Subgroups — C₂×#22588492 #31997831

Order 18176.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₇₁⋊C₈)	C₂	1	6	yes	maximal
2	#22588492	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₇₁⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	24	no	—
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	24	yes	—
7	C₂²×C₄²×C₇₁	C₄	1	3	yes	—
8	C₂²×C₄²×C₇₁	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₇₁	C₂²	1	3	yes	—
10	C₄³×C₇₁	C₄	1	4	yes	—
11	C₄³×C₇₁	C₂²	1	4	yes	—
12	C₂²×C₇₁⋊C₈	—	2	24	no	—
13	C₂²×C₇₁⋊C₈	C₂³	1	8	yes	—
14	C₄⋊(C₇₁⋊C₈)	—	2	48	no	—
15	C₂³×C₄×C₇₁	C₂×C₄	1	3	yes	—
16	C₂³×C₄×C₇₁	C₂×C₄	1	3	yes	—
17	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₇₁	—	2	12	no	—
19	C₂×C₄²×C₇₁	C₈	1	8	yes	complemented
20	C₂×C₄²×C₇₁	C₂×C₄	1	12	yes	—
21	C₂×C₄²×C₇₁	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇₁	D₄	1	12	yes	—
24	C₂×C₄²×C₇₁	Q₈	1	12	yes	—
25	C₂×C₄²×C₇₁	C₂³	1	6	yes	—
26	C₂×C₇₁⋊C₈	—	2	24	no	—
27	C₂×C₇₁⋊C₈	—	4	16	no	—
28	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	—	2	12	no	—
30	C₂²×C₄×C₇₁	C₄⋊C₄	1	12	yes	—
31	C₂²×C₄×C₇₁	C₄⋊C₄	1	12	yes	—
32	C₂²×C₄×C₇₁	C₂×C₈	1	12	yes	—
33	C₂²×C₄×C₇₁	M₁₆	1	12	yes	—
34	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₇₁	C₂×D₄	1	6	yes	—
38	C₂²×C₄×C₇₁	C₂×Q₈	1	6	yes	—
39	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
40	C₄²×C₇₁	—	2	24	no	—
41	C₄²×C₇₁	C₄⋊C₄	1	24	yes	—
42	C₄²×C₇₁	C₂×C₈	1	16	yes	complemented
43	C₄²×C₇₁	C₂×D₄	1	12	yes	—
44	C₄²×C₇₁	C₂×Q₈	1	12	yes	—
45	C₂³×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₂³×C₇₁	C₂²×C₈	1	2	yes	—
47	C₂³×C₇₁	C₂×M₁₆	1	6	yes	—
48	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	—	2	12	no	—
50	C₂×C₄×C₇₁	—	2	24	no	—
51	C₂×C₄×C₇₁	C₄⋊C₈	1	48	yes	complemented
52	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×C₄×C₇₁	#127	1	4	yes	—
56	C₂×C₄×C₇₁	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₇₁	C₂²×C₈	1	12	yes	—
58	C₂×C₄×C₇₁	C₂×M₁₆	1	12	yes	—
59	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₇₁	C₂²×Q₈	1	3	yes	—
61	C₇₁⋊C₈	—	4	8	no	—
62	C₂²×C₇₁	#384	1	4	yes	—
63	C₂²×C₇₁	C₂×C₄⋊C₈	1	12	yes	—
64	C₂²×C₇₁	#423	1	12	yes	—
65	C₂²×C₇₁	C₂²×C₄⋊C₄	1	3	yes	—
66	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
67	C₂²×C₇₁	C₂²×M₁₆	1	3	yes	—
68	C₄×C₇₁	—	2	12	no	—
69	C₄×C₇₁	#384	1	4	yes	—
70	C₄×C₇₁	C₂×C₄⋊C₈	1	24	yes	complemented
71	C₄×C₇₁	C₂×#127	1	1	yes	characteristic
72	C₄×C₇₁	C₂×C₄⋊Q₈	1	3	yes	—
73	C₂×#1842	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	#1842	1	8	yes	complemented
75	C₂×C₇₁	C₂×#384	1	1	yes	characteristic
76	C₂×C₇₁	C₂²×C₄⋊C₈	1	3	yes	—
77	C₂×C₇₁	C₂×#423	1	3	yes	—
78	C₂²×C₄⋊C₈	—	71	6	no	—
79	C₂×C₄³	D₇₁	1	1	yes	characteristic
80	#1842	—	71	8	no	—
81	C₇₁	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	71	4	no	—
83	C₂²×C₄²	Dic₇₁	1	3	yes	—
84	C₂²×C₄²	Dic₇₁	1	1	yes	characteristic
85	C₂²×C₄²	C₂×D₇₁	1	3	yes	—
86	C₂×C₄⋊C₈	—	71	24	no	—
87	C₂×C₄⋊C₈	—	142	24	no	—
88	C₄³	Dic₇₁	1	4	yes	—
89	C₄³	C₂×D₇₁	1	4	yes	—
90	C₂³×C₄	C₂×Dic₇₁	1	3	yes	—
91	C₂³×C₄	C₂×Dic₇₁	1	3	yes	—
92	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
93	C₂²×C₈	—	71	8	no	—
94	C₂²×C₈	—	142	24	no	—
95	C₂×C₄²	—	2	12	no	—
96	C₂×C₄²	C₇₁⋊C₈	1	8	yes	complemented
97	C₂×C₄²	Dic₁₄₂	1	12	yes	—
98	C₂×C₄²	D₂₈₄	1	12	yes	—
99	C₂×C₄²	C₂×Dic₇₁	1	12	yes	—
100	C₂×C₄²	C₂×Dic₇₁	1	6	yes	—
101	C₂×C₄²	C₂×Dic₇₁	1	4	yes	—
102	C₂×C₄²	C₂²×D₇₁	1	6	yes	—
103	C₄⋊C₈	—	142	48	no	—
104	C₂⁴	C₂²×Dic₇₁	1	1	yes	characteristic
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	C₄⋊Dic₇₁	1	12	yes	—
107	C₂²×C₄	C₄⋊Dic₇₁	1	12	yes	—
108	C₂²×C₄	C₂×C₇₁⋊C₈	1	12	yes	—
109	C₂²×C₄	#11765849	1	12	yes	—
110	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
111	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
112	C₂²×C₄	C₂²×Dic₇₁	1	6	yes	—
113	C₂²×C₄	C₂×D₂₈₄	1	6	yes	—
114	C₂²×C₄	C₂×Dic₁₄₂	1	6	yes	—
115	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
116	C₂×C₈	—	284	16	no	—
117	C₂×C₈	—	142	24	no	—
118	C₄²	—	2	24	no	—
119	C₄²	C₄⋊Dic₇₁	1	24	yes	—
120	C₄²	C₂×C₇₁⋊C₈	1	16	yes	complemented
121	C₄²	C₂×D₂₈₄	1	12	yes	—
122	C₄²	C₂×Dic₁₄₂	1	12	yes	—
123	C₂³	C₂×C₄⋊Dic₇₁	1	6	yes	—
124	C₂³	C₂²×C₇₁⋊C₈	1	2	yes	—
125	C₂³	C₂×#11765849	1	6	yes	—
126	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	24	no	—
129	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	48	yes	complemented
130	C₂×C₄	C₂×C₄⋊Dic₇₁	1	12	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₇₁	1	12	yes	—
133	C₂×C₄	#14556125	1	4	yes	—
134	C₂×C₄	#14556128	1	12	yes	—
135	C₂×C₄	C₂²×C₇₁⋊C₈	1	12	yes	—
136	C₂×C₄	C₂×#11765849	1	12	yes	—
137	C₂×C₄	C₂²×D₂₈₄	1	3	yes	—
138	C₂×C₄	C₂²×Dic₁₄₂	1	3	yes	—
139	C₈	—	284	8	no	—
140	C₂²	#17985869	1	4	yes	—
141	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	12	yes	—
142	C₂²	#17986023	1	12	yes	—
143	C₂²	C₂²×C₄⋊Dic₇₁	1	3	yes	—
144	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
145	C₂²	C₂²×#11765849	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	#17985869	1	4	yes	—
148	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	24	yes	complemented
149	C₄	C₂×#14556125	1	1	yes	characteristic
150	C₄	C₂×#14556128	1	3	yes	—
151	C₂	#22588492	1	8	yes	direct product factor
152	C₂	C₂×#17985869	1	1	yes	characteristic
153	C₂	C₂²×C₄⋊(C₇₁⋊C₈)	1	3	yes	—
154	C₂	C₂×#17986023	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×C₄³×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂²×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₄³	D₇₁
hypocenter	C₇₁	C₂×#1842
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #22588492	C₂	direct (N × H)
19. C₂×C₄²×C₇₁	C₈	semidirect
42. C₄²×C₇₁	C₂×C₈	semidirect
51. C₂×C₄×C₇₁	C₄⋊C₈	semidirect
70. C₄×C₇₁	C₂×C₄⋊C₈	semidirect
74. C₂×C₇₁	#1842	semidirect
81. C₇₁	C₂×#1842	semidirect
96. C₂×C₄²	C₇₁⋊C₈	semidirect
120. C₄²	C₂×C₇₁⋊C₈	semidirect
129. C₂×C₄	C₄⋊(C₇₁⋊C₈)	semidirect
148. C₄	C₂×C₄⋊(C₇₁⋊C₈)	semidirect
151. C₂	#22588492	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⋊(C₇₁⋊C₈)	2
#22588492	2
C₂×C₄³×C₇₁	2
C₂×#1842	71

Subgroups — C2×#22588492 #31997831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22588492 #31997831