Subgroups — #34115844

Order 20096.

388 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₅₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#23758154	C₂	1	4	yes	maximal, complemented
4	#117×D₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578964	—	2	1	no	—
6	C₂×#14578964	C₄	1	2	yes	complemented
7	D₁₅₇⋊M₁₆	—	2	1	no	—
8	D₁₅₇⋊M₁₆	—	2	1	no	—
9	D₁₅₇⋊M₁₆	—	2	1	no	—
10	D₁₅₇⋊M₁₆	—	2	1	no	—
11	D₁₅₇⋊M₁₆	—	2	1	no	—
12	D₁₅₇⋊M₁₆	—	2	1	no	—
13	D₁₅₇⋊M₁₆	C₄	1	2	yes	complemented
14	D₁₅₇⋊M₁₆	C₄	1	2	yes	complemented
15	C₂×#14578963	C₂²	1	2	yes	—
16	C₂×#14578963	C₂²	1	2	yes	—
17	C₂×D₁₅₇⋊C₈	—	2	1	no	—
18	C₂×D₁₅₇⋊C₈	C₄	1	2	yes	complemented
19	C₂²×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
20	#19097730	C₄	1	1	yes	characteristic
21	#19097729	—	2	1	no	—
22	#19097728	—	2	1	no	—
23	#19097727	C₂²	1	2	yes	—
24	C₄⋊C₄×D₁₅₇	—	2	1	no	—
25	C₂²⋊C₄×D₁₅₇	C₄	1	2	yes	—
26	C₄²×D₁₅₇	—	2	1	no	—
27	#117×C₁₅₇	C₄	1	1	yes	characteristic
28	#14578964	—	2	2	no	—
29	#14578964	—	2	2	no	—
30	#14578964	—	2	2	no	—
31	#14578964	—	2	2	no	—
32	#14578963	—	2	2	no	—
33	#14578963	—	2	2	no	—
34	#14578963	—	2	2	no	—
35	#14578963	C₂×C₄	1	4	yes	complemented
36	D₁₅₇⋊C₈	—	2	2	no	—
37	D₁₅₇⋊C₈	—	2	2	no	—
38	D₁₅₇⋊C₈	—	2	2	no	—
39	D₁₅₇⋊C₈	—	2	2	no	—
40	C₂×#11929441	—	2	2	no	—
41	C₂×#11929441	C₂×C₄	1	4	yes	complemented
42	C₂³×D₁₅₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₅₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₁₅₇	—	2	2	no	—
45	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
50	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₅₇	D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₅₇	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₁₅₇	Q₈	1	1	yes	characteristic
57	C₄⋊Dic₁₅₇	—	2	1	no	—
58	#14578933	—	2	2	no	—
59	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
60	#14578931	—	2	2	no	—
61	C₄×Dic₁₅₇	—	2	2	no	—
62	C₄×Dic₁₅₇	—	2	1	no	—
63	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₅₇	—	2	1	no	—
65	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
66	C₄²×C₁₅₇	—	2	1	no	—
67	#11929441	—	2	4	no	—
68	#11929441	—	2	4	no	—
69	C₂²×D₁₅₇	—	4	2	no	—
70	C₂²×D₁₅₇	C₄²	1	1	yes	characteristic
71	C₂²×D₁₅₇	C₄²	1	1	yes	characteristic
72	C₂²×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₁₅₇	—	2	2	no	—
77	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₅₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₅₇	C₂×Q₈	1	1	yes	characteristic
83	C₄×D₁₅₇	—	4	4	no	—
84	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
93	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
94	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
95	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
97	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
98	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₁₅₇	C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₅₇	—	2	2	no	—
102	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₅₇	—	4	1	no	—
109	C₂²×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
110	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×D₁₅₇	—	2	2	no	—
113	C₂×D₁₅₇	—	2	2	no	—
114	C₂×D₁₅₇	—	2	2	no	—
115	C₂×D₁₅₇	—	4	1	no	—
116	C₂×D₁₅₇	—	4	1	no	—
117	C₂×D₁₅₇	—	4	1	no	—
118	C₂×D₁₅₇	#95	1	1	yes	characteristic
119	C₂×D₁₅₇	#95	1	1	yes	characteristic
120	C₂×D₁₅₇	#95	1	1	yes	characteristic
121	C₂×D₁₅₇	#95	1	1	yes	characteristic
122	C₄×C₁₅₇	—	4	2	no	—
123	C₄×C₁₅₇	#95	1	1	yes	characteristic
124	C₄×C₁₅₇	#95	1	1	yes	characteristic
125	C₄×C₁₅₇	#95	1	1	yes	characteristic
126	C₄×C₁₅₇	#95	1	1	yes	characteristic
127	Dic₁₅₇	—	4	2	no	—
128	Dic₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	Dic₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Dic₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₁₅₇	—	2	1	no	—
133	C₂×C₁₅₇	—	2	1	no	—
134	C₂×C₁₅₇	—	2	1	no	—
135	C₂×C₁₅₇	C₂×#95	1	1	yes	characteristic
136	D₁₅₇	—	2	1	no	—
137	D₁₅₇	—	2	1	no	—
138	D₁₅₇	—	2	1	no	—
139	D₁₅₇	#344	1	2	yes	complemented
140	C₁₅₇	C₂×#344	1	1	yes	characteristic, complemented, Sylow
141	C₂×#344	—	157	1	no	maximal, Sylow
142	C₂²×M₁₆	—	157	1	no	—
143	C₂²×M₁₆	—	157	1	no	—
144	C₂×#117	—	157	1	no	—
145	#344	—	157	4	no	—
146	C₂³×C₄	—	157	1	no	—
147	C₂×M₁₆	—	314	1	no	—
148	C₂×M₁₆	—	314	1	no	—
149	C₂×M₁₆	—	314	1	no	—
150	C₂×M₁₆	—	314	1	no	—
151	C₂×M₁₆	—	314	1	no	—
152	C₂×M₁₆	—	314	1	no	—
153	C₂×M₁₆	—	314	1	no	—
154	C₂×M₁₆	—	157	2	no	—
155	C₂×M₁₆	—	157	2	no	—
156	C₂×M₁₆	—	157	2	no	—
157	C₂×M₁₆	—	157	2	no	—
158	C₂×M₁₆	—	157	2	no	—
159	C₂²×C₈	—	314	1	no	—
160	C₂²×C₈	—	157	2	no	—
161	#117	—	314	1	no	—
162	#117	—	314	1	no	—
163	#117	—	157	1	no	—
164	#117	—	157	2	no	—
165	#117	#10597845	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	314	1	no	—
167	C₂×C₂²⋊C₄	—	157	2	no	—
168	C₂×C₄²	—	314	1	no	—
169	C₂⁴	—	157	1	no	—
170	C₂²×C₄	—	314	2	no	—
171	C₂²×C₄	—	157	1	no	—
172	C₂²×C₄	—	157	1	no	—
173	C₂²×C₄	—	157	1	no	—
174	C₂²×C₄	—	157	1	no	—
175	C₂²×C₄	—	157	1	no	—
176	C₂²×C₄	—	157	1	no	—
177	C₂²×C₄	—	157	1	no	—
178	C₂²×C₄	—	157	1	no	—
179	C₂²×C₄	—	157	1	no	—
180	C₂²×C₄	—	157	1	no	—
181	C₂²×C₄	—	157	1	no	—
182	C₂²×C₄	—	157	1	no	—
183	C₂²×C₄	—	157	1	no	—
184	C₂²×C₄	C₂×#10597845	1	1	yes	characteristic
185	M₁₆	—	314	2	no	—
186	M₁₆	—	314	2	no	—
187	M₁₆	—	314	2	no	—
188	M₁₆	—	314	2	no	—
189	M₁₆	—	314	2	no	—
190	M₁₆	—	314	2	no	—
191	M₁₆	—	314	2	no	—
192	M₁₆	—	157	4	no	—
193	C₂×C₈	—	314	2	no	—
194	C₂×C₈	—	314	2	no	—
195	C₂×C₈	—	314	2	no	—
196	C₂×C₈	—	314	2	no	—
197	C₂×C₈	—	314	2	no	—
198	C₂×C₈	—	157	4	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	314	2	no	—
201	C₄⋊C₄	—	314	1	no	—
202	C₂²⋊C₄	—	314	2	no	—
203	C₂²⋊C₄	—	157	2	no	—
204	C₂²⋊C₄	C₂×#10597845	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	314	2	no	—
207	C₄²	—	314	1	no	—
208	C₂³	—	628	2	no	—
209	C₂³	—	157	1	no	—
210	C₂³	—	157	1	no	—
211	C₂³	—	157	1	no	—
212	C₂³	—	157	1	no	—
213	C₂³	—	157	1	no	—
214	C₂³	—	157	1	no	—
215	C₂³	C₂²×#10597845	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	314	2	no	—
218	C₂×C₄	—	628	4	no	—
219	C₂×C₄	—	157	1	no	—
220	C₂×C₄	—	157	1	no	—
221	C₂×C₄	—	157	1	no	—
222	C₂×C₄	—	157	1	no	—
223	C₂×C₄	—	157	1	no	—
224	C₂×C₄	—	157	1	no	—
225	C₂×C₄	—	157	1	no	—
226	C₂×C₄	—	157	1	no	—
227	C₂×C₄	—	157	1	no	—
228	C₂×C₄	—	157	1	no	—
229	C₂×C₄	—	157	1	no	—
230	C₂×C₄	—	157	1	no	—
231	C₂×C₄	—	157	1	no	—
232	C₂×C₄	—	157	1	no	—
233	C₂×C₄	—	157	1	no	—
234	C₂×C₄	—	157	1	no	—
235	C₂×C₄	—	157	1	no	—
236	C₂×C₄	—	157	1	no	—
237	C₂×C₄	—	157	1	no	—
238	C₂×C₄	—	157	1	no	—
239	C₂×C₄	—	157	1	no	—
240	C₂×C₄	—	157	1	no	—
241	C₂×C₄	C₄×#10597845	1	1	yes	characteristic
242	C₂×C₄	C₄×#10597845	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
245	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
246	C₂×C₄	C₄⋊#10597845	1	1	yes	characteristic
247	C₈	—	314	4	no	—
248	C₈	—	314	4	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	314	2	no	—
251	C₂²	—	314	2	no	—
252	C₂²	—	314	2	no	—
253	C₂²	—	628	1	no	—
254	C₂²	—	628	1	no	—
255	C₂²	—	628	1	no	—
256	C₂²	—	157	1	no	—
257	C₂²	—	157	1	no	—
258	C₂²	—	157	1	no	—
259	C₂²	—	157	1	no	—
260	C₂²	C₂×C₄×#10597845	1	1	yes	characteristic
261	C₂²	C₂×C₂²⋊#10597845	1	1	yes	characteristic
262	C₂²	C₂×C₄⋊#10597845	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	628	2	no	—
265	C₄	—	157	1	no	—
266	C₄	—	157	1	no	—
267	C₄	—	157	1	no	—
268	C₄	—	157	1	no	—
269	C₄	(C₂×#10597845)⋊C₄	1	1	yes	characteristic
270	C₄	(C₂×#10597845)⋊C₄	1	1	yes	characteristic
271	C₄	(C₂×#10597845)⋊C₄	1	1	yes	characteristic
272	C₄	(C₂×#10597845)⋊C₄	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	314	1	no	—
277	C₂	—	314	1	no	—
278	C₂	—	314	1	no	—
279	C₂	—	157	2	no	—
280	C₂	C₂×(C₂×#10597845)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×#10597845)⋊C₄
commutator	C₂²×C₁₅₇	C₂×C₄²
Frattini	C₂³	C₂²×#10597845
Fitting	#117×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₂×#95
prime radical	C₂³×D₁₅₇	C₂×C₄
cosocle	C₂²×Dic₁₅₇	C₂³
hypercenter	#117	#10597845
hypocenter	C₁₅₇	C₂×#344
LCS	G ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #23758154	C₂	semidirect
6. C₂×#14578964	C₄	semidirect
13. D₁₅₇⋊M₁₆	C₄	semidirect
14. D₁₅₇⋊M₁₆	C₄	semidirect
18. C₂×D₁₅₇⋊C₈	C₄	semidirect
35. #14578963	C₂×C₄	semidirect
41. C₂×#11929441	C₂×C₄	semidirect
139. D₁₅₇	#344	semidirect
140. C₁₅₇	C₂×#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₇⋊M₁₆	2
#23758154	2
#117×D₁₅₇	2
C₂×#344	157