Subgroups — C₂×#27161128 #38751319

Order 25216.

460 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691342	C₂	1	2	yes	maximal
2	#27161128	C₂	1	8	yes	maximal, direct product factor
3	C₂×#20263262	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	—	2	2	no	—
6	C₂×#15691342	—	2	4	no	—
7	C₂×#15691342	C₂²	1	4	yes	—
8	C₂×#15691342	C₂²	1	8	yes	—
9	C₂²×#13096802	—	2	2	no	—
10	#20263262	C₂²	1	16	yes	complemented
11	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₉₇	C₄	1	4	yes	—
14	C₂×M₁₆×C₁₉₇	C₂²	1	4	yes	—
15	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
16	#15691342	—	2	8	no	—
17	#15691342	—	2	4	no	—
18	#15691342	C₂³	1	8	yes	—
19	C₂×#13096802	—	2	8	no	—
20	C₂×#13096802	—	2	4	no	—
21	C₂⁴×C₁₉₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉₇	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₇	C₂³	1	2	yes	—
30	M₁₆×C₁₉₇	C₂×C₄	1	16	yes	—
31	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₁₉₇	C₂³	1	4	yes	—
34	#13096802	—	2	8	no	—
35	C₂³×C₁₉₇	—	4	2	no	—
36	C₂³×C₁₉₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₉₇	C₄⋊C₄	1	2	yes	—
38	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂³×C₁₉₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
53	C₈×C₁₉₇	C₂²×C₄	1	8	yes	—
54	C₂²×C₁₉₇	—	2	4	no	—
55	C₂²×C₁₉₇	—	2	2	no	—
56	C₂²×C₁₉₇	—	4	2	no	—
57	C₂²×C₁₉₇	—	4	2	no	—
58	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₁₉₇	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₁₉₇	—	2	4	no	—
68	C₂×C₁₉₇	—	2	1	no	—
69	C₂×C₁₉₇	—	2	1	no	—
70	C₂×C₁₉₇	#430	1	2	yes	complemented
71	C₂×C₁₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₁₉₇	C₂×#430	1	1	yes	characteristic, complemented, Sylow
73	C₂×#430	—	197	1	no	maximal, Sylow
74	C₂²×M₁₆	—	197	2	no	—
75	C₂²×M₁₆	D₁₉₇	1	1	yes	characteristic
76	#430	—	197	8	no	—
77	C₂×#108	—	197	4	no	—
78	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
79	C₂×M₁₆	—	394	4	no	—
80	C₂×M₁₆	—	394	2	no	—
81	C₂×M₁₆	—	197	4	no	—
82	C₂×M₁₆	—	197	8	no	—
83	C₂×M₁₆	Dic₁₉₇	1	4	yes	—
84	C₂×M₁₆	Dic₁₉₇	1	4	yes	—
85	C₂×M₁₆	C₂×D₁₉₇	1	4	yes	—
86	C₂²×C₈	—	394	2	no	—
87	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
88	#108	—	197	16	no	—
89	C₂⁴	Dic₃₉₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
92	C₂²×C₄	D₇₈₈	1	2	yes	—
93	C₂²×C₄	D₇₈₈	1	2	yes	—
94	C₂²×C₄	Dic₃₉₄	1	1	yes	characteristic
95	C₂²×C₄	Dic₃₉₄	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₉₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
98	M₁₆	—	394	8	no	—
99	M₁₆	—	394	4	no	—
100	M₁₆	—	197	8	no	—
101	M₁₆	C₂×Dic₁₉₇	1	16	yes	—
102	C₂×C₈	—	394	8	no	—
103	C₂×C₈	—	394	4	no	—
104	C₂×C₈	C₂×Dic₁₉₇	1	4	yes	—
105	C₂×C₈	C₂×Dic₁₉₇	1	4	yes	—
106	C₂×C₈	C₂²×D₁₉₇	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₁₉₇	1	2	yes	—
109	C₂³	C₄⋊Dic₁₉₇	1	2	yes	—
110	C₂³	C₂×Dic₃₉₄	1	1	yes	characteristic
111	C₂³	C₂×Dic₃₉₄	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₁₉₇	1	4	yes	—
113	C₂×C₄	C₄⋊Dic₁₉₇	1	2	yes	—
114	C₂×C₄	C₄⋊Dic₁₉₇	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₁₉₇	1	4	yes	—
116	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×Dic₁₉₇	1	4	yes	—
119	C₂×C₄	C₂×D₇₈₈	1	4	yes	—
120	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
122	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
123	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
125	C₈	—	394	8	no	—
126	C₈	C₂²×Dic₁₉₇	1	8	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₁₉₇	1	4	yes	—
133	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
134	C₂²	C₂²×Dic₃₉₄	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
136	C₄	C₂×C₄⋊Dic₁₉₇	1	4	yes	—
137	C₄	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
138	C₄	C₂³×Dic₁₉₇	1	1	yes	characteristic
139	C₄	C₂²×D₇₈₈	1	1	yes	characteristic
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#27161128	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₁₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₈₈
commutator	C₄×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂²×M₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉₇	Q₈
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₉₇
hypocenter	C₁₉₇	C₂×#430
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #27161128	C₂	direct (N × H)
3. C₂×#20263262	C₂	semidirect
10. #20263262	C₂²	semidirect
70. C₂×C₁₉₇	#430	semidirect
72. C₁₉₇	C₂×#430	semidirect
143. C₂	#27161128	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×#15691342	2
#27161128	2
C₂×#20263262	2
C₂²×M₁₆×C₁₉₇	2
C₂×#430	197

Subgroups — C2×#27161128 #38751319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161128 #38751319