Subgroups — #30647436

Order 16256.

492 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	C₂×#16861216	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550383	C₂	1	1	yes	maximal, characteristic
4	#22549951	C₂	1	4	yes	maximal, complemented
5	#22549944	C₂	1	4	yes	maximal, complemented
6	#22549902	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549839	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549834	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
11	#16861216	—	2	4	no	—
12	C₂×#14523232	C₂²	1	1	yes	characteristic
13	C₂×#14523232	C₂²	1	2	yes	complemented
14	C₂×#14523231	—	2	4	no	—
15	C₂×#14523231	C₂²	1	2	yes	—
16	C₂×#14523231	C₂²	1	2	yes	—
17	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
19	#16861190	C₄	1	2	yes	—
20	#16861185	C₄	1	2	yes	—
21	#16861180	C₄	1	2	yes	—
22	#16861177	C₄	1	2	yes	—
23	C₄×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
24	#16861168	C₂²	1	1	yes	characteristic
25	Dic₁₂₇⋊C₈	C₂²	1	4	yes	complemented
26	(C₂×D₁₂₇)⋊C₈	C₂²	1	4	yes	—
27	#16861119	C₂²	1	2	yes	complemented
28	C₈×Dic₁₂₇	C₂²	1	2	yes	complemented
29	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
33	C₂³×D₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	4	yes	—
35	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₂₇	—	2	2	no	—
37	C₂×C₄×D₁₂₇	—	2	2	no	—
38	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
40	#14523232	—	2	4	no	—
41	#14523232	—	2	2	no	—
42	#14523231	—	2	4	no	—
43	#14523231	—	2	4	no	—
44	C₂×C₁₂₇⋊C₈	—	2	4	no	—
45	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	complemented
46	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
50	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
51	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
53	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
54	C₄×Dic₁₂₇	C₂³	1	2	yes	—
55	#41×C₁₂₇	—	2	2	no	—
56	#41×C₁₂₇	—	2	2	no	—
57	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
62	M₁₆×C₁₂₇	—	2	2	no	—
63	C₂×C₈×C₁₂₇	—	2	2	no	—
64	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
65	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
66	C₂²×D₁₂₇	—	2	2	no	—
67	C₂²×D₁₂₇	—	4	2	no	—
68	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂²⋊D₁₂₇	—	4	4	no	—
70	C₂×Dic₁₂₇	—	2	2	no	—
71	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
73	C₄×D₁₂₇	—	2	4	no	—
74	C₄×D₁₂₇	—	2	2	no	—
75	C₄×D₁₂₇	—	2	2	no	—
76	C₁₂₇⋊C₈	—	2	2	no	—
77	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
78	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
79	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₁₂₇	—	2	2	no	—
82	D₄×C₁₂₇	—	2	4	no	—
83	D₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	#41	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₂₇	—	2	2	no	—
92	C₈×C₁₂₇	—	2	2	no	—
93	C₂²×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	2	2	no	—
95	C₂²×C₁₂₇	—	2	4	no	—
96	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
97	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₂₇	—	2	2	no	—
99	C₂×D₁₂₇	—	2	2	no	—
100	C₂×D₁₂₇	—	2	2	no	—
101	C₂×D₁₂₇	—	4	4	no	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
104	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
106	Dic₁₂₇	—	2	2	no	—
107	Dic₁₂₇	—	2	4	no	—
108	C₂×C₁₂₇	—	2	2	no	—
109	C₂×C₁₂₇	—	4	2	no	—
110	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₁₂₇	#568	1	2	yes	—
112	D₁₂₇	—	4	2	no	—
113	#2930	—	127	1	no	maximal, Sylow
114	C₁₂₇	#2930	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	127	1	no	—
116	C₂²×M₁₆	—	127	1	no	—
117	#525	—	127	1	no	—
118	#436	—	127	4	no	—
119	#435	—	127	4	no	—
120	#424	—	127	1	no	—
121	#408	D₁₂₇	1	1	yes	characteristic, complemented
122	#407	—	127	1	no	—
123	#405	—	127	1	no	—
124	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
125	C₂³×C₄	—	127	1	no	—
126	#131	—	254	4	no	—
127	C₂×M₁₆	—	127	1	no	—
128	C₂×M₁₆	—	127	2	no	—
129	C₂×M₁₆	—	127	2	no	—
130	C₂×M₁₆	—	127	2	no	—
131	C₂×M₁₆	—	254	4	no	—
132	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
133	C₂²×C₈	—	127	1	no	—
134	C₂²×C₈	—	127	2	no	—
135	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
136	#123	—	127	2	no	—
137	C₂²⋊Q₈	—	127	2	no	—
138	#121	—	127	2	no	—
139	#120	—	127	2	no	—
140	C₄×D₄	—	127	4	no	—
141	#117	—	127	1	no	—
142	C₄⋊C₈	—	127	4	no	—
143	C₂²⋊C₈	—	127	4	no	—
144	C₂²⋊C₈	C₂×D₁₂₇	1	4	yes	complemented
145	#97	—	127	2	no	—
146	C₄×C₈	—	127	2	no	—
147	C₂⁴	—	127	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
151	C₂×D₄	—	127	4	no	—
152	C₂×D₄	C₄×D₁₂₇	1	2	yes	—
153	C₂×D₄	C₄×D₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	—	127	1	no	—
155	C₂²×C₄	—	127	2	no	—
156	C₂²×C₄	—	127	2	no	—
157	C₂²×C₄	—	254	2	no	—
158	C₂²×C₄	—	254	2	no	—
159	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	254	4	no	—
163	M₁₆	—	254	4	no	—
164	M₁₆	—	254	4	no	—
165	M₁₆	—	254	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	127	4	no	—
168	C₂×C₈	—	127	1	no	—
169	C₂×C₈	—	127	1	no	—
170	C₂×C₈	—	127	2	no	—
171	C₂×C₈	—	254	4	no	—
172	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
173	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
174	C₄⋊C₄	—	127	4	no	—
175	C₄⋊C₄	—	127	2	no	—
176	C₂²⋊C₄	—	127	2	no	—
177	C₂²⋊C₄	—	127	4	no	—
178	C₂²⋊C₄	—	127	4	no	—
179	C₄²	—	127	2	no	—
180	C₂³	—	127	2	no	—
181	C₂³	—	508	2	no	—
182	C₂³	—	254	2	no	—
183	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
184	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	508	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	127	4	no	—
192	C₂×C₄	—	127	2	no	—
193	C₂×C₄	—	254	2	no	—
194	C₂×C₄	—	254	2	no	—
195	C₂×C₄	—	254	2	no	—
196	C₂×C₄	—	254	4	no	—
197	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
199	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
200	C₂×C₄	#14523247	1	2	yes	—
201	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	127	2	no	—
205	C₈	—	127	2	no	—
206	C₈	—	254	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	508	4	no	—
211	C₂²	—	254	2	no	—
212	C₂²	—	254	2	no	—
213	C₂²	—	254	2	no	—
214	C₂²	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	254	2	no	—
218	C₄	—	254	4	no	—
219	C₄	(C₂²⋊D₁₂₇)⋊C₄	1	2	yes	—
220	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
221	C₄	C₂×#14523247	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	508	2	no	—
225	C₂	C₂×(C₂²⋊D₁₂₇)⋊C₄	1	1	yes	characteristic
226	C₂	#22550641	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#408×C₁₂₇	C₂
prime radical	#22550383	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#408	D₁₂₇
hypocenter	C₁₂₇	#2930
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#16861216	C₂	semidirect
2. C₂²×#14523231	C₂	semidirect
4. #22549951	C₂	semidirect
5. #22549944	C₂	semidirect
6. #22549902	C₂	semidirect
7. #22549839	C₂	semidirect
8. #22549834	C₂	semidirect
9. #408×C₁₂₇	C₂	semidirect
13. C₂×#14523232	C₂²	semidirect
18. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
25. Dic₁₂₇⋊C₈	C₂²	semidirect
27. #16861119	C₂²	semidirect
28. C₈×Dic₁₂₇	C₂²	semidirect
32. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
45. C₂×C₁₂₇⋊C₈	D₄	semidirect
114. C₁₂₇	#2930	semidirect
121. #408	D₁₂₇	semidirect
144. C₂²⋊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
C₂×#16861216	2
C₂²×#14523231	2
#22550383	2
#22549951	2
#22549944	2
#22549902	2
#22549839	2
#22549834	2
#408×C₁₂₇	2
#2930	127