Subgroups — #322×D₁₂₇ #30637004

Order 16256.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×D₁₂₇	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549598	C₂	1	2	yes	maximal, complemented
4	#22549597	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₂₇	C₄	1	4	yes	—
7	C₂×C₈×D₁₂₇	C₄	1	2	yes	—
8	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
9	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₂₇	C₄	1	4	yes	—
12	C₈×Dic₁₂₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₈×D₁₂₇	—	2	8	no	—
20	C₈×D₁₂₇	C₈	1	8	yes	complemented
21	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₄²	1	1	yes	characteristic
27	C₂×Dic₁₂₇	C₄²	1	1	yes	characteristic
28	C₂×Dic₁₂₇	C₄²	1	2	yes	—
29	C₄×D₁₂₇	C₂×C₈	1	4	yes	—
30	C₄×D₁₂₇	M₁₆	1	8	yes	—
31	C₁₂₇⋊C₈	—	2	4	no	—
32	C₁₂₇⋊C₈	C₂×C₈	1	4	yes	complemented
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₈×C₁₂₇	—	2	4	no	—
36	C₈×C₁₂₇	C₂×C₈	1	4	yes	complemented
37	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
38	C₂×D₁₂₇	C₄×C₈	1	2	yes	—
39	C₂×D₁₂₇	#97	1	4	yes	—
40	C₄×C₁₂₇	C₂²×C₈	1	2	yes	—
41	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
42	Dic₁₂₇	C₄×C₈	1	2	yes	—
43	Dic₁₂₇	#97	1	4	yes	—
44	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂×#97	1	2	yes	—
46	D₁₂₇	#322	1	4	yes	direct product factor
47	C₂×#322	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂×#322	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₄×C₈	—	127	1	no	—
50	C₂×C₄×C₈	—	127	2	no	—
51	#322	—	127	1	no	—
52	#322	—	127	2	no	—
53	#322	D₁₂₇	1	1	yes	characteristic, direct product factor
54	C₂²×C₈	—	127	2	no	—
55	C₂²×C₈	—	127	4	no	—
56	C₂×C₄²	—	127	1	no	—
57	C₄×C₈	—	127	2	no	—
58	C₄×C₈	—	127	4	no	—
59	C₄×C₈	—	127	1	no	—
60	C₄×C₈	—	127	2	no	—
61	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
62	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	—	127	1	no	—
64	C₂²×C₄	—	127	2	no	—
65	C₂×C₈	—	127	2	no	—
66	C₂×C₈	—	127	8	no	—
67	C₂×C₈	—	127	4	no	—
68	C₂×C₈	—	254	8	no	—
69	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
70	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
71	C₄²	—	127	1	no	—
72	C₄²	—	127	2	no	—
73	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₂³	—	127	1	no	—
75	C₂×C₄	—	127	1	no	—
76	C₂×C₄	—	127	2	no	—
77	C₂×C₄	—	127	4	no	—
78	C₂×C₄	—	127	8	no	—
79	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	—	127	4	no	—
83	C₈	—	254	4	no	—
84	C₈	C₈×D₁₂₇	1	4	yes	complemented
85	C₂²	—	127	2	no	—
86	C₂²	—	127	4	no	—
87	C₂²	C₄²×D₁₂₇	1	1	yes	characteristic
88	C₄	—	127	2	no	—
89	C₄	—	127	4	no	—
90	C₄	C₂×C₈×D₁₂₇	1	2	yes	—
91	C₄	M₁₆×D₁₂₇	1	4	yes	—
92	C₂	—	127	4	no	—
93	C₂	C₄×C₈×D₁₂₇	1	1	yes	characteristic
94	C₂	#97×D₁₂₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Fitting	#322×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄²
prime radical	C₂²×D₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	C₂×#322
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #22549598	C₂	semidirect
4. #22549597	C₂	semidirect
5. #322×C₁₂₇	C₂	semidirect
20. C₈×D₁₂₇	C₈	semidirect
32. C₁₂₇⋊C₈	C₂×C₈	semidirect
36. C₈×C₁₂₇	C₂×C₈	semidirect
46. D₁₂₇	#322	direct (N × H)
48. C₁₂₇	C₂×#322	semidirect
53. #322	D₁₂₇	direct (N × H)
84. C₈	C₈×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×D₁₂₇	2
#22549598	2
#22549597	2
#322×C₁₂₇	2
C₂×#322	127

Subgroups — #322×D127 #30637004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #322×D₁₂₇ #30637004