Subgroups — C₂³×#13095525 #27159034

Order 12416.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095525	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#11715906	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095525	C₂²	1	28	yes	direct product factor
4	C₂²×#11715906	C₄	1	7	yes	—
5	C₂²×#11715906	C₂²	1	7	yes	—
6	C₂³×#87604	C₄	1	1	yes	characteristic
7	#13095525	C₂³	1	8	yes	direct product factor
8	C₂×#11715906	C₂×C₄	1	21	yes	—
9	C₂×#11715906	C₂³	1	7	yes	—
10	C₂²×#87604	C₈	1	7	yes	—
11	C₂²×#87604	C₂×C₄	1	7	yes	—
12	C₂³×D₉₇	C₈	1	1	yes	characteristic
13	C₂³×C₉₇	C₁₆	1	1	yes	characteristic, complemented
14	C₂²×D₉₇	C₁₆	1	7	yes	complemented
15	C₂²×D₉₇	C₂×C₈	1	7	yes	—
16	C₂×#87604	C₂×C₈	1	21	yes	—
17	C₂×#87604	C₂²×C₄	1	7	yes	—
18	#11715906	C₂²×C₄	1	7	yes	—
19	#11715906	C₂⁴	1	1	yes	characteristic
20	C₂²×C₉₇	C₂×C₁₆	1	7	yes	complemented
21	C₂×D₉₇	C₂×C₁₆	1	21	yes	complemented
22	C₂×D₉₇	C₂²×C₈	1	7	yes	—
23	#87604	C₂²×C₈	1	7	yes	—
24	#87604	C₂³×C₄	1	1	yes	characteristic
25	C₂×C₉₇	C₂²×C₁₆	1	7	yes	complemented
26	D₉₇	C₂²×C₁₆	1	7	yes	complemented
27	D₉₇	C₂³×C₈	1	1	yes	characteristic
28	C₂³×C₁₆	—	97	1	no	maximal, Sylow
29	C₉₇	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
30	C₂³×C₈	—	97	1	no	—
31	C₂²×C₁₆	—	97	14	no	—
32	C₂³×C₄	—	97	1	no	—
33	C₂²×C₈	—	97	7	no	—
34	C₂²×C₈	—	97	7	no	—
35	C₂×C₁₆	—	97	28	no	—
36	C₂⁴	—	97	1	no	—
37	C₂²×C₄	—	97	7	no	—
38	C₂²×C₄	—	97	7	no	—
39	C₂×C₈	—	97	21	no	—
40	C₂×C₈	—	97	7	no	—
41	C₁₆	—	97	8	no	—
42	C₂³	—	97	7	no	—
43	C₂³	—	97	7	no	—
44	C₂³	#13095525	1	1	yes	characteristic, direct product factor
45	C₂×C₄	—	97	7	no	—
46	C₂×C₄	—	97	21	no	—
47	C₈	—	97	1	no	—
48	C₈	—	97	7	no	—
49	C₂²	—	97	7	no	—
50	C₂²	—	97	21	no	—
51	C₂²	C₂×#13095525	1	7	yes	direct product factor
52	C₄	—	97	1	no	—
53	C₄	—	97	7	no	—
54	C₂	—	97	1	no	—
55	C₂	—	97	7	no	—
56	C₂	C₂²×#13095525	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#13095525
commutator, hypocenter	C₉₇	C₂³×C₁₆
Frattini	C₁	G
Fitting, socle	C₂³×C₉₇	C₁₆
prime radical	C₂³×D₉₇	C₈
cosocle	#11715906	C₂⁴
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#13095525	C₂	direct (N × H)
3. C₂×#13095525	C₂²	direct (N × H)
7. #13095525	C₂³	direct (N × H)
13. C₂³×C₉₇	C₁₆	semidirect
14. C₂²×D₉₇	C₁₆	semidirect
20. C₂²×C₉₇	C₂×C₁₆	semidirect
21. C₂×D₉₇	C₂×C₁₆	semidirect
25. C₂×C₉₇	C₂²×C₁₆	semidirect
26. D₉₇	C₂²×C₁₆	semidirect
29. C₉₇	C₂³×C₁₆	semidirect
44. C₂³	#13095525	direct (N × H)
51. C₂²	C₂×#13095525	direct (N × H)
56. C₂	C₂²×#13095525	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×#13095525	2
C₂³×#11715906	2
C₂³×C₁₆	97

Subgroups — C23×#13095525 #27159034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13095525 #27159034