Subgroups — #34108519

Order 20096.

408 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757179	C₂	1	1	yes	maximal, characteristic
2	#23757178	C₂	1	1	yes	maximal, characteristic
3	#23757178	C₂	1	1	yes	maximal, characteristic
4	#23757172	C₂	1	1	yes	maximal, characteristic
5	#23757169	C₂	1	1	yes	maximal, characteristic
6	#23757159	C₂	1	2	yes	maximal
7	#23757156	C₂	1	2	yes	maximal
8	#23757154	C₂	1	2	yes	maximal
9	#23757140	C₂	1	2	yes	maximal
10	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
11	#389×C₁₅₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₁₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×#14578933	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
22	#19097678	C₂²	1	2	yes	—
23	#19097678	C₂²	1	2	yes	—
24	#19097678	C₂²	1	2	yes	—
25	#19097678	C₂²	1	2	yes	—
26	#19097678	C₂²	1	2	yes	—
27	#19097678	C₂²	1	2	yes	—
28	#19097678	C₂²	1	2	yes	—
29	#19097678	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
34	#95×C₁₅₇	C₂²	1	2	yes	—
35	C₂×Dic₃₁₄	—	2	2	no	—
36	C₂×Dic₃₁₄	—	2	2	no	—
37	C₂×Dic₃₁₄	—	2	2	no	—
38	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
39	C₂²×Dic₁₅₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₅₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₅₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₅₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₅₇	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₅₇	—	2	4	no	—
45	C₄⋊Dic₁₅₇	—	2	2	no	—
46	#14578933	—	2	4	no	—
47	C₄×Dic₁₅₇	—	2	2	no	—
48	C₄×Dic₁₅₇	—	2	2	no	—
49	C₄×Dic₁₅₇	—	2	2	no	—
50	C₄×Dic₁₅₇	—	2	2	no	—
51	C₄×Dic₁₅₇	—	2	2	no	—
52	C₄×Dic₁₅₇	Q₈	1	4	yes	complemented
53	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₅₇	—	2	4	no	—
59	C₄⋊C₄×C₁₅₇	—	2	2	no	—
60	C₄⋊C₄×C₁₅₇	—	2	2	no	—
61	C₄²×C₁₅₇	—	2	2	no	—
62	Dic₃₁₄	—	4	4	no	—
63	C₂×Dic₁₅₇	—	2	1	no	—
64	C₂×Dic₁₅₇	—	2	1	no	—
65	C₂×Dic₁₅₇	—	2	1	no	—
66	C₂×Dic₁₅₇	—	2	1	no	—
67	C₂×Dic₁₅₇	—	2	1	no	—
68	C₂×Dic₁₅₇	—	2	1	no	—
69	C₂×Dic₁₅₇	—	2	1	no	—
70	C₂×Dic₁₅₇	—	2	1	no	—
71	C₂×Dic₁₅₇	—	2	2	no	—
72	C₂×Dic₁₅₇	—	2	2	no	—
73	C₂×Dic₁₅₇	—	2	2	no	—
74	C₂×Dic₁₅₇	—	2	2	no	—
75	C₂×Dic₁₅₇	—	2	2	no	—
76	C₂×Dic₁₅₇	—	2	2	no	—
77	C₂×Dic₁₅₇	C₂×Q₈	1	2	yes	—
78	C₂×Dic₁₅₇	C₂×Q₈	1	2	yes	—
79	C₂×Dic₁₅₇	#41	1	2	yes	—
80	C₂×Dic₁₅₇	#41	1	2	yes	—
81	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₁₅₇	—	2	1	no	—
83	C₂×C₄×C₁₅₇	—	2	1	no	—
84	C₂×C₄×C₁₅₇	—	2	1	no	—
85	C₂×C₄×C₁₅₇	—	2	1	no	—
86	C₂×C₄×C₁₅₇	—	2	2	no	—
87	C₂×C₄×C₁₅₇	—	2	2	no	—
88	C₂×C₄×C₁₅₇	—	2	2	no	—
89	C₂×C₄×C₁₅₇	—	2	2	no	—
90	C₂×C₄×C₁₅₇	—	2	2	no	—
91	C₂×C₄×C₁₅₇	—	2	2	no	—
92	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₁₅₇	#41	1	2	yes	—
94	C₂×C₄×C₁₅₇	#41	1	2	yes	—
95	C₂×C₄×C₁₅₇	#41	1	2	yes	—
96	C₂×C₄×C₁₅₇	#41	1	2	yes	—
97	C₂²×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
99	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
101	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₁₅₇	C₂×#41	1	1	yes	characteristic
104	C₄×C₁₅₇	—	2	2	no	—
105	C₄×C₁₅₇	—	2	2	no	—
106	C₄×C₁₅₇	—	4	2	no	—
107	C₄×C₁₅₇	—	4	2	no	—
108	C₄×C₁₅₇	#126	1	4	yes	—
109	Dic₁₅₇	—	2	2	no	—
110	Dic₁₅₇	—	2	2	no	—
111	Dic₁₅₇	—	2	2	no	—
112	Dic₁₅₇	—	2	2	no	—
113	Dic₁₅₇	—	4	2	no	—
114	Dic₁₅₇	—	4	2	no	—
115	C₂×C₁₅₇	C₂×#126	1	1	yes	characteristic
116	C₂×C₁₅₇	#529	1	1	yes	characteristic
117	C₂×C₁₅₇	#529	1	1	yes	characteristic
118	C₂×C₁₅₇	#529	1	1	yes	characteristic
119	C₂×C₁₅₇	#533	1	1	yes	characteristic
120	C₂×C₁₅₇	#533	1	1	yes	characteristic
121	C₂×C₁₅₇	#533	1	1	yes	characteristic
122	C₁₅₇	#2863	1	1	yes	characteristic, complemented, Sylow
123	#2863	—	157	1	no	maximal, Sylow
124	#391	—	157	1	no	—
125	#391	—	157	1	no	—
126	#391	—	157	1	no	—
127	#389	—	157	1	no	—
128	#389	—	157	1	no	—
129	#389	D₁₅₇	1	1	yes	characteristic
130	#387	—	157	2	no	—
131	#387	—	157	2	no	—
132	#387	—	157	2	no	—
133	#383	—	157	2	no	—
134	C₄³	—	157	1	no	—
135	C₂²×Q₈	—	157	1	no	—
136	C₂×C₄⋊C₄	—	157	2	no	—
137	C₂×C₄⋊C₄	—	157	2	no	—
138	C₂×C₄⋊C₄	—	157	1	no	—
139	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
140	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
142	C₂×C₄²	—	157	1	no	—
143	C₂×C₄²	—	157	1	no	—
144	C₂×C₄²	—	157	1	no	—
145	C₂×C₄²	—	157	1	no	—
146	C₂×C₄²	—	157	1	no	—
147	C₂×C₄²	—	157	1	no	—
148	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
149	#95	—	157	2	no	—
150	#95	—	157	2	no	—
151	#95	—	157	2	no	—
152	#95	—	157	2	no	—
153	#95	—	157	2	no	—
154	#95	—	157	2	no	—
155	#95	—	157	2	no	—
156	#95	—	157	2	no	—
157	#95	C₂×D₁₅₇	1	2	yes	—
158	C₂×Q₈	—	314	2	no	—
159	C₂×Q₈	—	314	2	no	—
160	C₂×Q₈	—	314	2	no	—
161	C₂²×C₄	—	157	2	no	—
162	C₂²×C₄	—	157	2	no	—
163	C₂²×C₄	—	157	1	no	—
164	C₂²×C₄	—	157	1	no	—
165	C₂²×C₄	—	157	1	no	—
166	C₂²×C₄	—	157	1	no	—
167	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	314	4	no	—
176	C₄⋊C₄	—	314	4	no	—
177	C₄⋊C₄	—	314	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	314	2	no	—
180	C₄²	—	314	2	no	—
181	C₄²	—	314	2	no	—
182	C₄²	—	314	2	no	—
183	C₄²	—	314	2	no	—
184	C₄²	—	157	4	no	—
185	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
186	Q₈	—	628	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
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	—	314	1	no	—
198	C₂×C₄	—	314	1	no	—
199	C₂×C₄	—	314	1	no	—
200	C₂×C₄	—	314	1	no	—
201	C₂×C₄	—	314	1	no	—
202	C₂×C₄	—	314	1	no	—
203	C₂×C₄	—	314	1	no	—
204	C₂×C₄	—	314	1	no	—
205	C₂×C₄	—	314	2	no	—
206	C₂×C₄	—	314	2	no	—
207	C₂×C₄	—	314	2	no	—
208	C₂×C₄	—	314	2	no	—
209	C₂×C₄	—	314	2	no	—
210	C₂×C₄	—	314	2	no	—
211	C₂×C₄	—	157	2	no	—
212	C₂×C₄	—	157	2	no	—
213	C₂×C₄	—	157	2	no	—
214	C₂×C₄	—	157	2	no	—
215	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
216	C₂×C₄	#14578953	1	2	yes	—
217	C₂×C₄	#14578953	1	2	yes	—
218	C₂×C₄	#14578954	1	2	yes	—
219	C₂×C₄	#14578955	1	2	yes	—
220	C₂²	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
221	C₂²	#41×D₁₅₇	1	1	yes	characteristic
222	C₂²	#41×D₁₅₇	1	1	yes	characteristic
223	C₂²	C₂×#14578953	1	1	yes	characteristic
224	C₂²	C₂×#14578953	1	1	yes	characteristic
225	C₂²	C₂×#14578954	1	1	yes	characteristic
226	C₂²	C₂×#14578955	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	314	2	no	—
232	C₄	—	314	2	no	—
233	C₄	—	314	2	no	—
234	C₄	—	314	2	no	—
235	C₄	—	628	2	no	—
236	C₄	—	628	2	no	—
237	C₄	#19097766	1	4	yes	—
238	C₂	C₂×#19097766	1	1	yes	characteristic
239	C₂	#23757804	1	1	yes	characteristic
240	C₂	#23757805	1	1	yes	characteristic
241	C₂	#23757808	1	1	yes	characteristic
242	C₂	#23757825	1	1	yes	characteristic
243	C₂	#23757827	1	1	yes	characteristic
244	C₂	#23757828	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator, socle, prime radical, cosocle	C₂³×C₁₅₇	C₂⁴
Fitting	#389×C₁₅₇	C₂
hypercenter	#389	D₁₅₇
hypocenter	C₁₅₇	#2863
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
52. C₄×Dic₁₅₇	Q₈	semidirect
122. C₁₅₇	#2863	semidirect

Maximal subgroups

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

subgroup	index
#23757179	2
#23757178	2
#23757172	2
#23757169	2
#23757159	2
#23757156	2
#23757154	2
#23757140	2
C₄²×Dic₁₅₇	2
#389×C₁₅₇	2
#2863	157