Subgroups — #30647364

Order 16256.

392 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549938	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549937	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549936	C₂	1	2	yes	maximal, complemented
5	#22549935	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549934	C₂	1	2	yes	maximal, complemented
7	#22549898	C₂	1	2	yes	maximal, complemented
8	#22549838	C₂	1	2	yes	maximal
9	#22549828	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#14523231	C₂²	1	2	yes	—
14	C₂×#14523231	C₂²	1	2	yes	—
15	#16861168	C₂²	1	1	yes	characteristic
16	#16861167	C₄	1	2	yes	—
17	#16861166	C₂²	1	1	yes	characteristic
18	#16861166	C₂²	1	1	yes	characteristic
19	#16861165	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₂₇	C₄	1	2	yes	—
21	C₂²⋊C₄×D₁₂₇	C₄	1	2	yes	—
22	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
25	Dic₁₂₇⋊C₈	C₂²	1	2	yes	complemented
26	Dic₁₂₇⋊C₈	C₂²	1	2	yes	complemented
27	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
30	#16861120	C₂²	1	1	yes	characteristic, complemented
31	#16861119	C₂²	1	2	yes	complemented
32	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
34	#117×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
37	#97×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₂₇	—	2	4	no	—
42	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
48	#14523231	—	2	4	no	—
49	#14523231	—	2	4	no	—
50	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
51	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
52	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
53	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₂₇	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	2	yes	—
59	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
64	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂²×D₁₂₇	—	2	2	no	—
70	C₂²×D₁₂₇	—	4	2	no	—
71	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₂₇	—	2	2	no	—
74	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₁₂₇	—	2	2	no	—
79	C₄×D₁₂₇	—	2	2	no	—
80	C₄×D₁₂₇	—	2	2	no	—
81	C₄×D₁₂₇	—	4	2	no	—
82	C₄×D₁₂₇	—	4	2	no	—
83	C₄×D₁₂₇	#41	1	4	yes	—
84	C₁₂₇⋊C₈	—	2	2	no	—
85	C₁₂₇⋊C₈	—	2	2	no	—
86	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₂₇	—	2	2	no	—
93	C₈×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	2	1	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₂₇	—	2	2	no	—
98	C₂×D₁₂₇	—	2	1	no	—
99	C₂×D₁₂₇	—	2	2	no	—
100	C₂×D₁₂₇	—	4	4	no	—
101	C₂×D₁₂₇	#117	1	2	yes	—
102	C₄×C₁₂₇	—	2	1	no	—
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	—	2	2	no	—
105	C₄×C₁₂₇	C₂×#41	1	2	yes	—
106	Dic₁₂₇	—	2	1	no	—
107	Dic₁₂₇	—	2	2	no	—
108	Dic₁₂₇	—	2	2	no	—
109	Dic₁₂₇	#117	1	2	yes	—
110	C₂×C₁₂₇	—	4	1	no	—
111	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
112	C₂×C₁₂₇	#568	1	2	yes	—
113	D₁₂₇	—	2	2	no	—
114	D₁₂₇	—	4	1	no	—
115	#2921	—	127	1	no	maximal, Sylow
116	C₁₂₇	#2921	1	1	yes	characteristic, complemented, Sylow
117	C₂×#117	—	127	1	no	—
118	#433	—	127	1	no	—
119	#433	—	127	2	no	—
120	#433	—	127	2	no	—
121	#433	—	127	1	no	—
122	#433	—	127	1	no	—
123	#433	D₁₂₇	1	1	yes	characteristic, complemented
124	#423	—	127	2	no	—
125	#407	—	127	2	no	—
126	C₄×M₁₆	—	127	1	no	—
127	C₄×M₁₆	—	127	1	no	—
128	C₂³×C₄	—	127	1	no	—
129	C₂×M₁₆	—	127	2	no	—
130	C₂×M₁₆	—	127	2	no	—
131	#117	—	127	1	no	—
132	#117	—	127	2	no	—
133	#117	—	127	2	no	—
134	#117	—	127	1	no	—
135	#117	—	127	1	no	—
136	#117	C₂×D₁₂₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	127	2	no	—
138	C₂×C₂²⋊C₄	—	127	2	no	—
139	C₂×C₄²	—	127	1	no	—
140	C₂×C₄²	—	127	1	no	—
141	C₄⋊C₈	—	127	2	no	—
142	C₄⋊C₈	—	127	2	no	—
143	C₄⋊C₈	—	127	2	no	—
144	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	complemented
145	C₂²⋊C₈	—	127	2	no	—
146	C₂²⋊C₈	—	127	2	no	—
147	C₂²⋊C₈	—	127	2	no	—
148	C₂²⋊C₈	C₂×D₁₂₇	1	2	yes	—
149	#97	—	127	2	no	—
150	#97	—	127	1	no	—
151	#97	C₂×D₁₂₇	1	1	yes	characteristic, complemented
152	C₄×C₈	—	127	2	no	—
153	C₄×C₈	—	127	1	no	—
154	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
155	C₂⁴	—	127	1	no	—
156	C₂²×C₄	—	127	1	no	—
157	C₂²×C₄	—	127	1	no	—
158	C₂²×C₄	—	127	2	no	—
159	C₂²×C₄	—	127	2	no	—
160	C₂²×C₄	—	127	1	no	—
161	C₂²×C₄	—	127	1	no	—
162	C₂²×C₄	—	127	1	no	—
163	C₂²×C₄	—	254	4	no	—
164	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
165	M₁₆	—	254	4	no	—
166	M₁₆	—	254	4	no	—
167	C₂×C₈	—	127	2	no	—
168	C₂×C₈	—	127	2	no	—
169	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
170	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
171	C₄⋊C₄	—	127	2	no	—
172	C₄⋊C₄	—	127	2	no	—
173	C₄⋊C₄	—	127	2	no	—
174	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
175	C₂²⋊C₄	—	127	2	no	—
176	C₂²⋊C₄	—	127	2	no	—
177	C₂²⋊C₄	—	127	2	no	—
178	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
179	C₄²	—	127	2	no	—
180	C₄²	—	127	2	no	—
181	C₄²	—	127	1	no	—
182	C₄²	—	127	1	no	—
183	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
184	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
185	C₂³	—	127	1	no	—
186	C₂³	—	127	1	no	—
187	C₂³	—	508	2	no	—
188	C₂³	—	254	2	no	—
189	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	127	1	no	—
192	C₂×C₄	—	127	4	no	—
193	C₂×C₄	—	127	2	no	—
194	C₂×C₄	—	127	2	no	—
195	C₂×C₄	—	127	1	no	—
196	C₂×C₄	—	508	2	no	—
197	C₂×C₄	—	508	2	no	—
198	C₂×C₄	—	254	2	no	—
199	C₂×C₄	—	254	2	no	—
200	C₂×C₄	—	254	2	no	—
201	C₂×C₄	—	254	2	no	—
202	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	254	2	no	—
209	C₈	—	254	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	127	2	no	—
213	C₂²	—	508	4	no	—
214	C₂²	—	254	1	no	—
215	C₂²	—	254	2	no	—
216	C₂²	—	254	2	no	—
217	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	127	2	no	—
222	C₄	—	254	1	no	—
223	C₄	—	254	2	no	—
224	C₄	—	254	2	no	—
225	C₄	#41×D₁₂₇	1	2	yes	—
226	C₂	—	4	1	no	—
227	C₂	—	508	1	no	—
228	C₂	—	254	2	no	—
229	C₂	#117×D₁₂₇	1	1	yes	characteristic
230	C₂	#22550641	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#433×C₁₂₇	C₂
prime radical	C₂³×D₁₂₇	C₂×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#433	D₁₂₇
hypocenter	C₁₂₇	#2921
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #22549938	C₂	semidirect
3. #22549937	C₂	semidirect
4. #22549936	C₂	semidirect
5. #22549935	C₂	semidirect
6. #22549934	C₂	semidirect
7. #22549898	C₂	semidirect
9. #22549828	C₂	semidirect
10. C₄×#14523231	C₂	semidirect
11. #433×C₁₂₇	C₂	semidirect
24. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
25. Dic₁₂₇⋊C₈	C₂²	semidirect
26. Dic₁₂₇⋊C₈	C₂²	semidirect
30. #16861120	C₂²	semidirect
31. #16861119	C₂²	semidirect
32. C₄×C₁₂₇⋊C₈	C₂²	semidirect
33. C₈×Dic₁₂₇	C₂²	semidirect
35. C₄⋊C₈×C₁₂₇	C₂²	semidirect
37. #97×C₁₂₇	C₂²	semidirect
38. C₄×C₈×C₁₂₇	C₂²	semidirect
116. C₁₂₇	#2921	semidirect
123. #433	D₁₂₇	semidirect
144. C₄⋊C₈	C₂×D₁₂₇	semidirect
151. #97	C₂×D₁₂₇	semidirect
154. C₄×C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#117×D₁₂₇	2
#22549938	2
#22549937	2
#22549936	2
#22549935	2
#22549934	2
#22549898	2
#22549838	2
#22549828	2
C₄×#14523231	2
#433×C₁₂₇	2
#2921	127