Subgroups — C₂×#22590124 #32150481

Order 18176.

694 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×#17986037	C₂	1	2	yes	maximal
2	#22590124	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17985827	C₂	1	2	yes	maximal, complemented
4	C₂×#17985825	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556087	—	2	2	no	—
7	#17986037	—	2	4	no	—
8	#17986037	C₂²	1	8	yes	—
9	C₂×#14556062	C₂²	1	4	yes	—
10	C₂×#14556060	—	2	2	no	—
11	#17985827	C₂²	1	8	yes	complemented
12	#17985825	C₂²	1	8	yes	complemented
13	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₁	C₄	1	8	yes	complemented
15	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₁	C₂²	1	4	yes	—
18	#14556087	—	2	8	no	—
19	#14556087	—	4	4	no	—
20	C₂×C₄⋊Dic₇₁	—	2	4	no	—
21	C₂×C₂²⋊Dic₇₁	—	4	2	no	—
22	C₂×C₄×Dic₇₁	—	4	2	no	—
23	#14556062	—	2	4	no	—
24	#14556062	C₂³	1	8	yes	—
25	#14556060	—	2	8	no	—
26	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	D₄	1	2	yes	—
30	C₂²×C₈×C₇₁	Q₈	1	4	yes	—
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
33	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₇₁	C₂×C₄	1	16	yes	complemented
36	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
37	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
38	C₂²×Dic₇₁	—	4	4	no	—
39	C₄⋊Dic₇₁	—	2	8	no	—
40	C₄⋊Dic₇₁	—	4	4	no	—
41	C₂²⋊Dic₇₁	—	4	8	no	—
42	C₄×Dic₇₁	—	4	8	no	—
43	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇₁	D₈	1	2	yes	—
46	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
59	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
60	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
61	C₂³×C₇₁	—	2	4	no	—
62	C₂³×C₇₁	#107	1	4	yes	—
63	C₂³×C₇₁	C₂×D₈	1	2	yes	—
64	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#107	1	8	yes	—
66	C₂×C₄×C₇₁	#107	1	4	yes	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
72	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₇₁	—	4	8	no	—
75	C₂×Dic₇₁	—	8	8	no	—
76	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
78	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₇₁	—	2	4	no	—
81	C₂²×C₇₁	—	2	8	no	—
82	C₂²×C₇₁	C₂×#107	1	4	yes	—
83	C₂²×C₇₁	C₂×#107	1	2	yes	—
84	C₂²×C₇₁	#508	1	4	yes	—
85	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₇₁	C₂×#107	1	4	yes	—
87	C₄×C₇₁	C₂×#107	1	2	yes	—
88	C₄×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
90	Dic₇₁	—	8	8	no	—
91	C₂×#2158	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	4	no	—
93	C₂×C₇₁	#2158	1	4	yes	complemented
94	C₂×C₇₁	C₂²×#107	1	1	yes	characteristic
95	C₂×C₇₁	C₂×#508	1	2	yes	—
96	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
97	C₂×#427	—	71	2	no	—
98	#2158	—	71	8	no	—
99	C₂×#367	—	71	2	no	—
100	C₂×#366	—	71	2	no	—
101	C₇₁	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
103	C₂×#117	—	142	2	no	—
104	C₂²×C₁₆	Dic₇₁	1	8	yes	complemented
105	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₇₁	1	4	yes	—
108	#427	—	71	8	no	—
109	#427	—	142	4	no	—
110	C₂×#107	—	71	4	no	—
111	C₂×#106	—	142	2	no	—
112	#367	—	71	8	no	—
113	#366	—	71	8	no	—
114	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₄₂	1	4	yes	—
116	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
117	C₂²×C₈	D₂₈₄	1	2	yes	—
118	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
121	#117	—	284	4	no	—
122	#117	—	142	8	no	—
123	C₂×C₄⋊C₄	—	142	4	no	—
124	C₂×C₂²⋊C₄	—	284	2	no	—
125	C₂×C₄²	—	284	2	no	—
126	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
127	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
128	C₂×C₁₆	C₂×Dic₇₁	1	16	yes	complemented
129	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
130	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
131	#107	—	71	8	no	—
132	#107	—	142	4	no	—
133	#106	—	142	8	no	—
134	C₂⁴	D₅₆₈	1	1	yes	characteristic
135	C₂²×C₄	—	284	4	no	—
136	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
137	C₂²×C₄	D₅₆₈	1	2	yes	—
138	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
139	C₂²×C₄	Dic₂₈₄	1	4	yes	—
140	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
142	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
143	C₂×C₈	C₄⋊Dic₇₁	1	8	yes	—
144	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
145	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
146	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
147	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
149	C₂×C₈	C₂×Dic₁₄₂	1	4	yes	—
150	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
151	C₄⋊C₄	—	284	4	no	—
152	C₄⋊C₄	—	142	8	no	—
153	C₂²⋊C₄	—	284	8	no	—
154	C₄²	—	284	8	no	—
155	C₁₆	C₂²×Dic₇₁	1	4	yes	—
156	C₁₆	C₂²×Dic₇₁	1	4	yes	—
157	C₂³	—	2	4	no	—
158	C₂³	#14556062	1	4	yes	—
159	C₂³	C₂×D₅₆₈	1	2	yes	—
160	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
161	C₂×C₄	—	284	8	no	—
162	C₂×C₄	—	568	8	no	—
163	C₂×C₄	#14556062	1	8	yes	—
164	C₂×C₄	#14556062	1	4	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
167	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
168	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₂₈₄	1	4	yes	—
170	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
171	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₇₁	1	2	yes	—
173	C₈	C₂×C₄⋊Dic₇₁	1	4	yes	—
174	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
176	C₂²	—	2	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂×#14556062	1	4	yes	—
179	C₂²	C₂×#14556062	1	2	yes	—
180	C₂²	#17986431	1	4	yes	—
181	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
182	C₄	—	568	8	no	—
183	C₄	C₂×#14556062	1	2	yes	—
184	C₄	C₂×#14556062	1	4	yes	—
185	C₄	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
186	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
187	C₂	—	2	4	no	—
188	C₂	#22590124	1	4	yes	direct product factor
189	C₂	C₂²×#14556062	1	1	yes	characteristic
190	C₂	C₂×#17986431	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₅₆₈
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂³×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#2158
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22590124	C₂	direct (N × H)
3. C₂×#17985827	C₂	semidirect
4. C₂×#17985825	C₂	semidirect
11. #17985827	C₂²	semidirect
12. #17985825	C₂²	semidirect
14. C₂²×C₁₆×C₇₁	C₄	semidirect
35. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
93. C₂×C₇₁	#2158	semidirect
101. C₇₁	C₂×#2158	semidirect
104. C₂²×C₁₆	Dic₇₁	semidirect
128. C₂×C₁₆	C₂×Dic₇₁	semidirect
188. C₂	#22590124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17986037	2
#22590124	2
C₂×#17985827	2
C₂×#17985825	2
C₂³×C₁₆×C₇₁	2
C₂×#2158	71

Subgroups — C2×#22590124 #32150481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590124 #32150481