Subgroups — #46845500

Order 34688.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774431	C₂	1	1	yes	maximal, characteristic
2	#31774026	C₂	1	2	yes	maximal, complemented
3	#31773968	C₂	1	2	yes	maximal, complemented
4	#31773967	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₂₇₁	C₂	1	2	yes	maximal, complemented
6	#31773957	C₂	1	2	yes	maximal, complemented
7	#31773954	C₂	1	1	yes	maximal, characteristic, complemented
8	#31773922	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16864637	C₂²	1	2	yes	—
11	C₂×C₈×D₂₇₁	C₂²	1	2	yes	—
12	#22576154	C₄	1	2	yes	—
13	#22576152	C₄	1	1	yes	characteristic
14	#22576148	C₄	1	1	yes	characteristic
15	#22576143	C₄	1	2	yes	—
16	#22576139	C₄	1	1	yes	characteristic
17	#22576134	C₄	1	1	yes	characteristic
18	C₄×Dic₅₄₂	C₂²	1	1	yes	characteristic
19	C₄×D₁₀₈₄	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₂₇₁	C₂²	1	2	yes	—
21	#22576117	C₂²	1	2	yes	—
22	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₂₇₁⋊C₈	C₂²	1	2	yes	—
25	Dic₂₇₁⋊C₈	C₂²	1	2	yes	—
26	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
27	(C₂×D₂₇₁)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₂₇₁)⋊C₈	C₂²	1	2	yes	—
29	#22576071	C₂²	1	2	yes	—
30	#22576070	C₂²	1	2	yes	—
31	C₈×Dic₂₇₁	C₂²	1	2	yes	—
32	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₂₇₁	C₂²	1	2	yes	—
36	C₄⋊C₈×C₂₇₁	C₂²	1	2	yes	—
37	C₂×Dic₅₄₂	C₂×C₄	1	1	yes	characteristic
38	C₂×D₁₀₈₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₂₇₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
41	#16864637	—	4	2	no	—
42	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
43	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
44	C₈×D₂₇₁	—	4	2	no	—
45	C₄⋊Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
47	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₂₇₁	C₂×C₄	1	2	yes	—
52	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₇₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₇₁	—	4	1	no	—
61	C₂×C₈×C₂₇₁	—	4	1	no	—
62	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
63	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
64	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
67	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
68	C₂²×D₂₇₁	C₂²×C₄	1	2	yes	—
69	Dic₅₄₂	—	4	1	no	—
70	D₁₀₈₄	—	4	1	no	—
71	C₂²⋊D₂₇₁	—	2	4	no	—
72	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
75	C₄×D₂₇₁	—	2	2	no	—
76	C₄×D₂₇₁	—	2	2	no	—
77	C₂₇₁⋊C₈	—	2	2	no	—
78	C₂₇₁⋊C₈	—	2	2	no	—
79	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₇₁	—	2	2	no	—
81	C₂×C₄×C₂₇₁	—	2	1	no	—
82	C₂×C₄×C₂₇₁	—	2	1	no	—
83	C₂×C₄×C₂₇₁	—	4	1	no	—
84	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
90	C₈×C₂₇₁	—	2	2	no	—
91	C₈×C₂₇₁	—	2	2	no	—
92	C₂²×C₂₇₁	—	2	1	no	—
93	C₂²×C₂₇₁	—	2	1	no	—
94	C₂²×C₂₇₁	#131	1	2	yes	—
95	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₂₇₁	—	2	2	no	—
97	C₂×D₂₇₁	—	2	2	no	—
98	C₂×D₂₇₁	—	2	2	no	—
99	C₄×C₂₇₁	—	2	1	no	—
100	C₄×C₂₇₁	—	2	1	no	—
101	C₄×C₂₇₁	—	2	1	no	—
102	C₄×C₂₇₁	—	2	1	no	—
103	C₄×C₂₇₁	—	2	1	no	—
104	C₄×C₂₇₁	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₂₇₁	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₂₇₁	—	2	2	no	—
107	Dic₂₇₁	—	2	2	no	—
108	Dic₂₇₁	—	2	2	no	—
109	C₂×C₂₇₁	—	2	2	no	—
110	C₂×C₂₇₁	#520	1	1	yes	characteristic
111	C₂×C₂₇₁	C₂×#131	1	1	yes	characteristic
112	C₂×C₂₇₁	#568	1	1	yes	characteristic
113	D₂₇₁	—	4	2	no	—
114	C₂₇₁	#2992	1	1	yes	characteristic, complemented, Sylow
115	#2992	—	271	1	no	maximal, Sylow
116	#529	—	271	1	no	—
117	#446	—	271	2	no	—
118	#435	—	271	2	no	—
119	#435	—	271	2	no	—
120	C₈×D₄	—	271	2	no	—
121	#433	—	271	2	no	—
122	#432	—	271	1	no	—
123	#424	—	271	2	no	—
124	C₂×C₄⋊C₈	D₂₇₁	1	1	yes	characteristic, complemented
125	C₂×M₁₆	—	271	2	no	—
126	C₂²×C₈	—	271	2	no	—
127	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
129	#126	—	271	2	no	—
130	#125	—	271	1	no	—
131	#124	—	271	1	no	—
132	#123	—	271	2	no	—
133	C₂²⋊Q₈	—	271	1	no	—
134	#121	—	271	1	no	—
135	C₄×Q₈	—	271	1	no	—
136	C₄×D₄	—	271	2	no	—
137	C₄×D₄	—	271	1	no	—
138	#117	—	271	2	no	—
139	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
140	C₄⋊C₈	—	271	2	no	—
141	C₄⋊C₈	—	271	2	no	—
142	C₄⋊C₈	—	271	1	no	—
143	C₄⋊C₈	—	271	1	no	—
144	C₄⋊C₈	C₂×D₂₇₁	1	2	yes	—
145	C₄⋊C₈	C₂×D₂₇₁	1	2	yes	—
146	C₂²⋊C₈	—	271	2	no	—
147	C₂²⋊C₈	—	271	2	no	—
148	C₂²⋊C₈	—	271	2	no	—
149	#97	—	271	2	no	—
150	#97	—	271	2	no	—
151	C₄×C₈	—	271	2	no	—
152	C₂×Q₈	—	271	1	no	—
153	C₂×D₄	—	271	2	no	—
154	C₂×D₄	—	271	1	no	—
155	C₂²×C₄	—	271	2	no	—
156	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
157	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
159	M₁₆	—	1084	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	271	2	no	—
163	C₂×C₈	—	271	2	no	—
164	C₂×C₈	—	1084	2	no	—
165	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
166	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
167	C₄⋊C₄	—	271	2	no	—
168	C₄⋊C₄	—	271	2	no	—
169	C₄⋊C₄	—	271	2	no	—
170	C₄⋊C₄	—	271	1	no	—
171	C₄⋊C₄	—	271	1	no	—
172	C₄⋊C₄	—	271	2	no	—
173	C₂²⋊C₄	—	271	2	no	—
174	C₂²⋊C₄	—	271	2	no	—
175	C₂²⋊C₄	—	271	2	no	—
176	C₂²⋊C₄	—	271	2	no	—
177	C₂²⋊C₄	—	271	2	no	—
178	C₄²	—	271	2	no	—
179	C₄²	C₄×D₂₇₁	1	1	yes	characteristic
180	C₄²	C₄×D₂₇₁	1	1	yes	characteristic
181	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
182	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
183	C₂³	—	271	2	no	—
184	C₂³	C₂×C₄×D₂₇₁	1	1	yes	characteristic
185	Q₈	—	1084	1	no	—
186	D₄	—	542	4	no	—
187	D₄	—	1084	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	271	2	no	—
193	C₂×C₄	—	271	2	no	—
194	C₂×C₄	—	271	2	no	—
195	C₂×C₄	—	542	2	no	—
196	C₂×C₄	—	542	2	no	—
197	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	—	542	2	no	—
206	C₈	—	542	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	542	2	no	—
210	C₂²	—	542	2	no	—
211	C₂²	—	542	2	no	—
212	C₂²	#22576166	1	2	yes	—
213	C₂²	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	542	2	no	—
220	C₄	—	542	2	no	—
221	C₄	—	542	2	no	—
222	C₄	#22576205	1	1	yes	characteristic
223	C₄	#22576208	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	1084	2	no	—
226	C₂	#31774378	1	1	yes	characteristic
227	C₂	C₂×#22576166	1	1	yes	characteristic
228	C₂	#31774663	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₂₇₁	C₂
prime radical	#22576134	C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₇₁
hypocenter	C₂₇₁	#2992
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #31774026	C₂	semidirect
3. #31773968	C₂	semidirect
4. #31773967	C₂	semidirect
5. C₈×C₂²⋊D₂₇₁	C₂	semidirect
6. #31773957	C₂	semidirect
7. #31773954	C₂	semidirect
8. #31773922	C₂	semidirect
9. C₂×C₄⋊C₈×C₂₇₁	C₂	semidirect
114. C₂₇₁	#2992	semidirect
124. C₂×C₄⋊C₈	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774431	2
#31774026	2
#31773968	2
#31773967	2
C₈×C₂²⋊D₂₇₁	2
#31773957	2
#31773954	2
#31773922	2
C₂×C₄⋊C₈×C₂₇₁	2
#2992	271