Subgroups — #30647697

Order 16256.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#22549954	C₂	1	2	yes	maximal, complemented
3	#22549948	C₂	1	4	yes	maximal, complemented
4	#22549929	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549929	C₂	1	2	yes	maximal, complemented
6	C₄×#14523232	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523232	C₂²	1	4	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
15	#16861120	C₂²	1	4	yes	—
16	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
19	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
20	C₂×#41×C₁₂₇	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₂₇	C₄	1	2	yes	—
24	C₄×D₄×C₁₂₇	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
27	#117×C₁₂₇	C₄	1	2	yes	—
28	#117×C₁₂₇	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
31	#14523232	—	2	8	no	—
32	C₂×C₁₂₇⋊C₈	—	2	4	no	—
33	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
35	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
36	#41×C₁₂₇	—	2	4	no	—
37	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
55	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
56	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₄²×C₁₂₇	C₂³	1	2	yes	—
59	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₁₂₇⋊C₈	—	2	2	no	—
61	C₁₂₇⋊C₈	—	2	4	no	—
62	C₁₂₇⋊C₈	#41	1	4	yes	complemented
63	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
64	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	Q₈×C₁₂₇	—	2	2	no	—
66	D₄×C₁₂₇	—	2	4	no	—
67	D₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	complemented
74	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
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	2	no	—
86	C₂²×C₁₂₇	—	2	2	no	—
87	C₂²×C₁₂₇	—	2	2	no	—
88	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
89	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₁₂₇	—	2	1	no	—
91	C₄×C₁₂₇	—	2	2	no	—
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	—	2	1	no	—
94	C₄×C₁₂₇	—	2	2	no	—
95	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
96	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
97	C₄×C₁₂₇	#131	1	2	yes	—
98	C₄×C₁₂₇	C₂×#41	1	2	yes	—
99	C₂×C₁₂₇	—	2	2	no	—
100	C₂×C₁₂₇	—	4	2	no	—
101	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
102	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
104	#2965	—	127	1	no	maximal, Sylow
105	C₁₂₇	#2965	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	D₁₂₇	1	1	yes	characteristic
107	#446	—	127	2	no	—
108	#436	—	127	2	no	—
109	#435	—	127	4	no	—
110	#431	—	127	2	no	—
111	#431	—	127	1	no	—
112	C₄×M₁₆	—	127	2	no	—
113	C₂×C₄×C₈	—	127	1	no	—
114	C₂×#41	Dic₁₂₇	1	1	yes	characteristic
115	C₂×M₁₆	—	127	4	no	—
116	C₂²×C₈	—	127	2	no	—
117	C₄×Q₈	Dic₁₂₇	1	1	yes	characteristic
118	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
119	C₄×D₄	Dic₁₂₇	1	2	yes	—
120	C₄×D₄	Dic₁₂₇	1	1	yes	characteristic
121	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
122	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
123	#117	Dic₁₂₇	1	2	yes	—
124	#117	Dic₁₂₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
126	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
127	C₄⋊C₈	—	127	4	no	—
128	C₄⋊C₈	—	127	2	no	—
129	C₂²⋊C₈	—	127	2	no	—
130	C₂²⋊C₈	—	127	4	no	—
131	#97	—	127	4	no	—
132	C₄×C₈	—	127	2	no	—
133	C₄×C₈	—	127	1	no	—
134	C₄×C₈	—	127	1	no	—
135	C₄×C₈	—	127	2	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
138	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
139	C₂×D₄	C₂×Dic₁₂₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
141	C₂²×C₄	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	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
146	M₁₆	—	254	8	no	—
147	C₂×C₈	—	127	2	no	—
148	C₂×C₈	—	127	2	no	—
149	C₂×C₈	—	127	4	no	—
150	C₂×C₈	—	254	4	no	—
151	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
158	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
159	C₄²	C₂×Dic₁₂₇	1	2	yes	—
160	C₄²	C₂×Dic₁₂₇	1	2	yes	—
161	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
162	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
163	C₄²	C₂²×D₁₂₇	1	2	yes	—
164	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
165	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
166	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	#14523232	1	4	yes	complemented
176	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
178	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
180	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
181	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
185	C₈	—	127	4	no	—
186	C₈	—	254	2	no	—
187	C₈	—	254	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂×#14523232	1	2	yes	—
194	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	C₂×#14523232	1	2	yes	—
201	C₄	C₂×#14523232	1	2	yes	—
202	C₄	#16861216	1	2	yes	—
203	C₄	#41×D₁₂₇	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	4	2	no	—
206	C₂	#41×Dic₁₂₇	1	1	yes	characteristic
207	C₂	C₂²×#14523232	1	1	yes	characteristic
208	C₂	C₂×#16861216	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×#41×C₁₂₇	C₂
prime radical	C₂×#41×C₁₂₇	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#2965
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. #22549954	C₂	semidirect
3. #22549948	C₂	semidirect
4. #22549929	C₂	semidirect
5. #22549929	C₂	semidirect
6. C₄×#14523232	C₂	semidirect
7. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
62. C₁₂₇⋊C₈	#41	semidirect
73. C₂×C₄×C₁₂₇	M₁₆	semidirect
105. C₁₂₇	#2965	semidirect
175. C₂×C₄	#14523232	semidirect

Maximal subgroups

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

subgroup	index
#22550007	2
#22549954	2
#22549948	2
#22549929	2
C₄×#14523232	2
C₂×C₄×C₁₂₇⋊C₈	2
C₄×#41×C₁₂₇	2
#2965	127