Subgroups — #34109259

Order 20096.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14578953	C₂	1	1	yes	maximal, characteristic
2	#23757403	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757400	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757344	C₂	1	2	yes	maximal, complemented
5	#23757341	C₂	1	2	yes	maximal, complemented
6	D₄×#11929434	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757327	C₂	1	2	yes	maximal, complemented
8	#23757324	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757295	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19097700	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578953	C₄	1	1	yes	characteristic
13	C₂×#14578937	C₂²	1	2	yes	—
14	C₂²×#11929434	C₂²	1	2	yes	—
15	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
16	#19097736	C₄	1	1	yes	characteristic
17	D₄×Dic₁₅₇	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₅₇	C₂²	1	2	yes	—
19	#19097731	C₄	1	2	yes	—
20	#19097729	C₄	1	1	yes	characteristic
21	#19097727	C₄	1	2	yes	—
22	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
23	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
24	C₄⋊#11929434	C₂²	1	1	yes	characteristic
25	#19097700	C₂²	1	1	yes	characteristic
26	#19097700	C₂²	1	1	yes	characteristic
27	#19097700	C₂²	1	2	yes	—
28	#19097700	C₂²	1	2	yes	—
29	#19097700	C₂²	1	2	yes	—
30	C₂²⋊#11929434	C₂²	1	2	yes	—
31	#19097683	C₂²	1	1	yes	characteristic
32	#19097683	C₂²	1	1	yes	characteristic
33	#19097681	C₂²	1	2	yes	—
34	#19097681	C₂²	1	2	yes	—
35	C₄×#11929434	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₅₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
41	#14578953	—	2	4	no	—
42	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₅₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
49	#14578937	—	4	2	no	—
50	C₂×#11929434	—	4	2	no	—
51	C₂×#11929434	C₂³	1	1	yes	characteristic
52	C₂×#11929434	C₂³	1	1	yes	characteristic
53	C₂×#11929434	C₂³	1	2	yes	—
54	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
55	#14578933	C₂×C₄	1	2	yes	—
56	#14578933	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
58	#14578931	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₅₇	C₂³	1	2	yes	—
64	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
67	C₂×C₈×C₁₅₇	—	4	2	no	—
68	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
71	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	2	yes	—
73	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₅₇	C₂²×C₄	1	1	yes	characteristic
75	Dic₃₁₄	—	2	2	no	—
76	C₂²⋊D₁₅₇	—	2	4	no	—
77	C₂×Dic₁₅₇	—	2	4	no	—
78	C₂×Dic₁₅₇	—	2	4	no	—
79	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₅₇	—	2	2	no	—
85	C₄×D₁₅₇	—	2	1	no	—
86	C₄×D₁₅₇	—	2	1	no	—
87	C₄×D₁₅₇	—	2	2	no	—
88	#11929434	—	2	1	no	—
89	#11929434	—	2	2	no	—
90	#11929434	—	2	1	no	—
91	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
92	D₄×C₁₅₇	M₁₆	1	4	yes	complemented
93	C₂×C₄×C₁₅₇	—	2	2	no	—
94	C₂×C₄×C₁₅₇	—	2	2	no	—
95	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₅₇	—	2	1	no	—
100	C₈×C₁₅₇	—	2	2	no	—
101	C₈×C₁₅₇	—	2	1	no	—
102	C₂²×C₁₅₇	—	2	2	no	—
103	C₂²×C₁₅₇	—	2	2	no	—
104	C₂²×C₁₅₇	C₂×M₁₆	1	4	yes	—
105	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₅₇	—	2	1	no	—
107	C₂×D₁₅₇	—	2	1	no	—
108	C₂×D₁₅₇	—	2	1	no	—
109	C₄×C₁₅₇	—	2	2	no	—
110	C₄×C₁₅₇	—	2	1	no	—
111	C₄×C₁₅₇	C₂×M₁₆	1	2	yes	—
112	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₁₅₇	—	2	1	no	—
115	Dic₁₅₇	—	2	1	no	—
116	Dic₁₅₇	—	2	2	no	—
117	Dic₁₅₇	—	2	2	no	—
118	Dic₁₅₇	#131	1	2	yes	—
119	C₂×C₁₅₇	—	2	4	no	—
120	C₂×C₁₅₇	#520	1	1	yes	characteristic
121	C₂×C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
123	D₁₅₇	—	4	1	no	—
124	C₁₅₇	#2990	1	1	yes	characteristic, complemented, Sylow
125	#2990	—	157	1	no	maximal, Sylow
126	C₄×#41	—	157	1	no	—
127	#446	—	157	1	no	—
128	#446	—	157	1	no	—
129	#435	—	157	2	no	—
130	#435	—	157	2	no	—
131	C₈×D₄	—	157	1	no	—
132	C₈×D₄	D₁₅₇	1	1	yes	characteristic, complemented
133	#432	—	157	2	no	—
134	#431	—	157	1	no	—
135	#423	—	157	1	no	—
136	C₂×C₄⋊C₈	—	157	2	no	—
137	C₂×#41	—	157	1	no	—
138	C₂×M₁₆	—	157	2	no	—
139	C₂²×C₈	—	157	2	no	—
140	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
141	C₄×Q₈	—	157	1	no	—
142	C₄×Q₈	—	157	1	no	—
143	C₄×D₄	—	157	1	no	—
144	C₄×D₄	—	157	2	no	—
145	C₄×D₄	—	157	2	no	—
146	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
147	#117	—	157	1	no	—
148	#117	—	157	2	no	—
149	C₂×C₄²	—	157	2	no	—
150	C₂×C₄²	—	157	1	no	—
151	C₄⋊C₈	—	157	1	no	—
152	C₄⋊C₈	—	157	2	no	—
153	C₄⋊C₈	—	157	2	no	—
154	C₄⋊C₈	—	157	2	no	—
155	C₄⋊C₈	—	157	1	no	—
156	C₄⋊C₈	—	157	1	no	—
157	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
158	C₂²⋊C₈	—	157	2	no	—
159	C₂²⋊C₈	—	157	1	no	—
160	C₂²⋊C₈	—	157	1	no	—
161	C₂²⋊C₈	C₂×D₁₅₇	1	2	yes	—
162	#97	—	157	2	no	—
163	#97	—	157	2	no	—
164	C₄×C₈	—	157	1	no	—
165	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
166	#41	—	314	4	no	—
167	C₂×Q₈	—	157	1	no	—
168	C₂×D₄	—	157	2	no	—
169	C₂×D₄	C₄×D₁₅₇	1	1	yes	characteristic
170	C₂²×C₄	—	157	2	no	—
171	C₂²×C₄	—	157	2	no	—
172	C₂²×C₄	—	157	1	no	—
173	C₂²×C₄	—	157	1	no	—
174	C₂²×C₄	—	157	1	no	—
175	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
176	M₁₆	—	628	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	628	2	no	—
179	C₂×C₈	—	157	1	no	—
180	C₂×C₈	—	157	1	no	—
181	C₂×C₈	—	157	2	no	—
182	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
185	C₄⋊C₄	—	157	2	no	—
186	C₄⋊C₄	—	157	2	no	—
187	C₄⋊C₄	—	157	1	no	—
188	C₄⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	157	2	no	—
190	C₂²⋊C₄	—	157	2	no	—
191	C₂²⋊C₄	C₄×D₁₅₇	1	2	yes	—
192	C₄²	—	157	2	no	—
193	C₄²	—	157	1	no	—
194	C₄²	—	157	1	no	—
195	C₄²	—	157	2	no	—
196	C₄²	—	157	2	no	—
197	C₄²	—	157	1	no	—
198	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
199	C₂³	—	157	1	no	—
200	C₂³	C₂×C₄×D₁₅₇	1	2	yes	—
201	Q₈	—	314	2	no	—
202	D₄	—	314	4	no	—
203	D₄	#14578937	1	4	yes	complemented
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	314	2	no	—
207	C₂×C₄	—	314	4	no	—
208	C₂×C₄	—	314	4	no	—
209	C₂×C₄	—	314	2	no	—
210	C₂×C₄	—	314	1	no	—
211	C₂×C₄	—	314	1	no	—
212	C₂×C₄	—	157	1	no	—
213	C₂×C₄	—	157	1	no	—
214	C₂×C₄	—	157	2	no	—
215	C₂×C₄	—	157	2	no	—
216	C₂×C₄	—	157	1	no	—
217	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	314	1	no	—
225	C₈	—	314	2	no	—
226	C₈	—	314	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	314	1	no	—
230	C₂²	—	314	1	no	—
231	C₂²	—	314	1	no	—
232	C₂²	C₂×#14578937	1	4	yes	—
233	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	314	1	no	—
237	C₄	—	314	1	no	—
238	C₄	—	314	2	no	—
239	C₄	—	314	2	no	—
240	C₄	—	157	2	no	—
241	C₄	C₂×#14578937	1	2	yes	—
242	C₄	#19097816	1	1	yes	characteristic
243	C₄	#19097818	1	1	yes	characteristic
244	C₂	—	2	4	no	—
245	C₂	—	628	1	no	—
246	C₂	#23757752	1	1	yes	characteristic
247	C₂	C₂²×#14578937	1	1	yes	characteristic
248	C₂	#131×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₈×D₄×C₁₅₇	C₂
prime radical	C₂×#14578953	C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₈×D₄	D₁₅₇
hypocenter	C₁₅₇	#2990
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #23757403	C₂	semidirect
3. #23757400	C₂	semidirect
4. #23757344	C₂	semidirect
5. #23757341	C₂	semidirect
6. D₄×#11929434	C₂	semidirect
7. #23757327	C₂	semidirect
8. #23757324	C₂	semidirect
9. #23757295	C₂	semidirect
10. C₂×#19097700	C₂	semidirect
11. C₈×D₄×C₁₅₇	C₂	semidirect
92. D₄×C₁₅₇	M₁₆	semidirect
124. C₁₅₇	#2990	semidirect
132. C₈×D₄	D₁₅₇	semidirect
203. D₄	#14578937	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14578953	2
#23757403	2
#23757400	2
#23757344	2
#23757341	2
D₄×#11929434	2
#23757327	2
#23757324	2
#23757295	2
C₂×#19097700	2
C₈×D₄×C₁₅₇	2
#2990	157