Subgroups — #30638710

Order 16256.

400 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
3	#22549631	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
6	C₄⋊(C₁₂₇⋊C₈)	—	2	4	no	—
7	C₂²⋊(C₁₂₇⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₂₇	—	2	4	no	—
12	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	—	2	4	no	—
17	C₂×C₁₂₇⋊C₈	—	2	4	no	—
18	C₂×C₁₂₇⋊C₈	—	2	4	no	—
19	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	—	2	2	no	—
21	C₂²×C₄×C₁₂₇	—	2	2	no	—
22	C₂²×C₄×C₁₂₇	—	2	2	no	—
23	C₂²×C₄×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	C₈	1	4	yes	complemented
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄²×C₁₂₇	—	2	2	no	—
37	C₄²×C₁₂₇	—	2	2	no	—
38	C₄²×C₁₂₇	—	2	2	no	—
39	C₄²×C₁₂₇	D₄	1	4	yes	—
40	C₁₂₇⋊C₈	—	4	4	no	—
41	C₂³×C₁₂₇	—	2	1	no	—
42	C₂³×C₁₂₇	—	2	1	no	—
43	C₂³×C₁₂₇	—	2	1	no	—
44	C₂³×C₁₂₇	—	2	1	no	—
45	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₂³×C₁₂₇	C₂×C₈	1	2	yes	—
47	C₂³×C₁₂₇	M₁₆	1	2	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	—	2	8	no	—
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	2	1	no	—
55	C₂×C₄×C₁₂₇	—	2	1	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	2	1	no	—
58	C₂×C₄×C₁₂₇	—	2	8	no	—
59	C₂×C₄×C₁₂₇	—	2	4	no	—
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
66	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
67	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	#41	1	2	yes	—
72	C₂²×C₁₂₇	—	2	2	no	—
73	C₂²×C₁₂₇	—	2	2	no	—
74	C₂²×C₁₂₇	—	2	2	no	—
75	C₂²×C₁₂₇	—	2	2	no	—
76	C₂²×C₁₂₇	—	2	2	no	—
77	C₂²×C₁₂₇	—	2	2	no	—
78	C₂²×C₁₂₇	C₄⋊C₈	1	4	yes	complemented
79	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	#117	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
83	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
86	C₄×C₁₂₇	—	2	2	no	—
87	C₄×C₁₂₇	—	2	2	no	—
88	C₄×C₁₂₇	—	2	2	no	—
89	C₄×C₁₂₇	—	2	2	no	—
90	C₄×C₁₂₇	C₂²⋊C₈	1	4	yes	complemented
91	C₄×C₁₂₇	#121	1	2	yes	—
92	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₁₂₇	—	2	4	no	—
94	C₂×C₁₂₇	#380	1	1	yes	characteristic
95	C₂×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₁₂₇	#407	1	1	yes	characteristic
97	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₁₂₇	#423	1	1	yes	characteristic
99	C₂×C₁₂₇	#431	1	1	yes	characteristic
100	C₂×C₁₂₇	#432	1	1	yes	characteristic
101	#1797	—	127	1	no	maximal, Sylow
102	C₁₂₇	#1797	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄²	D₁₂₇	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	127	2	no	—
105	C₂×C₂²⋊C₈	—	127	2	no	—
106	#336	—	127	2	no	—
107	C₂³×C₄	Dic₁₂₇	1	1	yes	characteristic
108	C₂³×C₄	Dic₁₂₇	1	1	yes	characteristic
109	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
110	C₂²×C₈	—	127	4	no	—
111	C₂×C₄²	—	2	4	no	—
112	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
113	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
116	C₄⋊C₈	—	254	4	no	—
117	C₂²⋊C₈	—	254	4	no	—
118	C₂⁴	C₂×Dic₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	C₁₂₇⋊C₈	1	4	yes	complemented
127	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
129	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
132	C₂²×C₄	D₅₀₈	1	2	yes	—
133	C₂²×C₄	Dic₂₅₄	1	2	yes	—
134	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
135	C₂×C₈	—	254	4	no	—
136	C₂×C₈	—	254	4	no	—
137	C₂×C₈	—	254	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂²⋊D₁₂₇	1	4	yes	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₄⋊Dic₁₂₇	1	2	yes	—
147	C₂³	C₂×C₁₂₇⋊C₈	1	2	yes	—
148	C₂³	#14523232	1	2	yes	—
149	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
163	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
164	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
165	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
166	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
167	C₂×C₄	#14523232	1	2	yes	—
168	C₂×C₄	#14523232	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
170	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
172	C₂×C₄	#14523248	1	2	yes	—
173	C₈	—	508	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	4	yes	complemented
181	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
182	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
183	C₂²	#16861168	1	1	yes	characteristic
184	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
185	C₂²	C₂×#14523232	1	1	yes	characteristic
186	C₂²	C₂×#14523232	1	1	yes	characteristic
187	C₂²	C₂×#14523232	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	C₂²⋊(C₁₂₇⋊C₈)	1	4	yes	complemented
193	C₄	#16861183	1	2	yes	—
194	C₄	#16861188	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	#22549736	1	1	yes	characteristic
197	C₂	C₂×C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
198	C₂	#22549840	1	1	yes	characteristic
199	C₂	C₂×C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
200	C₂	#22549900	1	1	yes	characteristic
201	C₂	#22549929	1	1	yes	characteristic
202	C₂	#22549933	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₂²×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×C₄²	D₁₂₇
hypocenter	C₁₂₇	#1797
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
3. #22549631	C₂	semidirect
27. C₂²×C₄×C₁₂₇	C₈	semidirect
78. C₂²×C₁₂₇	C₄⋊C₈	semidirect
90. C₄×C₁₂₇	C₂²⋊C₈	semidirect
102. C₁₂₇	#1797	semidirect
126. C₂²×C₄	C₁₂₇⋊C₈	semidirect
180. C₂²	C₄⋊(C₁₂₇⋊C₈)	semidirect
192. C₄	C₂²⋊(C₁₂₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
#22549631	2
C₂²×C₄²×C₁₂₇	2
#1797	127