Subgroups — C₂×#408×C₁₂₇ #30635830

Order 16256.

792 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. C₂³×C₈×C₁₂₇	C₂	semidirect
4. #408×C₁₂₇	C₂	direct (N × H)
5. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
16. C₂×M₁₆×C₁₂₇	C₂²	semidirect
20. C₂²×C₈×C₁₂₇	C₂²	semidirect
21. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
92. C₂×C₁₂₇	#408	direct (N × H)
95. C₂×#408	C₁₂₇	direct (N × H)
96. C₁₂₇	C₂×#408	direct (N × H)
98. C₂²×M₁₆	C₂×C₁₂₇	semidirect
99. C₂³×C₈	C₂×C₁₂₇	semidirect
100. #408	C₂×C₁₂₇	direct (N × H)
101. C₂×C₂²⋊C₈	C₂×C₁₂₇	semidirect
112. C₂×M₁₆	C₂²×C₁₂₇	semidirect
116. C₂²×C₈	C₂²×C₁₂₇	semidirect
117. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
188. C₂	#408×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×#41×C₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
C₂³×C₈×C₁₂₇	2
#408×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#408	127

Subgroups — C2×#408×C127 #30635830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#408×C₁₂₇ #30635830