Subgroups — #2857×C₂₇₁ #46833443

Order 34688.

644 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₂₇₁	C₂	1	2	yes	maximal
3	C₂²×C₄²×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#391×C₂₇₁	C₂	1	2	yes	maximal, complemented
5	#389×C₂₇₁	C₂	1	4	yes	maximal, complemented
6	#384×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#380×C₂₇₁	C₂	1	4	yes	maximal
8	C₂²×Q₈×C₂₇₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₇₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₇₁	—	2	4	no	—
12	C₂²⋊Q₈×C₂₇₁	—	2	8	no	—
13	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₄×C₂₇₁	C₂²	1	4	yes	—
18	C₂×C₄²×C₂₇₁	—	2	4	no	—
19	C₂×C₄²×C₂₇₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
22	#95×C₂₇₁	C₂²	1	8	yes	—
23	C₂⁴×C₂₇₁	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₂₇₁	—	2	4	no	—
25	C₂×Q₈×C₂₇₁	—	2	4	no	—
26	C₂×Q₈×C₂₇₁	—	2	4	no	—
27	C₂²×C₄×C₂₇₁	—	2	2	no	—
28	C₂²×C₄×C₂₇₁	—	2	4	no	—
29	C₂²×C₄×C₂₇₁	—	2	2	no	—
30	C₂²×C₄×C₂₇₁	—	2	4	no	—
31	C₂²×C₄×C₂₇₁	D₄	1	4	yes	—
32	C₂²×C₄×C₂₇₁	Q₈	1	8	yes	complemented
33	C₂²×C₄×C₂₇₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₇₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
39	C₄⋊C₄×C₂₇₁	—	2	4	no	—
40	C₄⋊C₄×C₂₇₁	—	2	8	no	—
41	C₄⋊C₄×C₂₇₁	—	2	8	no	—
42	C₄⋊C₄×C₂₇₁	—	2	8	no	—
43	C₂²⋊C₄×C₂₇₁	—	2	8	no	—
44	C₄²×C₂₇₁	—	2	2	no	—
45	C₄²×C₂₇₁	—	2	4	no	—
46	C₄²×C₂₇₁	D₄	1	4	yes	—
47	C₂³×C₂₇₁	—	2	1	no	—
48	C₂³×C₂₇₁	—	2	1	no	—
49	C₂³×C₂₇₁	—	2	2	no	—
50	C₂³×C₂₇₁	C₂×D₄	1	2	yes	—
51	C₂³×C₂₇₁	C₂×Q₈	1	4	yes	—
52	C₂³×C₂₇₁	C₂⁴	1	1	yes	characteristic
53	Q₈×C₂₇₁	—	4	8	no	—
54	C₂×C₄×C₂₇₁	—	2	1	no	—
55	C₂×C₄×C₂₇₁	—	2	2	no	—
56	C₂×C₄×C₂₇₁	—	2	1	no	—
57	C₂×C₄×C₂₇₁	—	2	2	no	—
58	C₂×C₄×C₂₇₁	—	2	8	no	—
59	C₂×C₄×C₂₇₁	—	2	16	no	—
60	C₂×C₄×C₂₇₁	—	2	4	no	—
61	C₂×C₄×C₂₇₁	—	2	4	no	—
62	C₂×C₄×C₂₇₁	—	2	4	no	—
63	C₂×C₄×C₂₇₁	—	2	8	no	—
64	C₂×C₄×C₂₇₁	—	2	4	no	—
65	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₂₇₁	C₂×D₄	1	4	yes	—
69	C₂×C₄×C₂₇₁	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₂₇₁	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₂₇₁	#41	1	2	yes	—
72	C₂×C₄×C₂₇₁	#41	1	4	yes	—
73	C₂²×C₂₇₁	—	2	2	no	—
74	C₂²×C₂₇₁	—	2	4	no	—
75	C₂²×C₂₇₁	—	2	2	no	—
76	C₂²×C₂₇₁	—	2	4	no	—
77	C₂²×C₂₇₁	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₂₇₁	C₂²×Q₈	1	2	yes	—
81	C₂²×C₂₇₁	C₂×#41	1	1	yes	characteristic
82	C₂²×C₂₇₁	C₂×#41	1	2	yes	—
83	C₄×C₂₇₁	—	2	2	no	—
84	C₄×C₂₇₁	—	2	4	no	—
85	C₄×C₂₇₁	—	4	4	no	—
86	C₄×C₂₇₁	—	4	4	no	—
87	C₄×C₂₇₁	#121	1	4	yes	—
88	C₄×C₂₇₁	C₂²⋊Q₈	1	8	yes	complemented
89	C₂×C₂₇₁	—	2	4	no	—
90	C₂×C₂₇₁	C₂×#121	1	1	yes	characteristic
91	C₂×C₂₇₁	C₂×C₂²⋊Q₈	1	2	yes	—
92	C₂×C₂₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₂₇₁	#532	1	1	yes	characteristic
94	C₂×C₂₇₁	#533	1	2	yes	—
95	C₂₇₁	#2857	1	1	yes	characteristic, direct product factor, Sylow
96	#2857	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₄⋊Q₈	C₂×C₂₇₁	1	1	yes	characteristic, complemented
98	C₂×C₂²⋊Q₈	C₂×C₂₇₁	1	2	yes	—
99	C₂²×C₄²	C₂×C₂₇₁	1	1	yes	characteristic
100	#391	C₂×C₂₇₁	1	2	yes	complemented
101	#389	C₂×C₂₇₁	1	4	yes	complemented
102	#384	C₂×C₂₇₁	1	1	yes	characteristic, complemented
103	#380	C₂×C₂₇₁	1	4	yes	—
104	C₂²×Q₈	C₂²×C₂₇₁	1	2	yes	—
105	C₂³×C₄	C₂²×C₂₇₁	1	1	yes	characteristic
106	C₂³×C₄	C₂²×C₂₇₁	1	2	yes	—
107	C₄⋊Q₈	—	2	4	no	—
108	C₂²⋊Q₈	—	2	8	no	—
109	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	4	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	4	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	4	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₂₇₁	1	4	yes	—
114	C₂×C₄²	—	2	4	no	—
115	C₂×C₄²	C₂²×C₂₇₁	1	2	yes	—
116	C₂×C₄²	C₂²×C₂₇₁	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₂₇₁	1	1	yes	characteristic
118	#95	C₂²×C₂₇₁	1	8	yes	—
119	C₂⁴	C₂³×C₂₇₁	1	1	yes	characteristic
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	D₄×C₂₇₁	1	4	yes	—
128	C₂²×C₄	Q₈×C₂₇₁	1	8	yes	complemented
129	C₂²×C₄	C₂³×C₂₇₁	1	4	yes	—
130	C₂²×C₄	C₂³×C₂₇₁	1	4	yes	—
131	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₂₇₁	1	2	yes	—
133	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₂₇₁	1	2	yes	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	2	8	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₂²⋊C₄	—	2	8	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	D₄×C₂₇₁	1	4	yes	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂×D₄×C₂₇₁	1	2	yes	—
147	C₂³	C₂×Q₈×C₂₇₁	1	4	yes	—
148	C₂³	C₂⁴×C₂₇₁	1	1	yes	characteristic
149	Q₈	—	4	8	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
163	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
164	C₂×C₄	C₂×D₄×C₂₇₁	1	4	yes	—
165	C₂×C₄	C₂×Q₈×C₂₇₁	1	4	yes	—
166	C₂×C₄	C₂×Q₈×C₂₇₁	1	4	yes	—
167	C₂×C₄	#41×C₂₇₁	1	2	yes	—
168	C₂×C₄	#41×C₂₇₁	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	C₄⋊Q₈×C₂₇₁	1	4	yes	complemented
174	C₂²	C₂²×D₄×C₂₇₁	1	1	yes	characteristic
175	C₂²	C₂²×D₄×C₂₇₁	1	1	yes	characteristic
176	C₂²	C₂²×Q₈×C₂₇₁	1	2	yes	—
177	C₂²	C₂×#41×C₂₇₁	1	1	yes	characteristic
178	C₂²	C₂×#41×C₂₇₁	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	4	4	no	—
182	C₄	—	4	4	no	—
183	C₄	#121×C₂₇₁	1	4	yes	—
184	C₄	C₂²⋊Q₈×C₂₇₁	1	8	yes	complemented
185	C₂	—	2	4	no	—
186	C₂	C₂×#121×C₂₇₁	1	1	yes	characteristic
187	C₂	C₂×C₂²⋊Q₈×C₂₇₁	1	2	yes	—
188	C₂	C₂×C₄⋊Q₈×C₂₇₁	1	1	yes	characteristic
189	C₂	#532×C₂₇₁	1	1	yes	characteristic
190	C₂	#533×C₂₇₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₇₁
prime radical	C₂⁴×C₂₇₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₂₇₁	C₂	semidirect
4. #391×C₂₇₁	C₂	semidirect
5. #389×C₂₇₁	C₂	semidirect
6. #384×C₂₇₁	C₂	semidirect
32. C₂²×C₄×C₂₇₁	Q₈	semidirect
77. C₂²×C₂₇₁	C₄⋊Q₈	semidirect
88. C₄×C₂₇₁	C₂²⋊Q₈	semidirect
95. C₂₇₁	#2857	direct (N × H)
96. #2857	C₂₇₁	direct (N × H)
97. C₂×C₄⋊Q₈	C₂×C₂₇₁	semidirect
100. #391	C₂×C₂₇₁	semidirect
101. #389	C₂×C₂₇₁	semidirect
102. #384	C₂×C₂₇₁	semidirect
128. C₂²×C₄	Q₈×C₂₇₁	semidirect
173. C₂²	C₄⋊Q₈×C₂₇₁	semidirect
184. C₄	C₂²⋊Q₈×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₂₇₁	2
C₂×C₂²⋊Q₈×C₂₇₁	2
C₂²×C₄²×C₂₇₁	2
#391×C₂₇₁	2
#389×C₂₇₁	2
#384×C₂₇₁	2
#380×C₂₇₁	2
#2857	271

Subgroups — #2857×C271 #46833443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2857×C₂₇₁ #46833443