Subgroups — #30647474

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861216	C₂	1	2	yes	maximal, complemented
2	#22549954	C₂	1	2	yes	maximal, complemented
3	#22549948	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549843	C₂	1	2	yes	maximal, complemented
7	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#16861216	—	2	8	no	—
10	C₂×#14523232	C₂²	1	1	yes	characteristic
11	C₂×#14523232	C₂²	1	4	yes	—
12	C₂×#14523232	C₂²	1	1	yes	characteristic
13	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
15	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
19	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
20	#16861120	C₂²	1	2	yes	—
21	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
22	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
23	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
24	#127×C₁₂₇	C₄	1	1	yes	characteristic
25	#124×C₁₂₇	C₄	1	2	yes	—
26	#121×C₁₂₇	C₄	1	2	yes	—
27	#121×C₁₂₇	C₄	1	2	yes	—
28	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
29	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
31	#14523232	—	2	8	no	—
32	#14523232	—	2	2	no	—
33	#14523232	D₄	1	4	yes	complemented
34	C₂×C₁₂₇⋊C₈	—	2	8	no	—
35	C₂×C₁₂₇⋊C₈	—	2	4	no	—
36	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
37	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
38	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
39	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
40	#41×C₁₂₇	—	2	4	no	—
41	#41×C₁₂₇	—	2	4	no	—
42	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
45	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
54	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
55	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₁₂₇⋊C₈	—	2	2	no	—
60	C₁₂₇⋊C₈	—	2	2	no	—
61	C₁₂₇⋊C₈	—	2	2	no	—
62	C₁₂₇⋊C₈	C₂×D₄	1	4	yes	—
63	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	Q₈×C₁₂₇	—	2	4	no	—
66	D₄×C₁₂₇	—	2	8	no	—
67	D₄×C₁₂₇	—	2	4	no	—
68	D₄×C₁₂₇	—	2	4	no	—
69	D₄×C₁₂₇	—	4	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	4	1	no	—
74	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₂₇	#41	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₂²×C₁₂₇	—	2	1	no	—
84	C₂²×C₁₂₇	—	2	1	no	—
85	C₂²×C₁₂₇	—	2	4	no	—
86	C₂²×C₁₂₇	—	2	4	no	—
87	C₂²×C₁₂₇	—	2	4	no	—
88	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
89	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	2	1	no	—
92	C₄×C₁₂₇	—	2	1	no	—
93	C₄×C₁₂₇	—	2	1	no	—
94	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
95	C₄×C₁₂₇	#131	1	2	yes	complemented
96	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₁₂₇	—	2	2	no	—
99	C₂×C₁₂₇	—	4	4	no	—
100	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
102	C₂×C₁₂₇	#568	1	1	yes	characteristic
103	#2935	—	127	1	no	maximal, Sylow
104	C₁₂₇	#2935	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	127	2	no	—
106	#532	D₁₂₇	1	1	yes	characteristic
107	#436	—	127	2	no	—
108	#435	—	127	4	no	—
109	C₈×D₄	—	127	2	no	—
110	C₂×C₄⋊C₈	—	127	1	no	—
111	#408	—	127	2	no	—
112	C₄×M₁₆	—	127	1	no	—
113	C₂×#41	C₂×D₁₂₇	1	2	yes	—
114	#131	—	254	8	no	—
115	C₂×M₁₆	—	127	1	no	—
116	C₂×M₁₆	—	127	4	no	—
117	C₂×M₁₆	—	127	1	no	—
118	C₂²×C₈	—	127	2	no	—
119	C₂²×C₈	—	127	4	no	—
120	C₄⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
121	#127	Dic₁₂₇	1	1	yes	characteristic
122	#124	Dic₁₂₇	1	2	yes	—
123	#121	Dic₁₂₇	1	2	yes	—
124	#121	Dic₁₂₇	1	2	yes	—
125	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
126	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
127	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
128	C₄⋊C₈	—	127	1	no	—
129	C₄⋊C₈	—	127	2	no	—
130	C₄⋊C₈	—	127	1	no	—
131	C₂²⋊C₈	—	127	4	no	—
132	C₂²⋊C₈	—	127	4	no	—
133	#97	—	127	2	no	—
134	C₄×C₈	—	127	2	no	—
135	#41	—	2	4	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
138	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
139	C₂×D₄	C₂×Dic₁₂₇	1	4	yes	—
140	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
141	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
142	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
145	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
146	M₁₆	—	127	4	no	—
147	M₁₆	—	254	8	no	—
148	M₁₆	—	254	2	no	—
149	C₂×C₈	—	127	2	no	—
150	C₂×C₈	—	127	2	no	—
151	C₂×C₈	—	127	2	no	—
152	C₂×C₈	—	127	2	no	—
153	C₂×C₈	—	254	8	no	—
154	C₂×C₈	—	254	4	no	—
155	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
158	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
159	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
160	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
161	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
162	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
163	C₂³	C₂²×Dic₁₂₇	1	4	yes	—
164	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
175	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
178	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
181	C₂×C₄	#14523247	1	2	yes	—
182	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
183	C₈	—	127	4	no	—
184	C₈	—	254	2	no	—
185	C₈	—	254	2	no	—
186	C₈	—	254	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	D₄×Dic₁₂₇	1	2	yes	—
193	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	D₄×Dic₁₂₇	1	2	yes	—
199	C₄	#16861216	1	2	yes	complemented
200	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
201	C₄	C₂×#14523247	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×D₄×Dic₁₂₇	1	1	yes	characteristic
205	C₂	C₂×#16861216	1	1	yes	characteristic
206	C₂	#22550643	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting, prime radical	#532×C₁₂₇	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#532	D₁₂₇
hypocenter	C₁₂₇	#2935
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#16861216	C₂	semidirect
2. #22549954	C₂	semidirect
3. #22549948	C₂	semidirect
4. D₄×C₁₂₇⋊C₈	C₂	semidirect
5. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
6. #22549843	C₂	semidirect
7. C₄×#14523232	C₂	semidirect
33. #14523232	D₄	semidirect
95. C₄×C₁₂₇	#131	semidirect
104. C₁₂₇	#2935	semidirect
199. C₄	#16861216	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861216	2
#22549954	2
#22549948	2
D₄×C₁₂₇⋊C₈	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549843	2
C₄×#14523232	2
#532×C₁₂₇	2
#2935	127