Subgroups — C₁₂₇×#2922 #30635874

Order 16256.

400 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#433×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	#431×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	#424×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#408×C₁₂₇	C₂	1	2	yes	maximal
6	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
10	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄×Q₈×C₁₂₇	C₄	1	2	yes	—
12	C₄×D₄×C₁₂₇	C₄	1	4	yes	—
13	C₄×D₄×C₁₂₇	C₄	1	2	yes	—
14	#117×C₁₂₇	C₂²	1	1	yes	characteristic
15	#117×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
18	C₄⋊C₈×C₁₂₇	C₂²	1	8	yes	—
19	C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	—
20	#97×C₁₂₇	C₂²	1	2	yes	—
21	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	#41×C₁₂₇	—	2	4	no	—
25	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₂₇	—	2	4	no	—
33	C₂×C₈×C₁₂₇	—	2	4	no	—
34	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
35	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
36	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₄²×C₁₂₇	C₂³	1	2	yes	—
44	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
46	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₂₇	—	2	2	no	—
48	D₄×C₁₂₇	—	2	4	no	—
49	D₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄×C₁₂₇	—	2	4	no	—
51	C₂×C₄×C₁₂₇	—	2	8	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	#41	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₂₇	—	2	4	no	—
61	C₈×C₁₂₇	—	2	4	no	—
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	—	2	2	no	—
64	C₂²×C₁₂₇	—	2	4	no	—
65	C₂²×C₁₂₇	#117	1	2	yes	—
66	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	2	4	no	—
70	C₄×C₁₂₇	#117	1	2	yes	—
71	C₄×C₁₂₇	#131	1	4	yes	—
72	C₄×C₁₂₇	C₂×#41	1	2	yes	—
73	C₂×C₁₂₇	—	2	2	no	—
74	C₂×C₁₂₇	—	4	2	no	—
75	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
76	C₂×C₁₂₇	C₂×#131	1	2	yes	—
77	#2922	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₁₂₇	#2922	1	1	yes	characteristic, direct product factor, Sylow
79	C₄×#41	C₂×C₁₂₇	1	1	yes	characteristic
80	#433	C₂×C₁₂₇	1	4	yes	complemented
81	#431	C₂×C₁₂₇	1	4	yes	complemented
82	#424	C₂×C₁₂₇	1	2	yes	complemented
83	#408	C₂×C₁₂₇	1	2	yes	—
84	C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
85	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
86	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
87	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
88	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
89	C₄×Q₈	C₄×C₁₂₇	1	2	yes	—
90	C₄×D₄	C₄×C₁₂₇	1	4	yes	—
91	C₄×D₄	C₄×C₁₂₇	1	2	yes	—
92	#117	C₂²×C₁₂₇	1	1	yes	characteristic
93	#117	C₂²×C₁₂₇	1	2	yes	—
94	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
96	C₄⋊C₈	C₂²×C₁₂₇	1	8	yes	—
97	C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	—
98	#97	C₂²×C₁₂₇	1	2	yes	—
99	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
100	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
101	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
104	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
105	C₂×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
107	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
108	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
109	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
113	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
114	C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
115	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
116	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
117	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
118	C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
119	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
120	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
121	C₄²	C₂³×C₁₂₇	1	2	yes	—
122	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
123	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
124	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
132	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
133	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
134	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
136	C₂×C₄	#41×C₁₂₇	1	4	yes	—
137	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	2	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#117×C₁₂₇	1	2	yes	—
144	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	4	no	—
148	C₄	#117×C₁₂₇	1	2	yes	—
149	C₄	#131×C₁₂₇	1	4	yes	—
150	C₄	C₂×#41×C₁₂₇	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#117×C₁₂₇	1	1	yes	characteristic
154	C₂	C₂×#131×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂×#41×C₁₂₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #433×C₁₂₇	C₂	semidirect
3. #431×C₁₂₇	C₂	semidirect
4. #424×C₁₂₇	C₂	semidirect
6. C₄×M₁₆×C₁₂₇	C₂	semidirect
7. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
77. #2922	C₁₂₇	direct (N × H)
78. C₁₂₇	#2922	direct (N × H)
80. #433	C₂×C₁₂₇	semidirect
81. #431	C₂×C₁₂₇	semidirect
82. #424	C₂×C₁₂₇	semidirect
84. C₄×M₁₆	C₂×C₁₂₇	semidirect
85. C₂×C₄×C₈	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#41×C₁₂₇	2
#433×C₁₂₇	2
#431×C₁₂₇	2
#424×C₁₂₇	2
#408×C₁₂₇	2
C₄×M₁₆×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
#2922	127

Subgroups — C127×#2922 #30635874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2922 #30635874