Subgroups — #27140039

Order 12416.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095495	C₂	1	2	yes	maximal
2	#20260457	C₂	1	2	yes	maximal, complemented
3	#20260456	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₄	1	4	yes	—
7	#15690721	C₄	1	4	yes	—
8	C₄×#11715895	C₂²	1	2	yes	—
9	C₄×#11715895	C₂²	1	2	yes	—
10	C₂×M₁₆×C₉₇	C₄	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	#97×C₉₇	C₄	1	2	yes	—
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	#13095495	—	2	8	no	—
16	C₂×#11715895	C₂×C₄	1	4	yes	—
17	C₂×#11715895	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	M₁₆×C₉₇	—	2	4	no	—
21	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
23	C₄²×C₉₇	C₂×C₄	1	2	yes	—
24	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₉₇	C₂³	1	1	yes	characteristic
26	C₂³×C₉₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	—	2	4	no	—
29	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₄²	1	2	yes	—
31	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₉₇	—	2	2	no	—
34	C₈×C₉₇	M₁₆	1	4	yes	complemented
35	#11715895	—	2	4	no	—
36	#11715895	M₁₆	1	8	yes	complemented
37	C₂²×C₉₇	—	2	2	no	—
38	C₂²×C₉₇	—	2	1	no	—
39	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
40	C₄×C₉₇	—	2	2	no	—
41	C₄×C₉₇	—	2	1	no	—
42	C₄×C₉₇	C₂×M₁₆	1	4	yes	—
43	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
44	C₂×C₉₇	—	4	1	no	—
45	C₂×C₉₇	C₄×M₁₆	1	2	yes	—
46	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
47	#1455	—	97	1	no	maximal, Sylow
48	C₉₇	#1455	1	1	yes	characteristic, complemented, Sylow
49	C₄×M₁₆	—	97	2	no	—
50	C₄×M₁₆	D₉₇	1	1	yes	characteristic
51	#322	—	97	1	no	—
52	#322	—	97	2	no	—
53	C₈²	—	97	1	no	—
54	C₂×M₁₆	—	97	4	no	—
55	C₂×M₁₆	Dic₉₇	1	2	yes	—
56	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
57	#97	—	97	4	no	—
58	#97	Dic₉₇	1	2	yes	—
59	C₄×C₈	—	97	2	no	—
60	C₄×C₈	—	97	2	no	—
61	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
62	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
63	C₂²×C₄	C₄×D₉₇	1	2	yes	—
64	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	194	8	no	—
67	C₂×C₈	—	97	4	no	—
68	C₂×C₈	—	97	4	no	—
69	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
70	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
71	C₄²	C₄×D₉₇	1	2	yes	—
72	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
73	C₄²	C₂²×D₉₇	1	1	yes	characteristic
74	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
78	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
80	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	97	8	no	—
83	C₈	—	194	4	no	—
84	C₈	#13095495	1	4	yes	complemented
85	C₂²	—	2	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	1	no	—
90	C₄	M₁₆×D₉₇	1	4	yes	—
91	C₄	C₂×#13095495	1	2	yes	—
92	C₂	—	4	1	no	—
93	C₂	M₁₆×Dic₉₇	1	2	yes	—
94	C₂	C₄×#13095495	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₄×M₁₆×C₉₇	C₂
prime radical	C₂³×C₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	#1455
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20260457	C₂	semidirect
3. #20260456	C₂	semidirect
4. C₈×#11715895	C₂	semidirect
34. C₈×C₉₇	M₁₆	semidirect
36. #11715895	M₁₆	semidirect
48. C₉₇	#1455	semidirect
84. C₈	#13095495	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095495	2
#20260457	2
#20260456	2
C₈×#11715895	2
C₄×M₁₆×C₉₇	2
#1455	97