Subgroups — C₂×#20260784 #27150343

Order 12416.

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×#13095495	C₂	1	2	yes	maximal
2	#20260784	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690746	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	—	2	2	no	—
6	C₂×#13095495	—	2	4	no	—
7	C₂×#13095495	C₂²	1	4	yes	—
8	C₂×#13095495	C₂²	1	8	yes	—
9	C₂²×#11715895	—	2	2	no	—
10	#15690746	C₂²	1	16	yes	complemented
11	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₉₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₉₇	C₄	1	4	yes	—
14	C₂×M₁₆×C₉₇	C₂²	1	4	yes	—
15	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
16	#13095495	—	2	8	no	—
17	#13095495	—	2	4	no	—
18	#13095495	C₂³	1	8	yes	—
19	C₂×#11715895	—	2	8	no	—
20	C₂×#11715895	—	2	4	no	—
21	C₂⁴×C₉₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
30	M₁₆×C₉₇	C₂×C₄	1	16	yes	—
31	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₉₇	C₂³	1	4	yes	—
34	C₂³×C₉₇	—	4	2	no	—
35	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₉₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
52	C₈×C₉₇	C₂²×C₄	1	8	yes	—
53	#11715895	—	2	8	no	—
54	C₂²×C₉₇	—	2	4	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	—	4	2	no	—
57	C₂²×C₉₇	—	4	2	no	—
58	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
60	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₉₇	—	2	4	no	—
68	C₂×C₉₇	—	2	1	no	—
69	C₂×C₉₇	—	2	1	no	—
70	C₂×C₉₇	#430	1	2	yes	complemented
71	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂×#430	—	97	1	no	maximal, Sylow
73	C₉₇	C₂×#430	1	1	yes	characteristic, complemented, Sylow
74	C₂²×M₁₆	—	97	2	no	—
75	C₂²×M₁₆	D₉₇	1	1	yes	characteristic
76	#430	—	97	8	no	—
77	C₂×#108	—	97	4	no	—
78	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
79	C₂×M₁₆	—	97	4	no	—
80	C₂×M₁₆	—	97	8	no	—
81	C₂×M₁₆	—	194	4	no	—
82	C₂×M₁₆	—	194	2	no	—
83	C₂×M₁₆	Dic₉₇	1	4	yes	—
84	C₂×M₁₆	Dic₉₇	1	4	yes	—
85	C₂×M₁₆	C₂×D₉₇	1	4	yes	—
86	C₂²×C₈	—	194	2	no	—
87	C₂²×C₈	C₂×D₉₇	1	2	yes	—
88	#108	—	97	16	no	—
89	C₂⁴	Dic₁₉₄	1	1	yes	characteristic
90	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
91	C₂²×C₄	Dic₁₉₄	1	2	yes	—
92	C₂²×C₄	D₃₈₈	1	2	yes	—
93	C₂²×C₄	D₃₈₈	1	2	yes	—
94	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
95	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
98	M₁₆	—	97	8	no	—
99	M₁₆	—	194	8	no	—
100	M₁₆	—	194	4	no	—
101	M₁₆	C₂×Dic₉₇	1	16	yes	—
102	C₂×C₈	—	194	8	no	—
103	C₂×C₈	—	194	4	no	—
104	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
105	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
106	C₂×C₈	C₂²×D₉₇	1	4	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	C₄⋊Dic₉₇	1	2	yes	—
109	C₂³	C₄⋊Dic₉₇	1	2	yes	—
110	C₂³	C₂×Dic₁₉₄	1	1	yes	characteristic
111	C₂³	C₂×Dic₁₉₄	1	2	yes	—
112	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
113	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
114	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
115	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
116	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
119	C₂×C₄	C₂×D₃₈₈	1	4	yes	—
120	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
122	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
123	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
125	C₈	—	194	8	no	—
126	C₈	C₂²×Dic₉₇	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⋊Dic₉₇	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₉₇	1	4	yes	—
133	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
134	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
135	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
136	C₄	C₂×C₄⋊Dic₉₇	1	4	yes	—
137	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
138	C₄	C₂³×Dic₉₇	1	1	yes	characteristic
139	C₄	C₂²×D₃₈₈	1	1	yes	characteristic
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#20260784	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂²×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₉₇	Q₈
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂²×M₁₆	D₉₇
hypocenter	C₉₇	C₂×#430
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
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. #20260784	C₂	direct (N × H)
3. C₂×#15690746	C₂	semidirect
10. #15690746	C₂²	semidirect
70. C₂×C₉₇	#430	semidirect
73. C₉₇	C₂×#430	semidirect
143. C₂	#20260784	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×#13095495	2
#20260784	2
C₂×#15690746	2
C₂²×M₁₆×C₉₇	2
C₂×#430	97

Subgroups — C2×#20260784 #27150343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20260784 #27150343