Subgroups — #2165×C₂₂₇ #44396121

Order 29056.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	#364×C₂₂₇	C₂	1	2	yes	maximal, complemented
3	#349×C₂₂₇	C₂	1	2	yes	maximal
4	#347×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₂₂₇	C₄	1	1	yes	characteristic
6	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₂₇	C₄	1	1	yes	characteristic
8	M₃₂×C₂₂₇	C₂²	1	4	yes	—
9	#97×C₂₂₇	C₄	1	1	yes	characteristic
10	#97×C₂₂₇	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₂₇	C₄	1	1	yes	characteristic
12	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₂₇	—	2	1	no	—
14	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
15	M₁₆×C₂₂₇	—	2	2	no	—
16	M₁₆×C₂₂₇	—	2	2	no	—
17	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
21	C₄²×C₂₂₇	—	2	1	no	—
22	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₂₇	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₂₂₇	—	2	4	no	—
25	C₂³×C₂₂₇	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₂₂₇	—	2	1	no	—
27	C₂×C₄×C₂₂₇	—	2	1	no	—
28	C₂×C₄×C₂₂₇	—	2	1	no	—
29	C₂×C₄×C₂₂₇	—	2	1	no	—
30	C₂×C₄×C₂₂₇	—	4	2	no	—
31	C₂×C₄×C₂₂₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₂₇	—	2	1	no	—
34	C₈×C₂₂₇	—	2	1	no	—
35	C₈×C₂₂₇	M₁₆	1	2	yes	—
36	C₈×C₂₂₇	#41	1	2	yes	—
37	C₂²×C₂₂₇	—	2	1	no	—
38	C₂²×C₂₂₇	—	4	1	no	—
39	C₂²×C₂₂₇	C₂×M₁₆	1	1	yes	characteristic
40	C₄×C₂₂₇	—	2	2	no	—
41	C₄×C₂₂₇	—	2	1	no	—
42	C₄×C₂₂₇	—	4	1	no	—
43	C₄×C₂₂₇	#117	1	1	yes	characteristic
44	C₄×C₂₂₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₂₂₇	—	2	1	no	—
46	C₂×C₂₂₇	—	4	1	no	—
47	C₂×C₂₂₇	#432	1	1	yes	characteristic
48	C₂₂₇	#2165	1	1	yes	characteristic, direct product factor, Sylow
49	#2165	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₄×M₁₆	C₂×C₂₂₇	1	1	yes	characteristic
51	#364	C₂×C₂₂₇	1	2	yes	complemented
52	#349	C₂×C₂₂₇	1	2	yes	—
53	#347	C₂×C₂₂₇	1	2	yes	complemented
54	C₂×M₁₆	C₄×C₂₂₇	1	1	yes	characteristic
55	C₂×M₁₆	C₂²×C₂₂₇	1	1	yes	characteristic
56	C₂×C₄²	C₄×C₂₂₇	1	1	yes	characteristic
57	M₃₂	C₂²×C₂₂₇	1	4	yes	—
58	#97	C₄×C₂₂₇	1	1	yes	characteristic
59	#97	C₂²×C₂₂₇	1	1	yes	characteristic
60	C₄×C₈	C₄×C₂₂₇	1	1	yes	characteristic
61	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	C₂×C₄×C₂₂₇	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	C₂×C₈	C₂×C₄×C₂₂₇	1	1	yes	characteristic
67	C₂×C₈	C₂×C₄×C₂₂₇	1	1	yes	characteristic
68	C₂×C₈	C₂×C₄×C₂₂₇	1	1	yes	characteristic
69	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
70	C₄²	—	2	1	no	—
71	C₄²	C₂×C₄×C₂₂₇	1	1	yes	characteristic
72	C₄²	C₂×C₄×C₂₂₇	1	1	yes	characteristic
73	C₁₆	—	2	4	no	—
74	C₂³	M₁₆×C₂₂₇	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	M₁₆×C₂₂₇	1	1	yes	characteristic
81	C₂×C₄	C₂²×C₄×C₂₂₇	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	1	no	—
84	C₈	M₁₆×C₂₂₇	1	2	yes	—
85	C₈	#41×C₂₂₇	1	2	yes	—
86	C₂²	—	2	1	no	—
87	C₂²	—	4	1	no	—
88	C₂²	C₂×M₁₆×C₂₂₇	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	#117×C₂₂₇	1	1	yes	characteristic
93	C₄	C₂×M₁₆×C₂₂₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#432×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₇	#117
commutator	C₂×C₄	C₂²×C₄×C₂₂₇
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₇
socle	C₂×C₂₂₇	#432
prime radical	C₂³×C₂₂₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₇ ◁ #97×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #364×C₂₂₇	C₂	semidirect
4. #347×C₂₂₇	C₂	semidirect
48. C₂₂₇	#2165	direct (N × H)
49. #2165	C₂₂₇	direct (N × H)
51. #364	C₂×C₂₂₇	semidirect
53. #347	C₂×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₁₆×C₂₂₇	2
#364×C₂₂₇	2
#349×C₂₂₇	2
#347×C₂₂₇	2
#2165	227

Subgroups — #2165×C227 #44396121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2165×C₂₂₇ #44396121