Subgroups — #27155141

Order 12416.

996 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261366	C₂	1	4	yes	maximal, complemented
2	#20261363	C₂	1	8	yes	maximal, complemented
3	#20261361	C₂	1	4	yes	maximal
4	#20261292	C₂	1	4	yes	maximal, complemented
5	#20261288	C₂	1	8	yes	maximal, complemented
6	C₂×#15690769	C₂	1	2	yes	maximal
7	C₂×#117×C₉₇	C₂	1	1	yes	maximal, characteristic
8	#15690810	C₂²	1	8	yes	complemented
9	#15690809	C₂²	1	16	yes	complemented
10	#15690802	C₂²	1	8	yes	complemented
11	#15690801	C₂²	1	16	yes	complemented
12	#15690789	C₂²	1	16	yes	—
13	#15690787	C₂²	1	32	yes	—
14	C₄×Dic₁₉₄	C₂²	1	16	yes	complemented
15	#15690769	C₂²	1	4	yes	—
16	#15690769	C₂²	1	4	yes	—
17	#15690769	C₂²	1	4	yes	—
18	#15690769	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊Dic₉₇	C₂²	1	4	yes	—
21	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
22	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
23	#117×C₉₇	C₂²	1	4	yes	—
24	#117×C₉₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
28	C₂×Dic₁₉₄	C₂³	1	16	yes	—
29	C₂²×Dic₉₇	C₂³	1	8	yes	—
30	C₄⋊Dic₉₇	C₂³	1	8	yes	—
31	C₄⋊Dic₉₇	C₂³	1	8	yes	—
32	C₄⋊Dic₉₇	C₂³	1	16	yes	—
33	#13095490	C₂³	1	32	yes	—
34	C₂²⋊Dic₉₇	C₂³	1	8	yes	—
35	C₂²⋊Dic₉₇	C₂³	1	8	yes	—
36	C₄×Dic₉₇	C₂³	1	8	yes	—
37	C₄×Dic₉₇	C₂³	1	8	yes	—
38	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₉₇	Q₈	1	8	yes	complemented
40	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
41	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₉₇	C₂³	1	8	yes	—
46	C₂²⋊C₄×C₉₇	C₂³	1	8	yes	—
47	C₄²×C₉₇	C₂³	1	4	yes	—
48	C₄²×C₉₇	C₂³	1	4	yes	—
49	C₂³×C₉₇	—	4	2	no	—
50	C₂³×C₉₇	C₂×Q₈	1	4	yes	—
51	C₂³×C₉₇	C₂⁴	1	2	yes	—
52	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₉₇	—	4	4	no	—
54	C₂×C₄×C₉₇	C₂×Q₈	1	8	yes	—
55	C₂×C₄×C₉₇	C₂×Q₈	1	8	yes	—
56	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₉₇	C₂⁴	1	8	yes	—
59	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
61	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₉₇	—	4	8	no	—
63	C₂×Dic₉₇	C₂⁴	1	16	yes	—
64	Dic₁₉₄	—	4	16	no	—
65	C₂²×C₉₇	—	2	4	no	—
66	C₂²×C₉₇	—	2	2	no	—
67	C₂²×C₉₇	—	4	4	no	—
68	C₂²×C₉₇	C₂²×Q₈	1	4	yes	—
69	C₂²×C₉₇	C₂²×Q₈	1	2	yes	—
70	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
71	C₄×C₉₇	—	2	8	no	—
72	C₄×C₉₇	C₂²×Q₈	1	2	yes	—
73	C₄×C₉₇	C₂²×Q₈	1	4	yes	—
74	C₄×C₉₇	C₂²×Q₈	1	2	yes	—
75	Dic₉₇	—	2	16	no	—
76	C₂×C₉₇	—	2	4	no	—
77	C₂×C₉₇	—	2	2	no	—
78	C₂×C₉₇	C₂³×Q₈	1	1	yes	characteristic
79	C₂×C₉₇	#585	1	2	yes	—
80	#3458	—	97	1	no	maximal, Sylow
81	C₉₇	#3458	1	1	yes	characteristic, complemented, Sylow
82	#544	—	97	8	no	—
83	#544	—	97	4	no	—
84	#543	—	97	4	no	—
85	#533	—	97	8	no	—
86	#533	—	97	4	no	—
87	C₂×#117	—	97	2	no	—
88	C₂×#117	D₉₇	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
90	C₄⋊Q₈	—	97	16	no	—
91	C₄⋊Q₈	—	97	8	no	—
92	#125	—	97	16	no	—
93	#125	—	97	8	no	—
94	C₂²⋊Q₈	—	97	16	no	—
95	C₂²⋊Q₈	—	97	32	no	—
96	C₄×Q₈	—	97	16	no	—
97	#117	—	97	4	no	—
98	#117	—	97	4	no	—
99	#117	—	97	4	no	—
100	#117	—	97	4	no	—
101	#117	C₂×D₉₇	1	4	yes	—
102	#117	C₂×D₉₇	1	4	yes	—
103	C₂×C₄⋊C₄	—	97	4	no	—
104	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
105	C₂×C₂²⋊C₄	—	97	4	no	—
106	C₂×C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
107	C₂×C₄²	—	97	4	no	—
108	C₂×C₄²	C₂×D₉₇	1	2	yes	—
109	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
110	C₂×Q₈	—	97	16	no	—
111	C₂²×C₄	—	97	8	no	—
112	C₂²×C₄	Dic₁₉₄	1	8	yes	complemented
113	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
114	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
115	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
117	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
118	C₄⋊C₄	—	97	32	no	—
119	C₄⋊C₄	—	97	8	no	—
120	C₄⋊C₄	—	97	8	no	—
121	C₄⋊C₄	—	97	16	no	—
122	C₄⋊C₄	C₂²×D₉₇	1	8	yes	—
123	C₂²⋊C₄	—	97	8	no	—
124	C₂²⋊C₄	—	97	8	no	—
125	C₂²⋊C₄	C₂²×D₉₇	1	8	yes	—
126	C₄²	—	97	8	no	—
127	C₄²	—	97	8	no	—
128	C₄²	C₂²×D₉₇	1	4	yes	—
129	C₄²	C₂²×D₉₇	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂×Dic₁₉₄	1	4	yes	—
132	C₂³	C₂³×D₉₇	1	1	yes	characteristic
133	C₂³	C₂³×D₉₇	1	2	yes	—
134	Q₈	—	388	16	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	97	16	no	—
137	C₂×C₄	—	388	8	no	—
138	C₂×C₄	C₂×Dic₁₉₄	1	8	yes	—
139	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
140	C₂×C₄	C₂×Dic₁₉₄	1	8	yes	—
141	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
142	C₂×C₄	C₂³×D₉₇	1	8	yes	—
143	C₂×C₄	C₂³×D₉₇	1	2	yes	—
144	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₂²×Dic₁₉₄	1	2	yes	—
150	C₂²	C₂²×Dic₁₉₄	1	4	yes	—
151	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	194	16	no	—
154	C₄	C₂²×Dic₁₉₄	1	4	yes	—
155	C₄	C₂²×Dic₁₉₄	1	2	yes	—
156	C₄	C₂²×Dic₁₉₄	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	2	4	no	—
159	C₂	C₂³×Dic₁₉₄	1	1	yes	characteristic
160	C₂	#20261571	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₉₇
Fitting	C₂×#117×C₉₇	C₂
prime radical	C₂⁴×C₉₇	C₂³
hypercenter	C₂×#117	D₉₇
hypocenter	C₉₇	#3458
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #20261366	C₂	semidirect
2. #20261363	C₂	semidirect
4. #20261292	C₂	semidirect
5. #20261288	C₂	semidirect
8. #15690810	C₂²	semidirect
9. #15690809	C₂²	semidirect
10. #15690802	C₂²	semidirect
11. #15690801	C₂²	semidirect
14. C₄×Dic₁₉₄	C₂²	semidirect
39. C₂²×C₄×C₉₇	Q₈	semidirect
81. C₉₇	#3458	semidirect
112. C₂²×C₄	Dic₁₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20261366	2
#20261363	2
#20261361	2
#20261292	2
#20261288	2
C₂×#15690769	2
C₂×#117×C₉₇	2
#3458	97