Subgroups — #30647682

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22550007	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550005	C₂	1	2	yes	maximal, complemented
4	#22550004	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550000	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549898	C₂	1	2	yes	maximal
8	#22549898	C₂	1	1	yes	maximal, characteristic
9	#22549828	C₂	1	2	yes	maximal
10	C₄×#14523231	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₂₇	C₄	1	1	yes	characteristic
13	C₂×#14523231	C₂²	1	1	yes	characteristic
14	C₂×#14523231	C₂²	1	2	yes	—
15	C₂×#14523231	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₂₇	C₄	1	1	yes	characteristic
17	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₅₄	C₂²	1	2	yes	—
19	#16861174	C₄	1	2	yes	—
20	#16861174	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₂₇	C₄	1	2	yes	—
22	C₄⋊C₄×D₁₂₇	C₄	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
24	C₄²×D₁₂₇	C₂²	1	2	yes	—
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
28	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
29	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
30	#16861120	C₂²	1	2	yes	—
31	#16861119	C₂²	1	1	yes	characteristic
32	#16861119	C₂²	1	1	yes	characteristic
33	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₂₇	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	2	yes	—
42	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₁₂₇	—	2	4	no	—
44	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
49	#14523231	—	2	2	no	—
50	#14523231	—	2	4	no	—
51	#14523231	Q₈	1	4	yes	complemented
52	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
57	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₂₇	C₂³	1	2	yes	—
65	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂³	1	2	yes	—
74	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
75	Dic₂₅₄	—	2	4	no	—
76	Dic₂₅₄	M₁₆	1	4	yes	complemented
77	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
82	C₄×D₁₂₇	—	2	2	no	—
83	C₄×D₁₂₇	—	2	4	no	—
84	C₄×D₁₂₇	—	4	1	no	—
85	C₄×D₁₂₇	—	4	2	no	—
86	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
87	C₄×D₁₂₇	#41	1	2	yes	—
88	C₁₂₇⋊C₈	—	2	2	no	—
89	C₁₂₇⋊C₈	—	2	1	no	—
90	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	complemented
91	Q₈×C₁₂₇	—	2	2	no	—
92	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₂₇	—	2	1	no	—
98	C₈×C₁₂₇	—	2	2	no	—
99	C₈×C₁₂₇	C₂×Q₈	1	2	yes	complemented
100	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₂₇	—	2	1	no	—
102	C₂×D₁₂₇	—	2	1	no	—
103	C₂×D₁₂₇	C₄×Q₈	1	2	yes	—
104	C₄×C₁₂₇	—	2	2	no	—
105	C₄×C₁₂₇	—	2	1	no	—
106	C₄×C₁₂₇	—	2	2	no	—
107	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
108	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
110	Dic₁₂₇	—	2	1	no	—
111	Dic₁₂₇	—	2	2	no	—
112	Dic₁₂₇	—	2	1	no	—
113	Dic₁₂₇	C₄×Q₈	1	2	yes	—
114	Dic₁₂₇	C₂×M₁₆	1	4	yes	—
115	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×C₁₂₇	#568	1	1	yes	characteristic
118	D₁₂₇	—	2	2	no	—
119	Q₈×M₁₆	—	127	1	no	maximal, Sylow
120	C₁₂₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	127	1	no	—
122	#446	—	127	1	no	—
123	#446	—	127	2	no	—
124	#446	—	127	2	no	—
125	#446	D₁₂₇	1	1	yes	characteristic, complemented
126	C₈×Q₈	—	127	1	no	—
127	C₈×Q₈	—	127	1	no	—
128	#423	—	127	2	no	—
129	#423	—	127	1	no	—
130	C₄×M₁₆	—	127	2	no	—
131	C₄×M₁₆	—	127	1	no	—
132	C₂²×Q₈	—	127	1	no	—
133	C₂×M₁₆	—	127	1	no	—
134	C₂×M₁₆	—	127	2	no	—
135	C₂×M₁₆	—	127	1	no	—
136	C₄×Q₈	—	127	1	no	—
137	C₄×Q₈	—	127	1	no	—
138	C₄×Q₈	—	127	2	no	—
139	C₄×Q₈	—	127	1	no	—
140	C₄×Q₈	—	127	2	no	—
141	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	127	2	no	—
143	C₂×C₄⋊C₄	—	127	1	no	—
144	C₂×C₄²	—	127	2	no	—
145	C₂×C₄²	—	127	1	no	—
146	C₄⋊C₈	—	127	2	no	—
147	C₄⋊C₈	—	127	2	no	—
148	C₄⋊C₈	—	127	2	no	—
149	C₄⋊C₈	—	127	1	no	—
150	C₄⋊C₈	—	127	2	no	—
151	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
153	#97	—	127	1	no	—
154	#97	—	127	1	no	—
155	#97	—	127	2	no	—
156	#97	C₂×D₁₂₇	1	2	yes	—
157	C₄×C₈	—	127	2	no	—
158	C₄×C₈	—	127	2	no	—
159	C₄×C₈	—	127	1	no	—
160	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
161	C₂×Q₈	—	127	1	no	—
162	C₂×Q₈	—	127	2	no	—
163	C₂×Q₈	—	254	4	no	—
164	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	—	127	1	no	—
166	C₂²×C₄	—	127	2	no	—
167	C₂²×C₄	—	127	1	no	—
168	C₂²×C₄	—	127	2	no	—
169	C₂²×C₄	—	127	1	no	—
170	M₁₆	—	127	4	no	—
171	M₁₆	—	254	4	no	—
172	M₁₆	—	254	2	no	—
173	C₂×C₈	—	127	1	no	—
174	C₂×C₈	—	127	2	no	—
175	C₂×C₈	—	127	1	no	—
176	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
177	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
179	C₄⋊C₄	—	127	2	no	—
180	C₄⋊C₄	—	127	2	no	—
181	C₄⋊C₄	—	127	2	no	—
182	C₄⋊C₄	—	127	2	no	—
183	C₄⋊C₄	—	127	1	no	—
184	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
185	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
186	C₄²	—	127	2	no	—
187	C₄²	—	127	2	no	—
188	C₄²	—	127	1	no	—
189	C₄²	—	127	1	no	—
190	C₄²	—	127	2	no	—
191	C₄²	—	127	1	no	—
192	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
193	C₄²	C₂²×D₁₂₇	1	2	yes	—
194	C₂³	—	127	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	127	4	no	—
197	Q₈	—	254	4	no	—
198	C₂×C₄	—	127	1	no	—
199	C₂×C₄	—	127	2	no	—
200	C₂×C₄	—	127	2	no	—
201	C₂×C₄	—	127	2	no	—
202	C₂×C₄	—	127	2	no	—
203	C₂×C₄	—	127	1	no	—
204	C₂×C₄	—	127	1	no	—
205	C₂×C₄	—	508	1	no	—
206	C₂×C₄	—	508	2	no	—
207	C₂×C₄	—	254	4	no	—
208	C₂×C₄	—	254	2	no	—
209	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	127	2	no	—
217	C₈	—	254	2	no	—
218	C₈	—	254	1	no	—
219	C₈	Q₈×D₁₂₇	1	2	yes	complemented
220	C₂²	—	127	2	no	—
221	C₂²	—	254	1	no	—
222	C₂²	—	254	1	no	—
223	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	127	2	no	—
228	C₄	—	127	4	no	—
229	C₄	—	254	1	no	—
230	C₄	—	254	2	no	—
231	C₄	—	254	1	no	—
232	C₄	M₁₆×D₁₂₇	1	2	yes	—
233	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
234	C₄	#41×D₁₂₇	1	1	yes	characteristic
235	C₂	—	254	2	no	—
236	C₂	C₄×Q₈×D₁₂₇	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₂₇	1	1	yes	characteristic
238	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	Q₈×M₁₆
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. #22550007	C₂	semidirect
3. #22550005	C₂	semidirect
4. #22550004	C₂	semidirect
5. C₈×Dic₂₅₄	C₂	semidirect
6. #22550000	C₂	semidirect
11. #446×C₁₂₇	C₂	semidirect
51. #14523231	Q₈	semidirect
76. Dic₂₅₄	M₁₆	semidirect
90. C₁₂₇⋊C₈	C₂×Q₈	semidirect
99. C₈×C₁₂₇	C₂×Q₈	semidirect
120. C₁₂₇	Q₈×M₁₆	semidirect
125. #446	D₁₂₇	semidirect
219. C₈	Q₈×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₂₇	2
#22550007	2
#22550005	2
#22550004	2
C₈×Dic₂₅₄	2
#22550000	2
#22549898	2
#22549828	2
C₄×#14523231	2
#446×C₁₂₇	2
Q₈×M₁₆	127