Subgroups — C₂×#400×C₁₂₇ #30635346

Order 16256.

604 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#95×C₁₂₇	C₂	1	2	yes	maximal
5	C₂×#95×C₁₂₇	C₂	1	2	yes	maximal
6	C₂×#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
16	#95×C₁₂₇	C₂²	1	4	yes	—
17	#95×C₁₂₇	C₂²	1	8	yes	—
18	#95×C₁₂₇	C₂²	1	8	yes	—
19	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	—	2	2	no	—
21	C₂²×C₄×C₁₂₇	—	2	2	no	—
22	C₂²×C₄×C₁₂₇	—	2	4	no	—
23	C₂²×C₄×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	—	2	4	no	—
25	C₂²×C₄×C₁₂₇	—	2	4	no	—
26	C₂²×C₄×C₁₂₇	—	2	8	no	—
27	C₂²×C₄×C₁₂₇	—	2	4	no	—
28	C₂²×C₄×C₁₂₇	—	2	4	no	—
29	C₂²×C₄×C₁₂₇	—	2	4	no	—
30	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
31	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
37	C₄⋊C₄×C₁₂₇	—	2	8	no	—
38	C₄⋊C₄×C₁₂₇	—	2	8	no	—
39	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
40	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
41	C₂³×C₁₂₇	#41	1	2	yes	—
42	C₂³×C₁₂₇	#41	1	4	yes	—
43	C₂³×C₁₂₇	#41	1	4	yes	—
44	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	—	2	8	no	—
46	C₂×C₄×C₁₂₇	—	2	8	no	—
47	C₂×C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄×C₁₂₇	—	2	4	no	—
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	8	no	—
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	2	4	no	—
56	C₂×C₄×C₁₂₇	—	2	4	no	—
57	C₂×C₄×C₁₂₇	—	4	4	no	—
58	C₂×C₄×C₁₂₇	—	4	8	no	—
59	C₂×C₄×C₁₂₇	—	4	8	no	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
63	C₂²×C₁₂₇	#123	1	4	yes	—
64	C₂²×C₁₂₇	#124	1	4	yes	—
65	C₂²×C₁₂₇	#125	1	8	yes	—
66	C₂²×C₁₂₇	#126	1	8	yes	—
67	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
70	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
71	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
72	C₄×C₁₂₇	—	2	4	no	—
73	C₄×C₁₂₇	—	2	4	no	—
74	C₄×C₁₂₇	—	4	2	no	—
75	C₄×C₁₂₇	—	4	4	no	—
76	C₄×C₁₂₇	—	4	4	no	—
77	C₂×C₁₂₇	#400	1	8	yes	direct product factor
78	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₁₂₇	C₂×#123	1	1	yes	characteristic
80	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
81	C₂×C₁₂₇	C₂×#125	1	2	yes	—
82	C₂×C₁₂₇	C₂×#126	1	2	yes	—
83	C₂×#400	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₂₇	C₂×#400	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
86	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
87	#400	C₂×C₁₂₇	1	8	yes	direct product factor
88	C₂×#95	C₂×C₁₂₇	1	2	yes	—
89	C₂×#95	C₂×C₁₂₇	1	2	yes	—
90	C₂×#95	C₂×C₁₂₇	1	1	yes	characteristic
91	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
93	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
94	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
99	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
100	#95	C₂²×C₁₂₇	1	4	yes	—
101	#95	C₂²×C₁₂₇	1	8	yes	—
102	#95	C₂²×C₁₂₇	1	8	yes	—
103	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
115	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
116	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
117	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
118	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
119	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
120	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
121	C₄⋊C₄	—	2	8	no	—
122	C₄⋊C₄	—	2	8	no	—
123	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
124	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
125	C₂³	#41×C₁₂₇	1	2	yes	—
126	C₂³	#41×C₁₂₇	1	4	yes	—
127	C₂³	#41×C₁₂₇	1	4	yes	—
128	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	—	4	8	no	—
144	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
145	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
146	C₂²	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
147	C₂²	#123×C₁₂₇	1	4	yes	—
148	C₂²	#124×C₁₂₇	1	4	yes	—
149	C₂²	#125×C₁₂₇	1	8	yes	—
150	C₂²	#126×C₁₂₇	1	8	yes	—
151	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
152	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
153	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
154	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
155	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	4	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	4	no	—
161	C₂	#400×C₁₂₇	1	8	yes	direct product factor
162	C₂	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
163	C₂	C₂×#123×C₁₂₇	1	1	yes	characteristic
164	C₂	C₂×#124×C₁₂₇	1	1	yes	characteristic
165	C₂	C₂×#125×C₁₂₇	1	2	yes	—
166	C₂	C₂×#126×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #400×C₁₂₇	C₂	direct (N × H)
77. C₂×C₁₂₇	#400	direct (N × H)
83. C₂×#400	C₁₂₇	direct (N × H)
84. C₁₂₇	C₂×#400	direct (N × H)
87. #400	C₂×C₁₂₇	direct (N × H)
161. C₂	#400×C₁₂₇	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×C₄⋊C₄×C₁₂₇	2
#400×C₁₂₇	2
C₂×#95×C₁₂₇	2
C₂×#400	127

Subgroups — C2×#400×C127 #30635346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#400×C₁₂₇ #30635346