Subgroups — #27141866

Order 12416.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
2	#20260601	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
7	C₄×#11715895	C₄	1	4	yes	complemented
8	#15690717	—	2	2	no	—
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	#95×C₉₇	—	2	2	no	—
12	C₂²×Dic₉₇	—	2	2	no	—
13	C₂×#11715895	—	2	2	no	—
14	C₂×#11715895	—	2	2	no	—
15	C₂×#11715895	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₉₇	—	2	4	no	—
17	C₂²×C₄×C₉₇	—	2	2	no	—
18	C₂²×C₄×C₉₇	D₄	1	2	yes	—
19	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	—	2	4	no	—
21	C₄²×C₉₇	C₂×C₄	1	2	yes	—
22	C₄²×C₉₇	Q₈	1	2	yes	—
23	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
24	C₂×C₄×C₉₇	—	2	4	no	—
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	—	4	4	no	—
27	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₉₇	#41	1	2	yes	—
31	C₂×C₄×C₉₇	#41	1	2	yes	—
32	C₂×Dic₉₇	—	4	2	no	—
33	C₂×Dic₉₇	—	4	4	no	—
34	#11715895	—	2	4	no	—
35	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₉₇	#124	1	1	yes	characteristic
37	C₂²×C₉₇	#124	1	1	yes	characteristic
38	C₂²×C₉₇	#135	1	4	yes	—
39	C₄×C₉₇	—	2	4	no	—
40	C₄×C₉₇	—	8	2	no	—
41	C₄×C₉₇	#117	1	2	yes	—
42	C₄×C₉₇	#125	1	2	yes	—
43	Dic₉₇	—	8	2	no	—
44	C₂×C₉₇	#382	1	1	yes	characteristic
45	C₂×C₉₇	#427	1	2	yes	—
46	C₂×C₉₇	#488	1	2	yes	—
47	C₂×C₉₇	#488	1	2	yes	—
48	#1835	—	97	1	no	maximal, Sylow
49	C₉₇	#1835	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄×C₈	—	97	1	no	—
51	#382	—	97	1	no	—
52	#382	D₉₇	1	1	yes	characteristic
53	#340	—	97	4	no	—
54	C₂²×C₈	—	97	2	no	—
55	C₂×C₄⋊C₄	—	97	2	no	—
56	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
57	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
58	C₄×C₈	—	97	4	no	—
59	#95	—	2	2	no	—
60	#95	—	194	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	194	2	no	—
63	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
64	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
65	C₂×C₈	—	97	4	no	—
66	C₂×C₈	—	194	2	no	—
67	C₂×C₈	—	194	2	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	194	4	no	—
70	C₄²	Dic₁₉₄	1	2	yes	—
71	C₄²	C₄×D₉₇	1	2	yes	—
72	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	388	2	no	—
77	C₂×C₄	—	388	4	no	—
78	C₂×C₄	#13095490	1	4	yes	—
79	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
81	C₂×C₄	#13095510	1	2	yes	—
82	C₂×C₄	#13095512	1	2	yes	—
83	C₈	—	194	4	no	—
84	C₂²	C₂×#13095490	1	1	yes	characteristic
85	C₂²	#15690797	1	1	yes	characteristic
86	C₂²	#15690799	1	1	yes	characteristic
87	C₂²	#15690830	1	4	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	—	776	2	no	—
91	C₄	#15690768	1	2	yes	—
92	C₄	#15690801	1	2	yes	—
93	C₂	#20260600	1	1	yes	characteristic
94	C₂	#20260771	1	2	yes	—
95	C₂	#20261089	1	2	yes	—
96	C₂	#20261092	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#382×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#382	D₉₇
hypocenter	C₉₇	#1835
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
7. C₄×#11715895	C₄	semidirect
49. C₉₇	#1835	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11715895	2
#20260601	2
#20260497	2
#382×C₉₇	2
#1835	97