Finite Group Databasebeta

Subgroups — #27153529

Order 12416.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261541	C₂	1	4	yes	maximal, complemented
2	#20260995	C₂	1	4	yes	maximal, complemented
3	#20260917	C₂	1	2	yes	maximal, complemented
4	#20260902	C₂	1	2	yes	maximal, complemented
5	#20260853	C₂	1	2	yes	maximal, complemented
6	#492×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#15690858	C₂²	1	2	yes	—
8	#15690855	C₂²	1	2	yes	complemented
9	#15690841	C₂²	1	4	yes	complemented
10	#15690839	—	2	4	no	—
11	#15690837	—	2	4	no	—
12	#15690836	C₂²	1	4	yes	—
13	#15690832	C₂²	1	2	yes	—
14	#15690831	—	2	4	no	—
15	SD₁₆×D₉₇	—	2	4	no	—
16	#15690816	C₂²	1	4	yes	complemented
17	#15690746	C₂²	1	2	yes	complemented
18	#15690738	C₂²	1	4	yes	complemented
19	#15690729	C₂²	1	2	yes	complemented
20	#15690727	C₂²	1	2	yes	complemented
21	C₂×SD₁₆×C₉₇	C₂²	1	4	yes	complemented
22	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
23	#127×C₉₇	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
25	#13095512	—	2	4	no	—
26	#13095511	—	4	2	no	—
27	#13095511	C₂³	1	4	yes	—
28	#13095510	—	2	4	no	—
29	Q₈×D₉₇	—	2	4	no	—
30	C₂×C₂²⋊D₉₇	—	4	2	no	—
31	D₄×D₉₇	—	2	4	no	—
32	Dic₃₈₈	D₄	1	2	yes	complemented
33	#13095501	—	2	4	no	—
34	#13095500	D₄	1	4	yes	complemented
35	#13095499	—	2	4	no	—
36	#13095498	—	2	4	no	—
37	D₇₇₆	D₄	1	2	yes	complemented
38	#13095496	—	2	4	no	—
39	#13095495	C₂³	1	4	yes	—
40	#13095494	—	2	4	no	—
41	C₈×D₉₇	—	2	4	no	—
42	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
43	C₂×D₄×C₉₇	—	4	1	no	—
44	C₂×D₄×C₉₇	C₂³	1	2	yes	—
45	SD₁₆×C₉₇	—	2	4	no	—
46	SD₁₆×C₉₇	—	2	4	no	—
47	C₂×C₈×C₉₇	C₂³	1	2	yes	—
48	C₄⋊C₄×C₉₇	—	4	1	no	—
49	C₄²×C₉₇	C₂³	1	1	yes	characteristic
50	C₂³×C₉₇	—	2	2	no	—
51	C₂²×D₉₇	—	4	2	no	—
52	Q₈×C₉₇	—	2	4	no	—
53	D₄×C₉₇	—	2	4	no	—
54	D₄×C₉₇	—	8	2	no	—
55	C₂×C₄×C₉₇	—	2	1	no	—
56	C₂×C₄×C₉₇	—	2	2	no	—
57	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₉₇	—	2	4	no	—
59	C₂²⋊D₉₇	—	4	4	no	—
60	C₂×Dic₉₇	—	4	2	no	—
61	D₃₈₈	—	4	2	no	—
62	D₃₈₈	C₂×D₄	1	4	yes	—
63	C₄×D₉₇	—	2	4	no	—
64	C₄×D₉₇	—	4	4	no	—
65	Dic₁₉₄	—	4	2	no	—
66	Dic₁₉₄	C₂×D₄	1	4	yes	—
67	C₈×C₉₇	C₂×D₄	1	2	yes	—
68	C₈×C₉₇	C₂×D₄	1	2	yes	—
69	#11715895	—	2	4	no	—
70	C₂²×C₉₇	—	4	2	no	—
71	C₂²×C₉₇	—	8	2	no	—
72	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×D₉₇	—	2	4	no	—
74	C₂×D₉₇	—	16	2	no	—
75	C₄×C₉₇	—	2	2	no	—
76	C₄×C₉₇	—	4	2	no	—
77	C₄×C₉₇	C₂²×D₄	1	2	yes	—
78	Dic₉₇	—	2	4	no	—
79	C₂×C₉₇	—	2	1	no	—
80	C₂×C₉₇	—	8	2	no	—
81	C₂×C₉₇	D₄²	1	1	yes	characteristic
82	D₉₇	—	4	4	no	—
83	#3291	—	97	1	no	maximal, Sylow
84	C₉₇	#3291	1	1	yes	characteristic, complemented, Sylow
85	#577	—	97	4	no	—
86	#492	D₉₇	1	1	yes	characteristic, complemented
87	#471	—	97	4	no	—
88	#456	—	97	2	no	—
89	#453	—	97	2	no	—
90	#443	—	97	2	no	—
91	2₋¹⁺⁴	—	97	2	no	—
92	H(5,𝔽₂)	—	97	2	no	—
93	#137	—	97	4	no	—
94	#137	—	194	4	no	—
95	AGL(1,ℤ/8)	—	97	4	no	—
96	AGL(1,ℤ/8)	—	194	4	no	—
97	#135	—	97	2	no	—
98	#135	—	194	4	no	—
99	C₂×SD₁₆	—	194	4	no	—
100	C₂×SD₁₆	C₂×D₉₇	1	4	yes	complemented
101	#131	—	97	4	no	—
102	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
103	#127	C₂×D₉₇	1	1	yes	characteristic
104	#108	—	97	2	no	—
105	#104	—	97	4	no	—
106	#101	—	97	2	no	—
107	#100	—	97	2	no	—
108	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
109	#41	—	97	4	no	—
110	#41	—	388	2	no	—
111	#41	—	194	4	no	—
112	#41	—	194	4	no	—
113	C₂×Q₈	—	194	4	no	—
114	C₂×Q₈	C₂²×D₉₇	1	2	yes	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	388	2	no	—
117	C₂×D₄	—	194	4	no	—
118	C₂×D₄	C₂²×D₉₇	1	2	yes	—
119	Q₁₆	—	97	2	no	—
120	Q₁₆	—	194	4	no	—
121	SD₁₆	—	2	4	no	—
122	SD₁₆	—	2	4	no	—
123	SD₁₆	—	97	4	no	—
124	SD₁₆	—	194	4	no	—
125	SD₁₆	—	194	4	no	—
126	D₈	—	97	2	no	—
127	D₈	—	194	4	no	—
128	M₁₆	—	97	4	no	—
129	M₁₆	—	194	4	no	—
130	C₂×C₈	—	194	4	no	—
131	C₂×C₈	C₂²×D₉₇	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄²	C₂²×D₉₇	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	388	2	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	97	4	no	—
138	Q₈	—	388	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	97	4	no	—
142	D₄	—	388	2	no	—
143	D₄	—	388	4	no	—
144	D₄	—	194	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	388	4	no	—
148	C₂×C₄	—	388	2	no	—
149	C₂×C₄	—	194	4	no	—
150	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
151	C₈	—	194	4	no	—
152	C₈	D₄×D₉₇	1	2	yes	—
153	C₈	D₄×D₉₇	1	2	yes	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	1552	2	no	—
157	C₂²	—	194	4	no	—
158	C₂²	#15690855	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	194	4	no	—
162	C₄	C₂×D₄×D₉₇	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	388	4	no	—
166	C₂	#20261370	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261370
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#492×C₉₇	C₂
socle	C₂×C₉₇	D₄²
hypercenter	#492	D₉₇
hypocenter	C₉₇	#3291
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #20261541	C₂	semidirect
2. #20260995	C₂	semidirect
3. #20260917	C₂	semidirect
4. #20260902	C₂	semidirect
5. #20260853	C₂	semidirect
6. #492×C₉₇	C₂	semidirect
8. #15690855	C₂²	semidirect
9. #15690841	C₂²	semidirect
16. #15690816	C₂²	semidirect
17. #15690746	C₂²	semidirect
18. #15690738	C₂²	semidirect
19. #15690729	C₂²	semidirect
20. #15690727	C₂²	semidirect
21. C₂×SD₁₆×C₉₇	C₂²	semidirect
22. C₄⋊Q₈×C₉₇	C₂²	semidirect
24. C₄×C₈×C₉₇	C₂²	semidirect
32. Dic₃₈₈	D₄	semidirect
34. #13095500	D₄	semidirect
37. D₇₇₆	D₄	semidirect
84. C₉₇	#3291	semidirect
86. #492	D₉₇	semidirect
100. C₂×SD₁₆	C₂×D₉₇	semidirect
102. C₄⋊Q₈	C₂×D₉₇	semidirect
108. C₄×C₈	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261541	2
#20260995	2
#20260917	2
#20260902	2
#20260853	2
#492×C₉₇	2
#3291	97