₇₁	—	2	2	no	—
88	C₂×C₄×C₂₇₁	—	2	1	no	—
89	C₂×C₄×C₂₇₁	—	4	1	no	—
90	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
99	C₂²×C₂₇₁	—	2	2	no	—
100	C₂²×C₂₇₁	—	2	1	no	—
101	C₂²×C₂₇₁	—	2	1	no	—
102	C₂²×C₂₇₁	—	2	2	no	—
103	C₂²×C₂₇₁	—	2	1	no	—
104	C₂²×C₂₇₁	—	2	2	no	—
105	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₂₇₁	—	2	2	no	—
107	C₄×C₂₇₁	—	2	1	no	—
108	C₄×C₂₇₁	—	2	1	no	—
109	C₄×C₂₇₁	—	2	2	no	—
110	C₄×C₂₇₁	—	2	1	no	—
111	C₄×C₂₇₁	—	2	1	no	—
112	C₄×C₂₇₁	—	2	2	no	—
113	C₄×C₂₇₁	#131	1	2	yes	—
114	C₄×C₂₇₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₂₇₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₂₇₁	—	4	1	no	—
117	C₂×C₂₇₁	—	4	2	no	—
118	C₂×C₂₇₁	#520	1	1	yes	characteristic
119	C₂×C₂₇₁	C₂×#131	1	1	yes	characteristic
120	C₂×C₂₇₁	#568	1	1	yes	characteristic
121	C₂₇₁	#2993	1	1	yes	characteristic, complemented, Sylow
122	#2993	—	271	1	no	maximal, Sylow
123	#529	D₂₇₁	1	1	yes	characteristic
124	#446	—	271	1	no	—
125	C₈×Q₈	—	271	1	no	—
126	#436	—	271	1	no	—
127	#436	—	271	2	no	—
128	#435	—	271	2	no	—
129	C₈×D₄	—	271	1	no	—
130	#433	—	271	2	no	—
131	#431	—	271	1	no	—
132	#424	—	271	2	no	—
133	#423	—	271	1	no	—
134	C₂×M₁₆	—	271	2	no	—
135	C₂×M₁₆	—	271	2	no	—
136	C₂²×C₈	—	271	2	no	—
137	#126	Dic₂₇₁	1	2	yes	—
138	#125	Dic₂₇₁	1	1	yes	characteristic
139	#124	Dic₂₇₁	1	1	yes	characteristic
140	#123	Dic₂₇₁	1	2	yes	—
141	C₂²⋊Q₈	Dic₂₇₁	1	1	yes	characteristic
142	#121	Dic₂₇₁	1	1	yes	characteristic
143	C₄×Q₈	C₂×D₂₇₁	1	1	yes	characteristic
144	C₄×D₄	C₂×D₂₇₁	1	1	yes	characteristic
145	C₄×D₄	C₂×D₂₇₁	1	2	yes	—
146	#117	C₂×D₂₇₁	1	2	yes	—
147	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
148	C₄⋊C₈	—	271	2	no	—
149	C₄⋊C₈	—	271	1	no	—
150	C₄⋊C₈	—	271	1	no	—
151	C₄⋊C₈	—	271	1	no	—
152	C₄⋊C₈	—	271	2	no	—
153	C₄⋊C₈	—	271	2	no	—
154	C₄⋊C₈	—	271	1	no	—
155	C₂²⋊C₈	—	271	1	no	—
156	C₂²⋊C₈	—	271	1	no	—
157	C₂²⋊C₈	—	271	2	no	—
158	C₂²⋊C₈	—	271	2	no	—
159	#97	—	271	2	no	—
160	C₄×C₈	—	271	1	no	—
161	C₄×C₈	—	271	1	no	—
162	C₄×C₈	—	271	2	no	—
163	C₂×Q₈	C₂×Dic₂₇₁	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₂₇₁	1	1	yes	characteristic
165	C₂×D₄	C₂×Dic₂₇₁	1	2	yes	—
166	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₇₁	1	2	yes	—
170	M₁₆	—	1084	2	no	—
171	M₁₆	—	1084	2	no	—
172	C₂×C₈	—	271	1	no	—
173	C₂×C₈	—	271	1	no	—
174	C₂×C₈	—	271	1	no	—
175	C₂×C₈	—	271	1	no	—
176	C₂×C₈	—	271	2	no	—
177	C₂×C₈	—	271	2	no	—
178	C₂×C₈	—	1084	2	no	—
179	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₂₇₁	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₂₇₁	1	2	yes	—
190	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
191	C₄²	C₂×Dic₂₇₁	1	1	yes	characteristic
192	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
193	C₄²	C₂²×D₂₇₁	1	2	yes	—
194	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
195	C₂³	C₂²×Dic₂₇₁	1	2	yes	—
196	C₂³	C₂²×Dic₂₇₁	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	4	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₂₇₁	1	2	yes	—
211	C₂×C₄	C₂²×Dic₂₇₁	1	2	yes	—
212	C₂×C₄	C₂²×Dic₂₇₁	1	2	yes	—
213	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
215	C₈	—	542	1	no	—
216	C₈	—	542	2	no	—
217	C₈	—	542	1	no	—
218	C₈	—	542	2	no	—
219	C₈	—	542	1	no	—
220	C₈	—	542	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₂³×Dic₂₇₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	#22576167	1	2	yes	—
236	C₄	#22576206	1	1	yes	characteristic
237	C₄	#22576207	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	4	2	no	—
240	C₂	#31774380	1	1	yes	characteristic
241	C₂	C₂×#22576167	1	1	yes	characteristic
242	C₂	#31774664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂³×C₄
Fitting	#529×C₂₇₁	C₂
prime radical	#121×C₂₇₁	C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	#529	D₂₇₁
hypocenter	C₂₇₁	#2993
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #31774028	C₂	semidirect
2. Q₈×C₂₇₁⋊C₈	C₂	semidirect
3. #31773975	C₂	semidirect
4. #31773975	C₂	semidirect
5. #31773969	C₂	semidirect
6. D₄×C₂₇₁⋊C₈	C₂	semidirect
7. #31773958	C₂	semidirect
8. #31773950	C₂	semidirect
9. #31773924	C₂	semidirect
10. #31773921	C₂	semidirect
121. C₂₇₁	#2993	semidirect

Maximal subgroups

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

subgroup	index
#31774028	2
Q₈×C₂₇₁⋊C₈	2
#31773975	2
#31773969	2
D₄×C₂₇₁⋊C₈	2
#31773958	2
#31773950	2
#31773924	2
#31773921	2
#529×C₂₇₁	2
#2993	271