Subgroups — #30641590

Order 16256.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549762	C₂	1	2	yes	maximal, complemented
2	#22549762	C₂	1	2	yes	maximal, complemented
3	#22549751	C₂	1	2	yes	maximal
4	#22549744	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549742	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549739	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×Dic₁₂₇	C₂	1	2	yes	maximal
9	#22549722	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic
11	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
18	C₁₂₇⋊#95	C₂²	1	2	yes	—
19	C₁₂₇⋊#95	C₂²	1	2	yes	—
20	C₁₂₇⋊#95	C₂²	1	2	yes	—
21	C₁₂₇⋊#95	C₂²	1	2	yes	—
22	C₁₂₇⋊#95	C₂²	1	2	yes	—
23	C₁₂₇⋊#95	C₂²	1	2	yes	—
24	C₁₂₇⋊#95	C₂²	1	1	yes	characteristic
25	C₁₂₇⋊#95	C₂²	1	1	yes	characteristic
26	C₁₂₇⋊#95	C₂²	1	1	yes	characteristic
27	C₁₂₇⋊#95	C₂²	1	1	yes	characteristic
28	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
29	#117×C₁₂₇	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₂₇	—	2	4	no	—
37	C₂²×Dic₁₂₇	—	2	4	no	—
38	C₂²×Dic₁₂₇	—	4	2	no	—
39	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
42	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
43	C₄⋊Dic₁₂₇	—	2	4	no	—
44	C₄⋊Dic₁₂₇	—	4	2	no	—
45	C₂²⋊Dic₁₂₇	—	2	4	no	—
46	C₄×Dic₁₂₇	—	2	4	no	—
47	C₄×Dic₁₂₇	—	4	2	no	—
48	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	—	2	4	no	—
50	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
60	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
61	C₄²×C₁₂₇	C₂×C₄	1	4	yes	complemented
62	C₄²×C₁₂₇	C₂×C₄	1	4	yes	complemented
63	C₂×Dic₁₂₇	—	2	2	no	—
64	C₂×Dic₁₂₇	—	2	2	no	—
65	C₂×Dic₁₂₇	—	2	2	no	—
66	C₂×Dic₁₂₇	—	2	2	no	—
67	C₂×Dic₁₂₇	—	2	2	no	—
68	C₂×Dic₁₂₇	—	2	2	no	—
69	C₂×Dic₁₂₇	—	2	2	no	—
70	C₂×Dic₁₂₇	—	2	2	no	—
71	C₂×Dic₁₂₇	—	2	2	no	—
72	C₂×Dic₁₂₇	—	2	2	no	—
73	C₂×Dic₁₂₇	—	2	2	no	—
74	C₂×Dic₁₂₇	—	4	8	no	—
75	C₂×Dic₁₂₇	#41	1	4	yes	—
76	C₂³×C₁₂₇	—	2	2	no	—
77	C₂³×C₁₂₇	—	2	1	no	—
78	C₂³×C₁₂₇	—	2	1	no	—
79	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₂₇	#41	1	4	yes	—
81	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	—	2	8	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×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	1	no	—
90	C₂×C₄×C₁₂₇	—	2	1	no	—
91	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
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	2	yes	—
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₁₂₇	—	2	2	no	—
98	C₂²×C₁₂₇	—	2	2	no	—
99	C₂²×C₁₂₇	—	2	4	no	—
100	C₂²×C₁₂₇	—	2	4	no	—
101	C₂²×C₁₂₇	#117	1	4	yes	—
102	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
104	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
105	C₂²×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	2	2	no	—
110	C₄×C₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	—	2	2	no	—
112	C₄×C₁₂₇	—	2	2	no	—
113	Dic₁₂₇	—	2	4	no	—
114	Dic₁₂₇	—	4	2	no	—
115	Dic₁₂₇	—	4	2	no	—
116	Dic₁₂₇	—	4	2	no	—
117	C₂×C₁₂₇	—	2	4	no	—
118	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
119	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₂₇	#520	1	1	yes	characteristic
121	C₂×C₁₂₇	#551	1	2	yes	—
122	C₂×C₁₂₇	#552	1	2	yes	—
123	#2343	—	127	1	no	maximal, Sylow
124	C₁₂₇	#2343	1	1	yes	characteristic, complemented, Sylow
125	C₂×#117	D₁₂₇	1	1	yes	characteristic
126	#387	—	127	2	no	—
127	#387	—	127	2	no	—
128	#385	—	127	2	no	—
129	#383	—	127	1	no	—
130	#382	—	127	1	no	—
131	#381	—	127	1	no	—
132	C₄×C₄⋊C₄	—	127	1	no	—
133	C₄×C₂²⋊C₄	—	127	2	no	—
134	#376	—	127	1	no	—
135	C₂×#95	—	127	1	no	—
136	C₂³×C₄	—	127	2	no	—
137	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
138	#117	Dic₁₂₇	1	8	yes	complemented
139	C₂×C₄⋊C₄	—	127	2	no	—
140	C₂×C₄⋊C₄	—	127	2	no	—
141	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	127	2	no	—
144	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
146	C₂×C₄²	—	127	2	no	—
147	C₂×C₄²	—	127	2	no	—
148	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
150	#95	—	127	1	no	—
151	#95	—	127	1	no	—
152	#95	—	127	2	no	—
153	#95	—	127	1	no	—
154	#95	—	127	2	no	—
155	#95	—	127	2	no	—
156	#95	—	127	2	no	—
157	#95	—	127	1	no	—
158	#95	—	127	2	no	—
159	#95	—	127	2	no	—
160	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	127	2	no	—
163	C₂²×C₄	—	127	2	no	—
164	C₂²×C₄	—	127	2	no	—
165	C₂²×C₄	—	127	2	no	—
166	C₂²×C₄	—	508	2	no	—
167	C₂²×C₄	—	254	4	no	—
168	C₂²×C₄	—	254	4	no	—
169	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
170	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
176	C₄⋊C₄	—	508	2	no	—
177	C₄⋊C₄	—	254	4	no	—
178	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	complemented
179	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	complemented
180	C₂²⋊C₄	—	254	4	no	—
181	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
182	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
183	C₄²	—	508	2	no	—
184	C₄²	—	254	4	no	—
185	C₄²	C₂×Dic₁₂₇	1	4	yes	complemented
186	C₄²	C₂×Dic₁₂₇	1	4	yes	complemented
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
191	C₂³	#14523248	1	4	yes	—
192	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	127	4	no	—
203	C₂×C₄	—	508	8	no	—
204	C₂×C₄	—	254	2	no	—
205	C₂×C₄	—	254	2	no	—
206	C₂×C₄	—	254	2	no	—
207	C₂×C₄	—	254	2	no	—
208	C₂×C₄	—	254	2	no	—
209	C₂×C₄	—	254	2	no	—
210	C₂×C₄	—	254	2	no	—
211	C₂×C₄	—	254	2	no	—
212	C₂×C₄	—	254	2	no	—
213	C₂×C₄	—	254	2	no	—
214	C₂×C₄	—	254	2	no	—
215	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
216	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
217	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
218	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
219	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
220	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#16861168	1	4	yes	—
226	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
227	C₂²	#41×D₁₂₇	1	2	yes	—
228	C₂²	C₂×#14523248	1	2	yes	—
229	C₂²	#16861255	1	1	yes	characteristic
230	C₂²	#16861256	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	508	2	no	—
238	C₄	—	508	2	no	—
239	C₄	—	508	2	no	—
240	C₄	—	254	4	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#16861168	1	1	yes	characteristic
243	C₂	#41×Dic₁₂₇	1	1	yes	characteristic
244	C₂	#22550359	1	1	yes	characteristic
245	C₂	#22550550	1	2	yes	—
246	C₂	#22550559	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂×#117×C₁₂₇	C₂
socle, cosocle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	#2343
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22549762	C₂	semidirect
2. #22549762	C₂	semidirect
4. #22549744	C₂	semidirect
5. #22549742	C₂	semidirect
7. C₄×C₄⋊Dic₁₂₇	C₂	semidirect
9. #22549722	C₂	semidirect
29. #117×C₁₂₇	C₄	semidirect
57. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
58. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
61. C₄²×C₁₂₇	C₂×C₄	semidirect
62. C₄²×C₁₂₇	C₂×C₄	semidirect
124. C₁₂₇	#2343	semidirect
138. #117	Dic₁₂₇	semidirect
178. C₄⋊C₄	C₂×Dic₁₂₇	semidirect
179. C₄⋊C₄	C₂×Dic₁₂₇	semidirect
185. C₄²	C₂×Dic₁₂₇	semidirect
186. C₄²	C₂×Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549762	2
#22549751	2
#22549744	2
#22549742	2
#22549739	2
C₄×C₄⋊Dic₁₂₇	2
C₂²⋊C₄×Dic₁₂₇	2
#22549722	2
C₂×C₁₂₇⋊#95	2
C₂×#117×C₁₂₇	2
#2343	127