Subgroups — #30647706

Order 16256.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550007	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549954	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549948	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549937	C₂	1	2	yes	maximal, complemented
8	#22549933	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549832	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14523232	C₂²	1	2	yes	—
13	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₁₂₇⋊C₈	C₂²	1	2	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	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
21	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
22	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
23	#16861120	C₂²	1	2	yes	—
24	#16861120	C₂²	1	2	yes	—
25	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
30	#126×C₁₂₇	C₄	1	2	yes	—
31	#125×C₁₂₇	C₄	1	1	yes	characteristic
32	#124×C₁₂₇	C₄	1	1	yes	characteristic
33	#123×C₁₂₇	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
35	#121×C₁₂₇	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
38	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
39	#117×C₁₂₇	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
41	#14523232	—	2	4	no	—
42	C₂×C₁₂₇⋊C₈	—	2	2	no	—
43	C₂×C₁₂₇⋊C₈	—	2	4	no	—
44	C₂×C₁₂₇⋊C₈	—	2	2	no	—
45	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
62	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
69	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂³	1	2	yes	—
74	C₁₂₇⋊C₈	—	2	1	no	—
75	C₁₂₇⋊C₈	—	2	1	no	—
76	C₁₂₇⋊C₈	—	2	2	no	—
77	C₁₂₇⋊C₈	—	2	2	no	—
78	C₁₂₇⋊C₈	#41	1	2	yes	—
79	C₁₂₇⋊C₈	#41	1	2	yes	—
80	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	Q₈×C₁₂₇	—	2	2	no	—
83	D₄×C₁₂₇	—	2	4	no	—
84	D₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	2	no	—
87	C₂×C₄×C₁₂₇	—	2	1	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	2	2	no	—
90	C₂×C₄×C₁₂₇	—	2	2	no	—
91	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
100	C₂²×C₁₂₇	—	2	1	no	—
101	C₂²×C₁₂₇	—	2	1	no	—
102	C₂²×C₁₂₇	—	2	2	no	—
103	C₂²×C₁₂₇	—	2	2	no	—
104	C₂²×C₁₂₇	—	2	2	no	—
105	C₂²×C₁₂₇	#131	1	2	yes	—
106	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₁₂₇	—	2	1	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	2	1	no	—
111	C₄×C₁₂₇	—	2	1	no	—
112	C₄×C₁₂₇	—	2	2	no	—
113	C₄×C₁₂₇	—	2	2	no	—
114	C₄×C₁₂₇	#131	1	2	yes	—
115	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₂₇	—	2	2	no	—
118	C₂×C₁₂₇	—	4	2	no	—
119	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
121	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
122	#2966	—	127	1	no	maximal, Sylow
123	C₁₂₇	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	D₁₂₇	1	1	yes	characteristic
125	#446	—	127	1	no	—
126	C₈×Q₈	—	127	1	no	—
127	#436	—	127	1	no	—
128	#435	—	127	2	no	—
129	C₈×D₄	—	127	1	no	—
130	C₈×D₄	—	127	2	no	—
131	#433	—	127	2	no	—
132	#432	—	127	1	no	—
133	#405	—	127	2	no	—
134	C₂×C₄×C₈	—	127	1	no	—
135	C₂×M₁₆	—	127	2	no	—
136	C₂²×C₈	—	127	1	no	—
137	C₂²×C₈	—	127	1	no	—
138	C₂²×C₈	—	127	2	no	—
139	#126	Dic₁₂₇	1	2	yes	—
140	#125	Dic₁₂₇	1	1	yes	characteristic
141	#124	Dic₁₂₇	1	1	yes	characteristic
142	#123	Dic₁₂₇	1	2	yes	—
143	C₂²⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
144	#121	Dic₁₂₇	1	1	yes	characteristic
145	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
146	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
147	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
148	#117	C₂×D₁₂₇	1	2	yes	—
149	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
150	C₄⋊C₈	—	127	1	no	—
151	C₄⋊C₈	—	127	1	no	—
152	C₄⋊C₈	—	127	2	no	—
153	C₄⋊C₈	—	127	2	no	—
154	C₂²⋊C₈	—	127	2	no	—
155	C₂²⋊C₈	—	127	2	no	—
156	C₂²⋊C₈	—	127	2	no	—
157	#97	—	127	2	no	—
158	#97	—	127	2	no	—
159	C₄×C₈	—	127	1	no	—
160	C₄×C₈	—	127	1	no	—
161	C₄×C₈	—	127	1	no	—
162	C₄×C₈	—	127	1	no	—
163	C₄×C₈	—	127	2	no	—
164	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
165	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
166	C₂×D₄	C₂×Dic₁₂₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
171	M₁₆	—	254	4	no	—
172	C₂×C₈	—	127	1	no	—
173	C₂×C₈	—	127	1	no	—
174	C₂×C₈	—	127	1	no	—
175	C₂×C₈	—	127	1	no	—
176	C₂×C₈	—	127	2	no	—
177	C₂×C₈	—	127	2	no	—
178	C₂×C₈	—	254	2	no	—
179	C₂×C₈	—	254	4	no	—
180	C₂×C₈	—	254	2	no	—
181	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
182	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
185	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
190	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
191	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
192	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
194	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
195	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
196	C₄²	C₂²×D₁₂₇	1	2	yes	—
197	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
198	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
213	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
214	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
215	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
217	C₈	—	127	2	no	—
218	C₈	—	127	2	no	—
219	C₈	—	254	1	no	—
220	C₈	—	254	1	no	—
221	C₈	—	254	2	no	—
222	C₈	—	254	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	#16861216	1	2	yes	—
229	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	#16861216	1	2	yes	—
238	C₄	#41×D₁₂₇	1	1	yes	characteristic
239	C₄	#41×D₁₂₇	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	#41×Dic₁₂₇	1	1	yes	characteristic
243	C₂	C₂×#16861216	1	1	yes	characteristic
244	C₂	C₂×#16861216	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#529×C₁₂₇	C₂
prime radical	#121×C₁₂₇	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#529	D₁₂₇
hypocenter	C₁₂₇	#2966
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. Q₈×C₁₂₇⋊C₈	C₂	semidirect
3. #22549954	C₂	semidirect
4. #22549948	C₂	semidirect
5. D₄×C₁₂₇⋊C₈	C₂	semidirect
6. D₄×C₁₂₇⋊C₈	C₂	semidirect
7. #22549937	C₂	semidirect
8. #22549933	C₂	semidirect
9. #22549832	C₂	semidirect
10. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
123. C₁₂₇	#2966	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
#22549954	2
#22549948	2
D₄×C₁₂₇⋊C₈	2
#22549937	2
#22549933	2
#22549832	2
C₂×C₄×C₁₂₇⋊C₈	2
#529×C₁₂₇	2
#2966	127