Subgroups — #30647826

Order 16256.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550430	C₂	1	1	yes	maximal, characteristic
2	#22550006	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550004	C₂	1	2	yes	maximal, complemented
4	#22550003	C₂	1	2	yes	maximal, complemented
5	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550000	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549938	C₂	1	2	yes	maximal
8	#22549936	C₂	1	2	yes	maximal
9	#22549932	C₂	1	1	yes	maximal, characteristic
10	#22549927	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16861208	C₄	1	1	yes	characteristic
13	#16861207	C₄	1	1	yes	characteristic
14	#16861199	C₄	1	2	yes	—
15	#16861189	C₄	1	1	yes	characteristic
16	#16861187	C₄	1	2	yes	—
17	#16861185	C₄	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₅₄	C₂²	1	2	yes	—
20	#16861166	C₂²	1	2	yes	—
21	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
24	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
27	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₂₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
31	#16861119	C₂²	1	2	yes	—
32	#16861119	C₂²	1	2	yes	—
33	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₁₂₇	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
39	#97×C₁₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
49	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
51	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₂₇	C₂³	1	2	yes	—
61	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₄²×C₁₂₇	C₂³	1	2	yes	—
70	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
71	Dic₂₅₄	—	2	2	no	—
72	Dic₂₅₄	—	4	2	no	—
73	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
78	C₄×D₁₂₇	—	2	2	no	—
79	C₄×D₁₂₇	—	2	1	no	—
80	C₄×D₁₂₇	—	2	1	no	—
81	C₄×D₁₂₇	—	4	1	no	—
82	C₁₂₇⋊C₈	—	2	1	no	—
83	C₁₂₇⋊C₈	—	2	1	no	—
84	C₁₂₇⋊C₈	—	2	2	no	—
85	Q₈×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₂₇	—	2	1	no	—
92	C₈×C₁₂₇	—	2	2	no	—
93	C₈×C₁₂₇	—	2	1	no	—
94	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₂₇	—	2	1	no	—
96	C₂×D₁₂₇	—	2	1	no	—
97	C₂×D₁₂₇	—	2	1	no	—
98	C₄×C₁₂₇	—	2	2	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	2	no	—
101	C₄×C₁₂₇	#131	1	2	yes	—
102	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₁₂₇	—	2	1	no	—
105	Dic₁₂₇	—	2	1	no	—
106	Dic₁₂₇	—	2	2	no	—
107	Dic₁₂₇	—	2	1	no	—
108	Dic₁₂₇	—	2	2	no	—
109	C₂×C₁₂₇	#519	1	1	yes	characteristic
110	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
111	C₂×C₁₂₇	#568	1	1	yes	characteristic
112	D₁₂₇	—	4	1	no	—
113	#2985	—	127	1	no	maximal, Sylow
114	C₁₂₇	#2985	1	1	yes	characteristic, complemented, Sylow
115	#533	—	127	1	no	—
116	#446	—	127	2	no	—
117	#446	—	127	2	no	—
118	#446	—	127	1	no	—
119	#446	D₁₂₇	1	1	yes	characteristic, complemented
120	C₈×Q₈	—	127	1	no	—
121	C₈×Q₈	—	127	1	no	—
122	#433	—	127	2	no	—
123	#433	—	127	2	no	—
124	#432	—	127	1	no	—
125	#431	—	127	1	no	—
126	C₄⋊Q₈	—	127	1	no	—
127	C₄⋊Q₈	—	127	1	no	—
128	#125	—	127	2	no	—
129	C₂²⋊Q₈	—	127	1	no	—
130	C₂²⋊Q₈	—	127	2	no	—
131	C₂²⋊Q₈	—	127	1	no	—
132	C₄×Q₈	—	127	1	no	—
133	C₄×Q₈	—	127	2	no	—
134	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
135	#117	—	127	2	no	—
136	C₂×C₄²	—	127	1	no	—
137	C₄⋊C₈	—	127	1	no	—
138	C₄⋊C₈	—	127	1	no	—
139	C₄⋊C₈	—	127	2	no	—
140	C₄⋊C₈	—	127	2	no	—
141	C₄⋊C₈	—	127	1	no	—
142	C₄⋊C₈	—	127	2	no	—
143	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
145	C₂²⋊C₈	—	127	1	no	—
146	C₂²⋊C₈	—	127	1	no	—
147	C₂²⋊C₈	—	127	2	no	—
148	#97	—	127	2	no	—
149	#97	—	127	2	no	—
150	#97	C₂×D₁₂₇	1	2	yes	—
151	C₄×C₈	—	127	2	no	—
152	C₄×C₈	—	127	1	no	—
153	C₄×C₈	—	127	2	no	—
154	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
155	C₂×Q₈	—	127	2	no	—
156	C₂×Q₈	—	127	1	no	—
157	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	—	127	1	no	—
159	C₂²×C₄	—	127	1	no	—
160	C₂²×C₄	—	127	1	no	—
161	C₂×C₈	—	127	1	no	—
162	C₂×C₈	—	127	1	no	—
163	C₂×C₈	—	127	2	no	—
164	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
165	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
167	C₄⋊C₄	—	127	2	no	—
168	C₄⋊C₄	—	127	2	no	—
169	C₄⋊C₄	—	127	2	no	—
170	C₄⋊C₄	—	127	2	no	—
171	C₄⋊C₄	—	127	2	no	—
172	C₄⋊C₄	—	127	2	no	—
173	C₄⋊C₄	—	127	1	no	—
174	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
175	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
176	C₂²⋊C₄	—	127	2	no	—
177	C₂²⋊C₄	—	127	2	no	—
178	C₄²	—	127	1	no	—
179	C₄²	—	127	1	no	—
180	C₄²	—	127	2	no	—
181	C₄²	—	127	1	no	—
182	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₄²	C₂²×D₁₂₇	1	2	yes	—
184	C₂³	—	127	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	508	2	no	—
187	Q₈	—	254	2	no	—
188	C₂×C₄	—	127	1	no	—
189	C₂×C₄	—	127	1	no	—
190	C₂×C₄	—	127	1	no	—
191	C₂×C₄	—	127	2	no	—
192	C₂×C₄	—	127	2	no	—
193	C₂×C₄	—	508	1	no	—
194	C₂×C₄	—	254	2	no	—
195	C₂×C₄	—	254	1	no	—
196	C₂×C₄	—	254	1	no	—
197	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	254	1	no	—
206	C₈	—	254	1	no	—
207	C₈	—	254	2	no	—
208	C₂²	—	254	1	no	—
209	C₂²	—	254	1	no	—
210	C₂²	—	254	1	no	—
211	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	254	1	no	—
216	C₄	—	254	1	no	—
217	C₄	—	254	2	no	—
218	C₄	—	254	1	no	—
219	C₄	—	254	2	no	—
220	C₄	#16861217	1	2	yes	—
221	C₄	#16861256	1	1	yes	characteristic
222	C₄	#16861257	1	1	yes	characteristic
223	C₂	—	508	1	no	—
224	C₂	#22550352	1	1	yes	characteristic
225	C₂	C₂×#16861217	1	1	yes	characteristic
226	C₂	#22550641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#446×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#446	D₁₂₇
hypocenter	C₁₂₇	#2985
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

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. #22550003	C₂	semidirect
5. Q₈×C₁₂₇⋊C₈	C₂	semidirect
6. #22550000	C₂	semidirect
11. #446×C₁₂₇	C₂	semidirect
114. C₁₂₇	#2985	semidirect
119. #446	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550430	2
#22550006	2
#22550004	2
#22550003	2
Q₈×C₁₂₇⋊C₈	2
#22550000	2
#22549938	2
#22549936	2
#22549932	2
#22549927	2
#446×C₁₂₇	2
#2985	127