Subgroups — #34108934

Order 20096.

492 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097791	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578954	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757383	C₂	1	2	yes	maximal, complemented
4	#23757378	C₂	1	2	yes	maximal, complemented
5	#23757373	C₂	1	2	yes	maximal, complemented
6	#23757371	C₂	1	2	yes	maximal, complemented
7	#23757309	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757269	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757267	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19097682	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	C₂²	1	1	yes	characteristic
13	#19097791	C₄	1	8	yes	complemented
14	C₂×#14578944	C₂²	1	1	yes	characteristic
15	C₂×#14578942	C₂²	1	1	yes	characteristic
16	C₂×#14578941	C₂²	1	1	yes	characteristic
17	C₂×#14578939	C₂²	1	1	yes	characteristic
18	C₂²×#11929434	C₂²	1	1	yes	characteristic
19	C₂²×#11929434	C₂²	1	1	yes	characteristic
20	C₄×Dic₃₁₄	C₂²	1	2	yes	—
21	C₄×D₆₂₈	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₅₇	—	2	2	no	—
23	#19097730	C₂²	1	1	yes	characteristic
24	#19097728	—	2	1	no	—
25	C₄²×D₁₅₇	—	2	1	no	—
26	#19097704	C₂²	1	2	yes	—
27	#19097702	C₂²	1	2	yes	—
28	#19097695	C₂²	1	2	yes	—
29	#19097694	C₂²	1	2	yes	—
30	#19097692	C₂²	1	2	yes	—
31	#19097691	C₂²	1	2	yes	—
32	#19097682	C₂²	1	2	yes	—
33	#19097682	C₂²	1	2	yes	—
34	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
36	C₄×D₄×C₁₅₇	—	2	2	no	—
37	C₄×D₄×C₁₅₇	C₂²	1	2	yes	—
38	#117×C₁₅₇	—	2	1	no	—
39	#117×C₁₅₇	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₅₇	—	2	1	no	—
41	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
42	#14578954	—	2	2	no	—
43	#14578954	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₃₁₄	C₂³	1	1	yes	characteristic
45	C₂×D₆₂₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₅₇	—	2	1	no	—
47	C₂×C₄×D₁₅₇	—	2	1	no	—
48	C₂×C₄×D₁₅₇	—	2	2	no	—
49	#14578944	C₂×C₄	1	4	yes	complemented
50	#14578942	C₂×C₄	1	4	yes	complemented
51	#14578941	C₂×C₄	1	4	yes	complemented
52	#14578939	C₂×C₄	1	4	yes	complemented
53	C₂×#11929434	—	2	2	no	—
54	C₂×#11929434	C₂×C₄	1	4	yes	complemented
55	C₂×#11929434	C₂³	1	2	yes	—
56	C₂×#11929434	C₂³	1	1	yes	characteristic
57	C₂×#11929434	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₅₇	C₂³	1	2	yes	—
59	#14578933	—	2	2	no	—
60	C₂²⋊Dic₁₅₇	—	2	1	no	—
61	#14578931	—	2	2	no	—
62	C₄×Dic₁₅₇	—	2	1	no	—
63	C₄×Dic₁₅₇	—	2	2	no	—
64	#41×C₁₅₇	—	2	2	no	—
65	#41×C₁₅₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₅₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₅₇	—	2	1	no	—
68	C₂×D₄×C₁₅₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₅₇	—	2	1	no	—
70	C₂²×C₄×C₁₅₇	—	2	2	no	—
71	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
72	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₅₇	—	2	2	no	—
74	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
77	C₄²×C₁₅₇	—	2	1	no	—
78	C₄²×C₁₅₇	—	2	2	no	—
79	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
80	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
81	C₄²×C₁₅₇	C₂³	1	2	yes	—
82	C₂²×D₁₅₇	—	2	1	no	—
83	Dic₃₁₄	—	2	1	no	—
84	Dic₃₁₄	C₂²×C₄	1	2	yes	—
85	D₆₂₈	—	2	1	no	—
86	D₆₂₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₅₇	—	2	4	no	—
88	C₂×Dic₁₅₇	—	2	1	no	—
89	C₂×Dic₁₅₇	—	2	2	no	—
90	C₄×D₁₅₇	—	2	4	no	—
91	C₄×D₁₅₇	—	4	4	no	—
92	#11929434	—	2	2	no	—
93	#11929434	C₂²×C₄	1	2	yes	—
94	#11929434	C₂²×C₄	1	2	yes	—
95	C₂³×C₁₅₇	—	2	1	no	—
96	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₅₇	—	2	1	no	—
98	Q₈×C₁₅₇	C₂²×C₄	1	2	yes	—
99	D₄×C₁₅₇	—	2	4	no	—
100	D₄×C₁₅₇	—	2	1	no	—
101	D₄×C₁₅₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₅₇	—	2	1	no	—
103	C₂×C₄×C₁₅₇	—	2	4	no	—
104	C₂×C₄×C₁₅₇	—	2	2	no	—
105	C₂×C₄×C₁₅₇	—	4	2	no	—
106	C₂×C₄×C₁₅₇	—	4	4	no	—
107	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₅₇	#41	1	2	yes	—
112	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₅₇	—	2	2	no	—
114	C₂²×C₁₅₇	—	2	2	no	—
115	C₂²×C₁₅₇	—	4	2	no	—
116	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
117	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₅₇	—	2	2	no	—
119	C₂×D₁₅₇	—	4	2	no	—
120	C₄×C₁₅₇	—	2	2	no	—
121	C₄×C₁₅₇	—	2	2	no	—
122	C₄×C₁₅₇	—	2	2	no	—
123	C₄×C₁₅₇	—	4	2	no	—
124	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
125	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₅₇	—	2	2	no	—
128	Dic₁₅₇	—	4	2	no	—
129	C₂×C₁₅₇	—	2	2	no	—
130	C₂×C₁₅₇	—	4	2	no	—
131	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₅₇	#575	1	2	yes	—
133	D₁₅₇	—	4	2	no	—
134	C₁₅₇	#2943	1	1	yes	characteristic, complemented, Sylow
135	#2943	—	157	1	no	maximal, Sylow
136	C₂×#135	—	157	1	no	—
137	C₄×#41	—	157	1	no	—
138	C₄×#41	D₁₅₇	1	1	yes	characteristic, complemented
139	#442	—	157	2	no	—
140	#441	—	157	2	no	—
141	#440	—	157	2	no	—
142	#440	—	157	2	no	—
143	#427	—	157	1	no	—
144	#416	—	157	1	no	—
145	#416	—	157	1	no	—
146	C₂×#97	—	157	1	no	—
147	C₂×#41	—	157	1	no	—
148	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
149	#135	—	157	8	no	—
150	C₂×Q₁₆	—	157	1	no	—
151	C₂×SD₁₆	—	157	1	no	—
152	C₂×SD₁₆	—	157	1	no	—
153	C₂×D₈	—	157	1	no	—
154	C₂²×C₈	—	157	1	no	—
155	C₂²×C₈	—	157	1	no	—
156	C₄×Q₈	—	157	2	no	—
157	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	314	2	no	—
160	C₄×D₄	—	157	2	no	—
161	C₄×D₄	C₂×D₁₅₇	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	314	1	no	—
164	#117	—	157	1	no	—
165	#117	C₂×D₁₅₇	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	314	1	no	—
168	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
169	#107	—	157	2	no	—
170	#106	—	157	2	no	—
171	#103	—	157	2	no	—
172	#103	—	157	2	no	—
173	#102	—	157	2	no	—
174	#102	—	157	2	no	—
175	#97	—	157	2	no	—
176	#97	—	157	2	no	—
177	#41	—	2	2	no	—
178	#41	—	314	2	no	—
179	#41	—	157	4	no	—
180	#41	C₄×D₁₅₇	1	4	yes	complemented
181	C₂×Q₈	—	157	1	no	—
182	C₂×Q₈	C₂²×D₁₅₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	314	1	no	—
185	C₂×D₄	—	157	1	no	—
186	C₂×D₄	C₂²×D₁₅₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	314	1	no	—
190	C₂²×C₄	—	314	2	no	—
191	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
193	Q₁₆	—	157	4	no	—
194	SD₁₆	—	157	4	no	—
195	SD₁₆	—	157	4	no	—
196	D₈	—	157	4	no	—
197	C₂×C₈	—	314	2	no	—
198	C₂×C₈	—	157	2	no	—
199	C₂×C₈	—	157	4	no	—
200	C₂×C₈	—	157	1	no	—
201	C₂×C₈	—	157	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	314	2	no	—
204	C₄⋊C₄	—	157	2	no	—
205	C₄⋊C₄	C₂²×D₁₅₇	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	314	1	no	—
209	C₂²⋊C₄	—	314	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	314	2	no	—
213	C₄²	—	314	1	no	—
214	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
216	C₄²	C₂²×D₁₅₇	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	314	1	no	—
219	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	314	1	no	—
222	Q₈	—	157	2	no	—
223	Q₈	C₂×C₄×D₁₅₇	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	314	4	no	—
227	D₄	—	314	1	no	—
228	D₄	—	157	2	no	—
229	D₄	C₂×C₄×D₁₅₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	314	1	no	—
236	C₂×C₄	—	314	4	no	—
237	C₂×C₄	—	314	2	no	—
238	C₂×C₄	—	628	4	no	—
239	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
243	C₂×C₄	#14578954	1	2	yes	—
244	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
245	C₈	—	314	2	no	—
246	C₈	—	157	2	no	—
247	C₈	—	157	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	314	2	no	—
252	C₂²	—	628	2	no	—
253	C₂²	C₄×C₂²⋊D₁₅₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	314	2	no	—
260	C₄	—	628	2	no	—
261	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
262	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
263	C₄	C₂×#14578954	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	628	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₅₇	1	1	yes	characteristic
268	C₂	#23758071	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₄×#41×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂×#19097791	C₂
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₄×#41	D₁₅₇
hypocenter	C₁₅₇	#2943
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14578954	C₂	semidirect
3. #23757383	C₂	semidirect
4. #23757378	C₂	semidirect
5. #23757373	C₂	semidirect
6. #23757371	C₂	semidirect
7. #23757309	C₂	semidirect
8. #23757269	C₂	semidirect
9. #23757267	C₂	semidirect
10. C₂×#19097682	C₂	semidirect
11. C₄×#41×C₁₅₇	C₂	semidirect
13. #19097791	C₄	semidirect
43. #14578954	C₂×C₄	semidirect
49. #14578944	C₂×C₄	semidirect
50. #14578942	C₂×C₄	semidirect
51. #14578941	C₂×C₄	semidirect
52. #14578939	C₂×C₄	semidirect
54. C₂×#11929434	C₂×C₄	semidirect
65. #41×C₁₅₇	C₂×C₄	semidirect
134. C₁₅₇	#2943	semidirect
138. C₄×#41	D₁₅₇	semidirect
180. #41	C₄×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097791	2
C₄×#14578954	2
#23757383	2
#23757378	2
#23757373	2
#23757371	2
#23757309	2
#23757269	2
#23757267	2
C₂×#19097682	2
C₄×#41×C₁₅₇	2
#2943	157