Subgroups — C₂×#20260498 #27141450

Order 12416.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₉₇	C₂	1	2	yes	maximal
2	#20260498	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₉₇	—	2	2	no	—
5	C₂×C₄⋊Dic₉₇	—	2	2	no	—
6	C₂×C₄⋊Dic₉₇	—	2	4	no	—
7	C₂×C₄⋊Dic₉₇	C₄	1	8	yes	complemented
8	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
9	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉₇	—	2	4	no	—
11	C₂²×C₈×C₉₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
13	C₂²×Dic₉₇	—	2	8	no	—
14	C₂²×Dic₉₇	—	2	4	no	—
15	C₂²×Dic₉₇	—	4	4	no	—
16	C₂²×Dic₉₇	—	4	4	no	—
17	C₄⋊Dic₉₇	—	2	8	no	—
18	C₄⋊Dic₉₇	—	2	4	no	—
19	C₄⋊Dic₉₇	C₂×C₄	1	8	yes	complemented
20	C₂⁴×C₉₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₉₇	D₄	1	4	yes	—
25	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₉₇	—	2	8	no	—
28	C₂×C₈×C₉₇	—	2	4	no	—
29	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	complemented
30	C₂³×C₉₇	C₂²⋊C₄	1	4	yes	—
31	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
32	C₂³×C₉₇	D₈	1	2	yes	—
33	C₂³×C₉₇	SD₁₆	1	4	yes	—
34	C₂³×C₉₇	Q₁₆	1	2	yes	—
35	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₄²	1	4	yes	—
37	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
39	C₂×C₄×C₉₇	C₄⋊C₄	1	8	yes	—
40	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₉₇	—	2	8	no	—
46	C₂×Dic₉₇	—	4	16	no	—
47	C₂×Dic₉₇	—	4	4	no	—
48	C₂×Dic₉₇	—	4	4	no	—
49	C₈×C₉₇	—	2	4	no	—
50	C₂²×C₉₇	#95	1	4	yes	—
51	C₂²×C₉₇	#102	1	8	yes	—
52	C₂²×C₉₇	#103	1	8	yes	—
53	C₂²×C₉₇	#106	1	4	yes	—
54	C₂²×C₉₇	#107	1	4	yes	—
55	C₂²×C₉₇	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
59	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₉₇	#95	1	4	yes	—
61	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
64	Dic₉₇	—	4	8	no	—
65	C₂×C₉₇	#340	1	8	yes	complemented
66	C₂×C₉₇	C₂×#95	1	1	yes	characteristic
67	C₂×C₉₇	C₂×#102	1	2	yes	—
68	C₂×C₉₇	C₂×#103	1	2	yes	—
69	C₂×C₉₇	C₂×#106	1	1	yes	characteristic
70	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
71	C₂×#340	—	97	1	no	maximal, Sylow
72	C₉₇	C₂×#340	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₈	D₉₇	1	1	yes	characteristic
74	C₂²×C₄⋊C₄	—	97	2	no	—
75	#340	—	97	4	no	—
76	C₂³×C₄	—	194	2	no	—
77	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	Dic₉₇	1	4	yes	complemented
80	C₂²×C₈	C₂×D₉₇	1	2	yes	—
81	C₂×C₄⋊C₄	—	97	8	no	—
82	C₂×C₄⋊C₄	—	97	4	no	—
83	C₂×C₄⋊C₄	—	194	2	no	—
84	C₂×C₄⋊C₄	—	194	4	no	—
85	C₂⁴	D₃₈₈	1	1	yes	characteristic
86	C₂²×C₄	—	388	4	no	—
87	C₂²×C₄	—	388	4	no	—
88	C₂²×C₄	—	194	8	no	—
89	C₂²×C₄	—	194	4	no	—
90	C₂²×C₄	Dic₁₉₄	1	2	yes	—
91	C₂²×C₄	C₄×D₉₇	1	4	yes	—
92	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₉₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×Dic₉₇	1	4	yes	complemented
99	C₄⋊C₄	—	97	8	no	—
100	C₄⋊C₄	—	194	8	no	—
101	C₄⋊C₄	—	194	4	no	—
102	C₂³	#13095488	1	4	yes	—
103	C₂³	C₄⋊Dic₉₇	1	2	yes	—
104	C₂³	D₇₇₆	1	2	yes	—
105	C₂³	#13095500	1	4	yes	—
106	C₂³	Dic₃₈₈	1	2	yes	—
107	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
108	C₂×C₄	—	388	4	no	—
109	C₂×C₄	—	388	4	no	—
110	C₂×C₄	—	388	16	no	—
111	C₂×C₄	—	194	8	no	—
112	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
113	C₂×C₄	#13095488	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
115	C₂×C₄	#13095490	1	8	yes	—
116	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
118	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
120	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	#15690717	1	4	yes	—
123	C₂²	#15690732	1	8	yes	—
124	C₂²	#15690735	1	8	yes	—
125	C₂²	#15690742	1	4	yes	—
126	C₂²	#15690744	1	4	yes	—
127	C₂²	C₂×#13095488	1	2	yes	—
128	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
129	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
130	C₂²	C₂×#13095500	1	2	yes	—
131	C₂²	C₂×Dic₃₈₈	1	1	yes	characteristic
132	C₄	—	388	8	no	—
133	C₄	#15690717	1	4	yes	—
134	C₄	C₂×C₄×Dic₉₇	1	1	yes	characteristic
135	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
136	C₄	C₂×#13095490	1	2	yes	—
137	C₂	#20260498	1	8	yes	direct product factor
138	C₂	C₂×#15690717	1	1	yes	characteristic
139	C₂	C₂×#15690732	1	2	yes	—
140	C₂	C₂×#15690735	1	2	yes	—
141	C₂	C₂×#15690742	1	1	yes	characteristic
142	C₂	C₂×#15690744	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃₈₈
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂³×C₈×C₉₇	C₂
socle, prime radical	C₂⁴×C₉₇	D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂³×C₈	D₉₇
hypocenter	C₉₇	C₂×#340
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #20260498	C₂	direct (N × H)
7. C₂×C₄⋊Dic₉₇	C₄	semidirect
11. C₂²×C₈×C₉₇	C₄	semidirect
19. C₄⋊Dic₉₇	C₂×C₄	semidirect
29. C₂×C₈×C₉₇	C₂×C₄	semidirect
65. C₂×C₉₇	#340	semidirect
72. C₉₇	C₂×#340	semidirect
79. C₂²×C₈	Dic₉₇	semidirect
98. C₂×C₈	C₂×Dic₉₇	semidirect
137. C₂	#20260498	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⋊Dic₉₇	2
#20260498	2
C₂³×C₈×C₉₇	2
C₂×#340	97

Subgroups — C2×#20260498 #27141450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20260498 #27141450