Subgroups — #30647677

Order 16256.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550433	C₂	1	1	yes	maximal, characteristic
2	#22550006	C₂	1	2	yes	maximal, complemented
3	#22550004	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₅₄	C₂	1	2	yes	maximal, complemented
5	#22549904	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#22549834	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	#16861209	C₄	1	1	yes	characteristic
11	#16861209	C₄	1	1	yes	characteristic
12	#16861201	C₄	1	2	yes	—
13	#16861188	C₄	1	2	yes	—
14	#16861188	C₄	1	2	yes	—
15	C₄×Dic₂₅₄	C₂²	1	2	yes	—
16	C₄×Dic₂₅₄	C₂²	1	2	yes	—
17	#16861168	C₂²	1	2	yes	—
18	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
21	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
22	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
23	#16861119	C₂²	1	4	yes	—
24	C₈×Dic₁₂₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
28	#97×C₁₂₇	C₂²	1	2	yes	—
29	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
30	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₂₇⋊C₈	—	2	4	no	—
33	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
35	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
36	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
39	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
40	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₂₇	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₂₇	—	2	2	no	—
47	M₁₆×C₁₂₇	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
50	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
54	Dic₂₅₄	—	2	4	no	—
55	Dic₂₅₄	—	4	2	no	—
56	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
58	C₁₂₇⋊C₈	—	2	2	no	—
59	C₁₂₇⋊C₈	—	2	2	no	—
60	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	4	1	no	—
63	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₁₂₇	#41	1	2	yes	—
70	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
71	C₈×C₁₂₇	—	2	2	no	—
72	C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₁₂₇	—	2	1	no	—
74	C₂²×C₁₂₇	—	2	1	no	—
75	C₂²×C₁₂₇	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₂₇	—	2	1	no	—
78	C₄×C₁₂₇	—	2	1	no	—
79	C₄×C₁₂₇	—	2	1	no	—
80	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
81	C₄×C₁₂₇	#131	1	2	yes	—
82	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
84	Dic₁₂₇	—	2	4	no	—
85	Dic₁₂₇	—	2	4	no	—
86	C₂×C₁₂₇	—	2	2	no	—
87	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
89	C₂×C₁₂₇	#568	1	1	yes	characteristic
90	#2962	—	127	1	no	maximal, Sylow
91	C₁₂₇	#2962	1	1	yes	characteristic, complemented, Sylow
92	#533	—	127	1	no	—
93	#446	—	127	4	no	—
94	#446	—	127	2	no	—
95	C₈×Q₈	—	127	2	no	—
96	#424	—	127	2	no	—
97	C₂×C₄⋊C₈	—	127	1	no	—
98	#405	—	127	2	no	—
99	C₄×M₁₆	D₁₂₇	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
102	C₂²×C₈	—	127	2	no	—
103	C₄⋊Q₈	—	127	1	no	—
104	C₄⋊Q₈	—	127	1	no	—
105	#125	—	127	2	no	—
106	C₂²⋊Q₈	—	127	2	no	—
107	C₂²⋊Q₈	—	127	2	no	—
108	C₄×Q₈	—	127	2	no	—
109	C₄×Q₈	—	127	2	no	—
110	#117	—	127	2	no	—
111	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
112	C₄⋊C₈	—	127	4	no	—
113	C₄⋊C₈	—	127	4	no	—
114	C₄⋊C₈	—	127	1	no	—
115	C₄⋊C₈	—	127	1	no	—
116	C₄⋊C₈	—	127	2	no	—
117	#97	—	127	4	no	—
118	#97	C₂×D₁₂₇	1	2	yes	—
119	C₄×C₈	—	127	4	no	—
120	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
121	C₂×Q₈	—	127	2	no	—
122	C₂×Q₈	—	127	2	no	—
123	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₂₅₄	1	4	yes	complemented
128	C₂×C₈	—	127	2	no	—
129	C₂×C₈	—	127	2	no	—
130	C₂×C₈	—	254	4	no	—
131	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
132	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
133	C₄⋊C₄	—	127	4	no	—
134	C₄⋊C₄	—	127	4	no	—
135	C₄⋊C₄	—	127	2	no	—
136	C₄⋊C₄	—	127	2	no	—
137	C₄⋊C₄	—	127	2	no	—
138	C₄⋊C₄	—	127	2	no	—
139	C₂²⋊C₄	—	127	4	no	—
140	C₄²	—	127	4	no	—
141	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
142	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
143	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
144	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
146	Q₈	—	508	2	no	—
147	Q₈	—	254	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	127	4	no	—
151	C₂×C₄	—	127	4	no	—
152	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
158	C₂×C₄	#14523248	1	2	yes	—
159	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	254	2	no	—
162	C₈	—	254	2	no	—
163	C₈	C₂×Dic₂₅₄	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₂₅₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	254	4	no	—
172	C₄	—	254	4	no	—
173	C₄	C₄×Dic₂₅₄	1	2	yes	—
174	C₄	#16861217	1	2	yes	—
175	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
176	C₄	C₂×#14523248	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₂₅₄	1	1	yes	characteristic
179	C₂	C₂×#16861217	1	1	yes	characteristic
180	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₄×M₁₆×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#2962
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22550006	C₂	semidirect
3. #22550004	C₂	semidirect
4. C₈×Dic₂₅₄	C₂	semidirect
47. M₁₆×C₁₂₇	Q₈	semidirect
91. C₁₂₇	#2962	semidirect
127. M₁₆	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
#22550433	2
#22550006	2
#22550004	2
C₈×Dic₂₅₄	2
#22549904	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549834	2
C₄×M₁₆×C₁₂₇	2
#2962	127