Subgroups — C₂×#27160836 #38742413

Order 25216.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263285	C₂	1	2	yes	maximal
2	#27160836	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#20263285	—	2	2	no	—
5	#20263285	—	2	2	no	—
6	#20263285	C₄	1	4	yes	complemented
7	#20263285	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₉₇	—	2	2	no	—
9	C₂×C₂²⋊Dic₁₉₇	—	2	2	no	—
10	C₂×C₄×Dic₁₉₇	C₄	1	4	yes	complemented
11	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉₇	—	2	2	no	—
13	C₂×M₁₆×C₁₉₇	—	2	2	no	—
14	C₂×M₁₆×C₁₉₇	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₉₇	C₄	1	2	yes	complemented
17	C₂²×Dic₁₉₇	—	2	4	no	—
18	C₄⋊Dic₁₉₇	—	2	4	no	—
19	C₄⋊Dic₁₉₇	—	2	4	no	—
20	C₂²⋊Dic₁₉₇	—	2	4	no	—
21	C₂²⋊Dic₁₉₇	—	2	4	no	—
22	C₄×Dic₁₉₇	—	2	4	no	—
23	C₄×Dic₁₉₇	C₂×C₄	1	8	yes	complemented
24	C₂⁴×C₁₉₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉₇	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₇	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₉₇	—	2	4	no	—
36	M₁₆×C₁₉₇	—	2	4	no	—
37	C₂×C₈×C₁₉₇	—	2	2	no	—
38	C₂×C₈×C₁₉₇	—	2	2	no	—
39	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	complemented
40	C₂×Dic₁₉₇	—	2	8	no	—
41	C₂×Dic₁₉₇	—	4	8	no	—
42	C₂³×C₁₉₇	—	4	2	no	—
43	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂³×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	C₄²	1	2	yes	—
49	C₂×C₄×C₁₉₇	C₄²	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
66	C₈×C₁₉₇	—	2	4	no	—
67	C₂²×C₁₉₇	—	2	4	no	—
68	C₂²×C₁₉₇	—	2	2	no	—
69	C₂²×C₁₉₇	—	4	2	no	—
70	C₂²×C₁₉₇	—	4	1	no	—
71	C₂²×C₁₉₇	—	4	1	no	—
72	C₂²×C₁₉₇	#95	1	1	yes	characteristic
73	C₂²×C₁₉₇	#95	1	2	yes	—
74	C₂²×C₁₉₇	#95	1	1	yes	characteristic
75	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₁₉₇	#95	1	2	yes	—
78	C₄×C₁₉₇	#95	1	1	yes	characteristic
79	C₄×C₁₉₇	#95	1	1	yes	characteristic
80	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
81	C₄×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
83	Dic₁₉₇	—	4	8	no	—
84	C₂×C₁₉₇	—	2	2	no	—
85	C₂×C₁₉₇	—	2	2	no	—
86	C₂×C₁₉₇	—	2	1	no	—
87	C₂×C₁₉₇	—	2	1	no	—
88	C₂×C₁₉₇	H(3,ℤ/4)	1	2	yes	complemented
89	C₂×C₁₉₇	C₂×#95	1	1	yes	characteristic
90	C₁₉₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
91	C₂×H(3,ℤ/4)	—	197	1	no	maximal, Sylow
92	C₂²×M₁₆	D₁₉₇	1	1	yes	characteristic
93	C₂×#117	—	197	2	no	—
94	H(3,ℤ/4)	—	197	4	no	—
95	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	Dic₁₉₇	1	2	yes	complemented
99	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
100	C₂²×C₈	Dic₁₉₇	1	2	yes	complemented
101	#117	—	394	2	no	—
102	#117	—	394	2	no	—
103	#117	—	197	4	no	—
104	#117	—	197	4	no	—
105	C₂×C₄⋊C₄	—	394	2	no	—
106	C₂×C₂²⋊C₄	—	394	2	no	—
107	C₂×C₄²	—	197	4	no	—
108	C₂⁴	D₇₈₈	1	1	yes	characteristic
109	C₂²×C₄	—	394	4	no	—
110	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
111	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
112	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
116	C₂²×C₄	D₇₈₈	1	1	yes	characteristic
117	C₂²×C₄	Dic₃₉₄	1	1	yes	characteristic
118	C₂²×C₄	Dic₃₉₄	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	4	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×Dic₁₉₇	1	4	yes	complemented
125	C₄⋊C₄	—	394	4	no	—
126	C₄⋊C₄	—	394	4	no	—
127	C₂²⋊C₄	—	394	4	no	—
128	C₂²⋊C₄	—	394	4	no	—
129	C₄²	—	394	4	no	—
130	C₄²	—	197	8	no	—
131	C₂³	—	4	2	no	—
132	C₂³	#15691335	1	2	yes	—
133	C₂³	#15691335	1	2	yes	—
134	C₂³	C₄⋊Dic₁₉₇	1	1	yes	characteristic
135	C₂³	C₄⋊Dic₁₉₇	1	1	yes	characteristic
136	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
137	C₂×C₄	—	788	8	no	—
138	C₂×C₄	—	394	8	no	—
139	C₂×C₄	C₄×Dic₁₉₇	1	1	yes	characteristic
140	C₂×C₄	C₄×Dic₁₉₇	1	2	yes	—
141	C₂×C₄	C₄×Dic₁₉₇	1	1	yes	characteristic
142	C₂×C₄	#15691335	1	2	yes	—
143	C₂×C₄	#15691335	1	2	yes	—
144	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
147	C₂×C₄	#15691337	1	2	yes	—
148	C₂×C₄	#15691337	1	2	yes	—
149	C₂×C₄	#15691337	1	2	yes	—
150	C₂×C₄	#15691337	1	2	yes	—
151	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
152	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
154	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
156	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₂²	—	2	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	#20263233	1	1	yes	characteristic
164	C₂²	#20263233	1	2	yes	—
165	C₂²	#20263233	1	1	yes	characteristic
166	C₂²	C₂×#15691335	1	2	yes	—
167	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
168	C₄	—	788	8	no	—
169	C₄	#20263233	1	2	yes	—
170	C₄	#20263233	1	1	yes	characteristic
171	C₄	#20263233	1	1	yes	characteristic
172	C₄	C₂×C₄×Dic₁₉₇	1	1	yes	characteristic
173	C₄	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
174	C₄	C₂×#15691337	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#27160836	1	2	yes	direct product factor
180	C₂	C₂×#20263233	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₉₇
commutator	C₄×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂²×M₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#95
prime radical	C₂⁴×C₁₉₇	D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂²×M₁₆	D₁₉₇
hypocenter	C₁₉₇	C₂×H(3,ℤ/4)
LCS	G ▷ C₄×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #27160836	C₂	direct (N × H)
6. #20263285	C₄	semidirect
10. C₂×C₄×Dic₁₉₇	C₄	semidirect
14. C₂×M₁₆×C₁₉₇	C₄	semidirect
16. C₂²×C₈×C₁₉₇	C₄	semidirect
23. C₄×Dic₁₉₇	C₂×C₄	semidirect
39. C₂×C₈×C₁₉₇	C₂×C₄	semidirect
88. C₂×C₁₉₇	H(3,ℤ/4)	semidirect
90. C₁₉₇	C₂×H(3,ℤ/4)	semidirect
98. C₂×M₁₆	Dic₁₉₇	semidirect
100. C₂²×C₈	Dic₁₉₇	semidirect
124. C₂×C₈	C₂×Dic₁₉₇	semidirect
179. C₂	#27160836	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×#20263285	2
#27160836	2
C₂²×M₁₆×C₁₉₇	2
C₂×H(3,ℤ/4)	197

Subgroups — C2×#27160836 #38742413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27160836 #38742413