Subgroups — C₂²×#333×C₇₁ #31838568

Order 18176.

1334 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₇₁	C₂	1	2	yes	maximal
3	C₂×#333×C₇₁	C₂	1	12	yes	maximal, direct product factor
4	C₂³×Q₈×C₇₁	C₄	1	2	yes	—
5	C₂³×C₈×C₇₁	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₇₁	—	2	6	no	—
7	C₂×C₄⋊Q₈×C₇₁	C₄	1	6	yes	—
8	C₂×C₄⋊Q₈×C₇₁	C₂²	1	6	yes	—
9	C₂²×C₄⋊C₄×C₇₁	—	2	2	no	—
10	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₇₁	—	2	6	no	—
12	C₂×C₄⋊C₈×C₇₁	C₂²	1	12	yes	—
13	#333×C₇₁	C₂²	1	16	yes	direct product factor
14	C₂²×Q₈×C₇₁	—	2	6	no	—
15	C₂²×Q₈×C₇₁	—	4	8	no	—
16	C₂²×Q₈×C₇₁	C₂×C₄	1	12	yes	—
17	C₂³×C₄×C₇₁	—	2	2	no	—
18	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₇₁	D₄	1	2	yes	—
20	C₂²×C₈×C₇₁	—	2	12	no	—
21	C₂²×C₈×C₇₁	—	4	8	no	—
22	C₄⋊Q₈×C₇₁	—	2	12	no	—
23	C₄⋊Q₈×C₇₁	—	2	12	no	—
24	C₄⋊Q₈×C₇₁	C₂×C₄	1	12	yes	—
25	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
26	C₂×C₄⋊C₄×C₇₁	—	2	12	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	12	no	—
28	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₇₁	D₄	1	6	yes	—
30	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
31	C₄⋊C₈×C₇₁	—	2	12	no	—
32	C₄⋊C₈×C₇₁	C₂³	1	8	yes	—
33	C₂⁴×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Q₈×C₇₁	—	2	12	no	—
35	C₂×Q₈×C₇₁	—	4	24	no	—
36	C₂×Q₈×C₇₁	—	4	24	no	—
37	C₂×Q₈×C₇₁	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₇₁	—	2	4	no	—
39	C₂²×C₄×C₇₁	—	2	12	no	—
40	C₂²×C₄×C₇₁	—	4	8	no	—
41	C₂²×C₄×C₇₁	C₂²⋊C₄	1	3	yes	—
42	C₂²×C₄×C₇₁	C₂²⋊C₄	1	6	yes	—
43	C₂²×C₄×C₇₁	SD₁₆	1	8	yes	—
44	C₂²×C₄×C₇₁	Q₁₆	1	8	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	6	yes	—
47	C₂×C₈×C₇₁	—	2	8	no	—
48	C₂×C₈×C₇₁	—	4	24	no	—
49	C₄⋊C₄×C₇₁	—	2	24	no	—
50	C₄⋊C₄×C₇₁	—	2	8	no	—
51	C₄²×C₇₁	C₂²⋊C₄	1	6	yes	—
52	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
53	C₄²×C₇₁	C₂×D₄	1	6	yes	—
54	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂³×C₇₁	#100	1	4	yes	—
56	C₂³×C₇₁	#103	1	8	yes	—
57	C₂³×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
58	Q₈×C₇₁	—	4	8	no	—
59	Q₈×C₇₁	—	4	24	no	—
60	C₂×C₄×C₇₁	—	2	6	no	—
61	C₂×C₄×C₇₁	—	2	6	no	—
62	C₂×C₄×C₇₁	—	2	8	no	—
63	C₂×C₄×C₇₁	—	4	24	no	—
64	C₂×C₄×C₇₁	#103	1	24	yes	—
65	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	3	yes	—
66	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	6	yes	—
67	C₂×C₄×C₇₁	C₂×SD₁₆	1	12	yes	—
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	12	yes	—
69	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
71	C₈×C₇₁	—	4	8	no	—
72	C₂²×C₇₁	#333	1	16	yes	direct product factor
73	C₂²×C₇₁	C₂×#100	1	6	yes	—
74	C₂²×C₇₁	C₂×#103	1	12	yes	—
75	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₇₁	—	2	3	no	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	4	8	no	—
79	C₄×C₇₁	C₂×#103	1	12	yes	—
80	C₄×C₇₁	C₂²×SD₁₆	1	2	yes	—
81	C₄×C₇₁	C₂²×Q₁₆	1	2	yes	—
82	C₂²×#333	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₇₁	C₂×#333	1	12	yes	direct product factor
84	C₂×C₇₁	C₂²×#100	1	1	yes	characteristic
85	C₂×C₇₁	C₂²×#103	1	2	yes	—
86	C₂²×C₄⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
87	C₂²×C₄⋊C₈	C₂×C₇₁	1	2	yes	—
88	C₂×#333	C₂×C₇₁	1	12	yes	direct product factor
89	C₇₁	C₂²×#333	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×Q₈	C₄×C₇₁	1	2	yes	—
91	C₂³×C₈	—	2	2	no	—
92	C₂×C₄⋊Q₈	—	2	6	no	—
93	C₂×C₄⋊Q₈	C₄×C₇₁	1	6	yes	—
94	C₂×C₄⋊Q₈	C₂²×C₇₁	1	6	yes	—
95	C₂²×C₄⋊C₄	—	2	2	no	—
96	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	2	6	no	—
98	C₂×C₄⋊C₈	C₂²×C₇₁	1	12	yes	—
99	#333	C₂²×C₇₁	1	16	yes	direct product factor
100	C₂²×Q₈	—	2	6	no	—
101	C₂²×Q₈	—	4	8	no	—
102	C₂²×Q₈	C₂×C₄×C₇₁	1	12	yes	—
103	C₂³×C₄	—	2	2	no	—
104	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
105	C₂³×C₄	D₄×C₇₁	1	2	yes	—
106	C₂²×C₈	—	2	12	no	—
107	C₂²×C₈	—	4	8	no	—
108	C₄⋊Q₈	—	2	12	no	—
109	C₄⋊Q₈	—	2	12	no	—
110	C₄⋊Q₈	C₂×C₄×C₇₁	1	12	yes	—
111	C₄⋊Q₈	C₂³×C₇₁	1	4	yes	—
112	C₂×C₄⋊C₄	—	2	12	no	—
113	C₂×C₄⋊C₄	—	2	12	no	—
114	C₂×C₄²	C₂×C₄×C₇₁	1	3	yes	—
115	C₂×C₄²	D₄×C₇₁	1	6	yes	—
116	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
117	C₄⋊C₈	—	2	12	no	—
118	C₄⋊C₈	C₂³×C₇₁	1	8	yes	—
119	C₂⁴	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
120	C₂×Q₈	—	2	12	no	—
121	C₂×Q₈	—	4	24	no	—
122	C₂×Q₈	—	4	24	no	—
123	C₂×Q₈	C₂²×C₄×C₇₁	1	8	yes	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	—	4	8	no	—
127	C₂²×C₄	C₂²⋊C₄×C₇₁	1	3	yes	—
128	C₂²×C₄	C₂²⋊C₄×C₇₁	1	6	yes	—
129	C₂²×C₄	SD₁₆×C₇₁	1	8	yes	—
130	C₂²×C₄	Q₁₆×C₇₁	1	8	yes	—
131	C₂²×C₄	C₂²×C₄×C₇₁	1	3	yes	—
132	C₂²×C₄	C₂×D₄×C₇₁	1	6	yes	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	—	4	24	no	—
135	C₄⋊C₄	—	2	24	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄²	C₂²⋊C₄×C₇₁	1	6	yes	—
138	C₄²	C₂²×C₄×C₇₁	1	3	yes	—
139	C₄²	C₂×D₄×C₇₁	1	6	yes	—
140	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
141	C₂³	#100×C₇₁	1	4	yes	—
142	C₂³	#103×C₇₁	1	8	yes	—
143	C₂³	C₂×C₂²⋊C₄×C₇₁	1	3	yes	—
144	Q₈	—	4	8	no	—
145	Q₈	—	4	24	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	4	24	no	—
150	C₂×C₄	#103×C₇₁	1	24	yes	—
151	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	3	yes	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	6	yes	—
153	C₂×C₄	C₂×SD₁₆×C₇₁	1	12	yes	—
154	C₂×C₄	C₂×Q₁₆×C₇₁	1	12	yes	—
155	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
157	C₈	—	4	8	no	—
158	C₂²	#333×C₇₁	1	16	yes	direct product factor
159	C₂²	C₂×#100×C₇₁	1	6	yes	—
160	C₂²	C₂×#103×C₇₁	1	12	yes	—
161	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
162	C₄	—	2	3	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	8	no	—
165	C₄	C₂×#103×C₇₁	1	12	yes	—
166	C₄	C₂²×SD₁₆×C₇₁	1	2	yes	—
167	C₄	C₂²×Q₁₆×C₇₁	1	2	yes	—
168	C₂	C₂×#333×C₇₁	1	12	yes	direct product factor
169	C₂	C₂²×#100×C₇₁	1	1	yes	characteristic
170	C₂	C₂²×#103×C₇₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇₁	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₄²	C₂⁴×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#333×C₇₁	C₂	direct (N × H)
13. #333×C₇₁	C₂²	direct (N × H)
72. C₂²×C₇₁	#333	direct (N × H)
82. C₂²×#333	C₇₁	direct (N × H)
83. C₂×C₇₁	C₂×#333	direct (N × H)
88. C₂×#333	C₂×C₇₁	direct (N × H)
89. C₇₁	C₂²×#333	direct (N × H)
99. #333	C₂²×C₇₁	direct (N × H)
158. C₂²	#333×C₇₁	direct (N × H)
168. C₂	C₂×#333×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₇₁	2
C₂²×C₄⋊C₈×C₇₁	2
C₂×#333×C₇₁	2
C₂²×#333	71

Subgroups — C22×#333×C71 #31838568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#333×C₇₁ #31838568