Subgroups — #30647679

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	#22550007	C₂	1	2	yes	maximal, complemented
2	#22550007	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	#22549903	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#22549832	C₂	1	2	yes	maximal
7	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523232	C₂²	1	1	yes	characteristic
10	C₂×#14523232	C₂²	1	1	yes	characteristic
11	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
16	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
17	#16861120	C₂²	1	2	yes	—
18	#16861120	C₂²	1	4	yes	—
19	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
20	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
23	#125×C₁₂₇	C₄	1	2	yes	—
24	C₂²⋊Q₈×C₁₂₇	C₄	1	2	yes	—
25	C₂²⋊Q₈×C₁₂₇	C₄	1	2	yes	—
26	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
28	#117×C₁₂₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
30	#14523232	—	2	2	no	—
31	#14523232	Q₈	1	4	yes	complemented
32	C₂×C₁₂₇⋊C₈	—	2	4	no	—
33	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
36	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
37	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
45	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₄²×C₁₂₇	C₂³	1	4	yes	—
54	C₁₂₇⋊C₈	—	2	2	no	—
55	C₁₂₇⋊C₈	—	2	2	no	—
56	C₁₂₇⋊C₈	—	2	2	no	—
57	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
58	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	Q₈×C₁₂₇	—	2	4	no	—
60	Q₈×C₁₂₇	—	4	2	no	—
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	4	yes	—
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₂₇	#41	1	2	yes	—
72	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
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	4	no	—
79	C₄×C₁₂₇	—	2	1	no	—
80	C₄×C₁₂₇	—	2	1	no	—
81	C₄×C₁₂₇	—	2	4	no	—
82	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
83	C₄×C₁₂₇	#131	1	2	yes	—
84	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
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	D₁₂₇	1	1	yes	characteristic
93	#446	—	127	2	no	—
94	#446	—	127	4	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	—	127	1	no	—
100	C₂×M₁₆	—	127	1	no	—
101	C₂×M₁₆	—	127	1	no	—
102	C₂²×C₈	—	127	2	no	—
103	C₄⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
104	C₄⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
105	#125	Dic₁₂₇	1	2	yes	—
106	C₂²⋊Q₈	Dic₁₂₇	1	2	yes	—
107	C₂²⋊Q₈	Dic₁₂₇	1	2	yes	—
108	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
109	C₄×Q₈	C₂×D₁₂₇	1	2	yes	—
110	#117	C₂×D₁₂₇	1	2	yes	—
111	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
112	C₄⋊C₈	—	127	1	no	—
113	C₄⋊C₈	—	127	1	no	—
114	C₄⋊C₈	—	127	2	no	—
115	C₄⋊C₈	—	127	4	no	—
116	C₄⋊C₈	—	127	4	no	—
117	#97	—	127	2	no	—
118	#97	—	127	4	no	—
119	C₄×C₈	—	127	2	no	—
120	C₄×C₈	—	127	4	no	—
121	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
122	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
123	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
126	M₁₆	—	127	4	no	—
127	M₁₆	—	254	2	no	—
128	C₂×C₈	—	127	2	no	—
129	C₂×C₈	—	127	2	no	—
130	C₂×C₈	—	127	2	no	—
131	C₂×C₈	—	127	2	no	—
132	C₂×C₈	—	254	4	no	—
133	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
135	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
136	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
137	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
139	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
140	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
141	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
142	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
143	C₄²	C₂²×D₁₂₇	1	4	yes	—
144	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
145	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
154	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
155	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
156	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
158	C₂×C₄	#14523249	1	2	yes	—
159	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
160	C₈	—	127	4	no	—
161	C₈	—	254	2	no	—
162	C₈	—	254	2	no	—
163	C₈	—	254	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×Dic₁₂₇	1	2	yes	—
167	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	Q₈×Dic₁₂₇	1	2	yes	—
174	C₄	#16861216	1	2	yes	—
175	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
176	C₄	C₂×#14523249	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₁₂₇	1	1	yes	characteristic
179	C₂	C₂×#16861216	1	1	yes	characteristic
180	C₂	#22550643	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#533×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#533	D₁₂₇
hypocenter	C₁₂₇	#2962
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. #22550007	C₂	semidirect
3. Q₈×C₁₂₇⋊C₈	C₂	semidirect
31. #14523232	Q₈	semidirect
91. C₁₂₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#22550007	2
Q₈×C₁₂₇⋊C₈	2
#22549903	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549832	2
C₄×#14523232	2
#533×C₁₂₇	2
#2962	127