Subgroups — #30652063

Order 16256.

1556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523248	C₂	1	2	yes	maximal, complemented
2	#22550457	C₂	1	4	yes	maximal, complemented
3	#22550452	C₂	1	4	yes	maximal, complemented
4	#22550445	C₂	1	4	yes	maximal, complemented
5	#22550438	C₂	1	4	yes	maximal, complemented
6	#22550426	C₂	1	4	yes	maximal, complemented
7	#22550388	C₂	1	8	yes	maximal, complemented
8	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	—	2	16	no	—
10	C₂×#14523248	C₂²	1	4	yes	—
11	C₂×#14523248	C₂²	1	4	yes	—
12	C₂×#14523248	C₂²	1	4	yes	complemented
13	C₂×#14523248	C₂²	1	4	yes	complemented
14	C₂²×Dic₂₅₄	C₂²	1	2	yes	complemented
15	C₂²×D₅₀₈	C₂²	1	2	yes	complemented
16	C₂²×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
17	C₂²×C₄×D₁₂₇	C₂²	1	4	yes	complemented
18	#16861209	C₂²	1	4	yes	complemented
19	#16861206	C₂²	1	4	yes	complemented
20	#16861197	C₂²	1	8	yes	complemented
21	#16861193	C₂²	1	16	yes	complemented
22	#16861189	C₂²	1	16	yes	complemented
23	#16861184	C₂²	1	16	yes	complemented
24	#16861183	C₂²	1	16	yes	complemented
25	#16861179	C₂²	1	16	yes	complemented
26	C₄×D₅₀₈	C₂²	1	16	yes	complemented
27	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	#117×C₁₂₇	C₂²	1	4	yes	complemented
29	#117×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	complemented
31	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	complemented
33	C₂³×D₁₂₇	C₂³	1	4	yes	—
34	#14523248	—	2	16	no	—
35	#14523248	—	2	16	no	—
36	#14523248	—	2	16	no	—
37	#14523248	—	2	16	no	—
38	C₂×Dic₂₅₄	—	2	8	no	—
39	C₂×Dic₂₅₄	C₂³	1	4	yes	—
40	C₂×Dic₂₅₄	C₂³	1	4	yes	—
41	C₂×D₅₀₈	—	2	8	no	—
42	C₂×D₅₀₈	C₂³	1	4	yes	—
43	C₂×D₅₀₈	C₂³	1	4	yes	—
44	C₂×D₅₀₈	C₂³	1	16	yes	—
45	C₂×C₂²⋊D₁₂₇	—	2	16	no	—
46	C₂×C₂²⋊D₁₂₇	C₂³	1	8	yes	—
47	C₂×C₂²⋊D₁₂₇	C₂³	1	8	yes	—
48	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
49	C₂×C₄×D₁₂₇	—	2	16	no	—
50	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
51	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
52	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
53	(C₂×D₁₂₇)⋊C₄	C₂³	1	32	yes	—
54	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	D₄	1	8	yes	complemented
56	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
62	C₂²⋊C₄×C₁₂₇	C₂³	1	8	yes	—
63	C₄²×C₁₂₇	C₂³	1	4	yes	—
64	C₄²×C₁₂₇	C₂³	1	4	yes	—
65	C₂²×D₁₂₇	—	2	8	no	—
66	C₂²×D₁₂₇	—	4	8	no	—
67	C₂²×D₁₂₇	C₂⁴	1	8	yes	—
68	Dic₂₅₄	—	2	8	no	—
69	Dic₂₅₄	—	2	8	no	—
70	D₅₀₈	—	2	8	no	—
71	D₅₀₈	—	2	8	no	—
72	D₅₀₈	—	4	16	no	—
73	C₂²⋊D₁₂₇	—	2	16	no	—
74	C₂²⋊D₁₂₇	—	2	16	no	—
75	C₂×Dic₁₂₇	—	2	8	no	—
76	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
77	C₄×D₁₂₇	—	2	16	no	—
78	C₄×D₁₂₇	—	2	16	no	—
79	C₂³×C₁₂₇	—	4	2	no	—
80	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
81	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
82	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	—	4	4	no	—
84	C₂×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
89	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₂²×C₁₂₇	—	2	2	no	—
93	C₂²×C₁₂₇	—	2	4	no	—
94	C₂²×C₁₂₇	—	4	4	no	—
95	C₂²×C₁₂₇	C₂²×D₄	1	4	yes	—
96	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
97	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
98	C₂×D₁₂₇	—	2	8	no	—
99	C₂×D₁₂₇	—	2	16	no	—
100	C₂×D₁₂₇	—	4	16	no	—
101	C₄×C₁₂₇	—	2	8	no	—
102	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
103	C₄×C₁₂₇	C₂²×D₄	1	4	yes	—
104	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
105	Dic₁₂₇	—	2	8	no	—
106	C₂×C₁₂₇	—	2	4	no	—
107	C₂×C₁₂₇	—	2	2	no	—
108	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
109	C₂×C₁₂₇	#585	1	2	yes	—
110	D₁₂₇	—	4	8	no	—
111	#3449	—	127	1	no	maximal, Sylow
112	C₁₂₇	#3449	1	1	yes	characteristic, complemented, Sylow
113	C₂²×#41	—	127	2	no	—
114	#537	—	127	4	no	—
115	#536	—	127	4	no	—
116	#535	—	127	4	no	—
117	#534	—	127	4	no	—
118	#532	—	127	4	no	—
119	#525	—	127	8	no	—
120	C₂×#117	D₁₂₇	1	1	yes	characteristic, complemented
121	C₂×#41	—	127	4	no	—
122	C₂×#41	—	127	4	no	—
123	C₂×#41	—	127	4	no	—
124	C₂×#41	—	127	4	no	—
125	C₂×#41	—	254	16	no	—
126	C₂²×Q₈	—	127	2	no	—
127	C₂²×D₄	—	127	4	no	—
128	C₂²×D₄	—	127	2	no	—
129	C₂³×C₄	—	127	4	no	—
130	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	127	4	no	—
132	#127	—	127	4	no	—
133	#124	—	127	8	no	—
134	#123	—	127	16	no	—
135	C₂²⋊Q₈	—	127	16	no	—
136	#121	—	127	16	no	—
137	#121	—	127	16	no	—
138	#120	—	127	16	no	—
139	C₄×D₄	—	127	16	no	—
140	#117	C₂×D₁₂₇	1	4	yes	complemented
141	#117	C₂×D₁₂₇	1	4	yes	complemented
142	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	complemented
143	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
144	C₂×C₄²	C₂×D₁₂₇	1	2	yes	complemented
145	C₂⁴	—	127	4	no	—
146	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
147	#41	—	254	16	no	—
148	#41	—	254	16	no	—
149	#41	—	254	16	no	—
150	#41	—	254	16	no	—
151	C₂×Q₈	—	127	4	no	—
152	C₂×Q₈	—	127	4	no	—
153	C₂×Q₈	—	254	8	no	—
154	C₂×D₄	—	127	8	no	—
155	C₂×D₄	—	127	8	no	—
156	C₂×D₄	—	127	16	no	—
157	C₂×D₄	—	127	4	no	—
158	C₂×D₄	—	127	4	no	—
159	C₂×D₄	—	254	16	no	—
160	C₂×D₄	—	254	8	no	—
161	C₂²×C₄	—	127	4	no	—
162	C₂²×C₄	—	127	8	no	—
163	C₂²×C₄	—	127	8	no	—
164	C₂²×C₄	—	254	16	no	—
165	C₂²×C₄	D₅₀₈	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
167	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
171	C₄⋊C₄	—	127	16	no	—
172	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
173	C₂²⋊C₄	—	127	32	no	—
174	C₂²⋊C₄	C₂²×D₁₂₇	1	8	yes	—
175	C₄²	C₂²×D₁₂₇	1	4	yes	—
176	C₄²	C₂²×D₁₂₇	1	4	yes	—
177	C₂³	—	4	2	no	—
178	C₂³	—	127	8	no	—
179	C₂³	—	508	8	no	—
180	C₂³	—	254	8	no	—
181	C₂³	C₂×D₅₀₈	1	4	yes	—
182	C₂³	C₂³×D₁₂₇	1	2	yes	—
183	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
184	Q₈	—	254	8	no	—
185	Q₈	—	254	8	no	—
186	D₄	—	508	16	no	—
187	D₄	—	254	16	no	—
188	D₄	—	254	16	no	—
189	D₄	—	254	8	no	—
190	D₄	—	254	8	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	127	8	no	—
193	C₂×C₄	—	254	8	no	—
194	C₂×C₄	—	254	16	no	—
195	C₂×C₄	—	254	16	no	—
196	C₂×C₄	C₂×D₅₀₈	1	8	yes	—
197	C₂×C₄	C₂×D₅₀₈	1	8	yes	—
198	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
199	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
200	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
201	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
203	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	508	16	no	—
208	C₂²	—	254	8	no	—
209	C₂²	—	254	16	no	—
210	C₂²	C₂²×D₅₀₈	1	4	yes	—
211	C₂²	C₂²×D₅₀₈	1	2	yes	—
212	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	254	8	no	—
215	C₄	C₂²×D₅₀₈	1	2	yes	—
216	C₄	C₂²×D₅₀₈	1	4	yes	—
217	C₄	C₂²×D₅₀₈	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	—	2	2	no	—
220	C₂	—	508	8	no	—
221	C₂	C₂³×D₅₀₈	1	1	yes	characteristic
222	C₂	#22550712	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#3449
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523248	C₂	semidirect
2. #22550457	C₂	semidirect
3. #22550452	C₂	semidirect
4. #22550445	C₂	semidirect
5. #22550438	C₂	semidirect
6. #22550426	C₂	semidirect
7. #22550388	C₂	semidirect
8. C₂×#117×C₁₂₇	C₂	semidirect
12. C₂×#14523248	C₂²	semidirect
13. C₂×#14523248	C₂²	semidirect
14. C₂²×Dic₂₅₄	C₂²	semidirect
15. C₂²×D₅₀₈	C₂²	semidirect
17. C₂²×C₄×D₁₂₇	C₂²	semidirect
18. #16861209	C₂²	semidirect
19. #16861206	C₂²	semidirect
20. #16861197	C₂²	semidirect
21. #16861193	C₂²	semidirect
22. #16861189	C₂²	semidirect
23. #16861184	C₂²	semidirect
24. #16861183	C₂²	semidirect
25. #16861179	C₂²	semidirect
26. C₄×D₅₀₈	C₂²	semidirect
28. #117×C₁₂₇	C₂²	semidirect
29. #117×C₁₂₇	C₂²	semidirect
30. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
32. C₂×C₄²×C₁₂₇	C₂²	semidirect
55. C₂²×C₄×C₁₂₇	D₄	semidirect
112. C₁₂₇	#3449	semidirect
120. C₂×#117	D₁₂₇	semidirect
140. #117	C₂×D₁₂₇	semidirect
141. #117	C₂×D₁₂₇	semidirect
142. C₂×C₄⋊C₄	C₂×D₁₂₇	semidirect
144. C₂×C₄²	C₂×D₁₂₇	semidirect
165. C₂²×C₄	D₅₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523248	2
#22550457	2
#22550452	2
#22550445	2
#22550438	2
#22550426	2
#22550388	2
C₂×#117×C₁₂₇	2
#3449	127