Subgroups — C₂³×#13095514 #27158695

Order 12416.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095514	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#11715895	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	C₂²	1	28	yes	direct product factor
4	C₂²×#11715895	C₄	1	7	yes	—
5	C₂²×#11715895	C₂²	1	7	yes	—
6	C₂³×C₄×C₉₇	C₄	1	1	yes	characteristic
7	#13095514	C₂³	1	8	yes	direct product factor
8	C₂×#11715895	C₂×C₄	1	21	yes	—
9	C₂×#11715895	C₂³	1	7	yes	—
10	C₂⁴×C₉₇	C₈	1	1	yes	characteristic
11	C₂²×C₄×C₉₇	C₈	1	7	yes	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	7	yes	—
13	C₂³×C₉₇	C₁₆	1	8	yes	complemented
14	C₂³×C₉₇	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₉₇	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₉₇	C₂²×C₄	1	7	yes	—
17	#11715895	C₂²×C₄	1	7	yes	—
18	#11715895	C₂⁴	1	1	yes	characteristic
19	C₂²×C₉₇	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₉₇	C₂²×C₈	1	7	yes	—
21	C₄×C₉₇	C₂²×C₈	1	7	yes	—
22	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₉₇	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₉₇	C₂³×C₈	1	1	yes	characteristic
25	C₂³×C₁₆	—	97	1	no	maximal, Sylow
26	C₉₇	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₈	—	97	1	no	—
28	C₂²×C₁₆	—	97	14	no	—
29	C₂³×C₄	#87604	1	1	yes	characteristic
30	C₂²×C₈	—	97	7	no	—
31	C₂²×C₈	—	97	7	no	—
32	C₂×C₁₆	—	97	28	no	—
33	C₂⁴	#11715897	1	1	yes	characteristic
34	C₂²×C₄	#11715897	1	7	yes	—
35	C₂²×C₄	C₂×#87604	1	7	yes	—
36	C₂×C₈	—	97	21	no	—
37	C₂×C₈	—	97	7	no	—
38	C₁₆	—	97	8	no	—
39	C₂³	#13095514	1	8	yes	direct product factor
40	C₂³	C₂×#11715897	1	7	yes	—
41	C₂×C₄	C₂×#11715897	1	21	yes	—
42	C₂×C₄	C₂²×#87604	1	7	yes	—
43	C₈	—	97	1	no	—
44	C₈	—	97	7	no	—
45	C₂²	C₂×#13095514	1	28	yes	direct product factor
46	C₂²	C₂²×#11715897	1	7	yes	—
47	C₄	C₂²×#11715897	1	7	yes	—
48	C₄	C₂³×#87604	1	1	yes	characteristic
49	C₂	C₂²×#13095514	1	14	yes	direct product factor
50	C₂	C₂³×#11715897	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	#87604
commutator, hypocenter	C₉₇	C₂³×C₁₆
Frattini	C₄	C₂³×#87604
Fitting	C₂³×C₄×C₉₇	C₄
socle, prime radical	C₂⁴×C₉₇	C₈
cosocle	#11715895	C₂⁴
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13095514	C₂	direct (N × H)
3. C₂×#13095514	C₂²	direct (N × H)
7. #13095514	C₂³	direct (N × H)
13. C₂³×C₉₇	C₁₆	semidirect
19. C₂²×C₉₇	C₂×C₁₆	semidirect
23. C₂×C₉₇	C₂²×C₁₆	semidirect
26. C₉₇	C₂³×C₁₆	semidirect
39. C₂³	#13095514	direct (N × H)
45. C₂²	C₂×#13095514	direct (N × H)
49. C₂	C₂²×#13095514	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×#13095514	2
C₂³×#11715895	2
C₂³×C₁₆	97

Subgroups — C23×#13095514 #27158695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13095514 #27158695