Subgroups — C₄×#19097802 #34108955

Order 20096.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097802	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578954	C₂	1	1	yes	maximal, characteristic
3	#23757378	C₂	1	2	yes	maximal, complemented
4	#23757371	C₂	1	2	yes	maximal, complemented
5	C₄×#14578944	C₂	1	2	yes	maximal, complemented
6	C₄×#14578941	C₂	1	2	yes	maximal, complemented
7	#23757313	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757282	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757272	C₂	1	1	yes	maximal, characteristic
10	C₄×#14578938	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	C₂²	1	1	yes	characteristic
13	#19097802	C₄	1	8	yes	direct product factor
14	C₂×#14578944	C₂²	1	2	yes	—
15	C₂×#14578941	C₂²	1	2	yes	—
16	C₂×#14578938	C₂²	1	1	yes	characteristic
17	C₂×#14578938	C₂²	1	1	yes	characteristic
18	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
20	C₄×D₆₂₈	C₂²	1	1	yes	characteristic
21	C₄×D₆₂₈	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₅₇	—	2	1	no	—
23	C₄×C₂²⋊D₁₅₇	—	2	1	no	—
24	#19097730	C₂²	1	1	yes	characteristic
25	#19097728	—	2	1	no	—
26	C₄²×D₁₅₇	—	2	1	no	—
27	#19097704	C₂²	1	2	yes	—
28	#19097702	C₂²	1	2	yes	—
29	#19097695	C₂²	1	2	yes	—
30	#19097695	C₂²	1	2	yes	—
31	#19097694	C₂²	1	2	yes	—
32	#19097691	C₂²	1	2	yes	—
33	#19097682	C₂²	1	2	yes	—
34	C₄×#11929434	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₅₇	—	2	1	no	—
37	C₄×Q₈×C₁₅₇	—	2	1	no	—
38	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
39	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₁₅₇	—	2	1	no	—
43	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
44	#14578954	—	2	2	no	—
45	#14578954	C₂×C₄	1	4	yes	—
46	C₂×Dic₃₁₄	C₂³	1	1	yes	characteristic
47	C₂×D₆₂₈	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₅₇	—	2	1	no	—
49	C₂×C₄×D₁₅₇	—	2	1	no	—
50	C₂×C₄×D₁₅₇	—	2	1	no	—
51	C₂×C₄×D₁₅₇	—	2	1	no	—
52	#14578944	C₂×C₄	1	8	yes	complemented
53	#14578941	C₂×C₄	1	8	yes	complemented
54	#14578938	—	2	2	no	—
55	#14578938	C₂×C₄	1	4	yes	complemented
56	C₂×#11929434	C₂³	1	2	yes	—
57	C₂×#11929434	C₂³	1	2	yes	—
58	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
60	#14578933	—	2	1	no	—
61	#14578933	—	2	1	no	—
62	C₂²⋊Dic₁₅₇	—	2	1	no	—
63	#14578931	—	2	1	no	—
64	#14578931	—	2	1	no	—
65	C₄×Dic₁₅₇	—	2	1	no	—
66	C₄×Dic₁₅₇	—	2	1	no	—
67	C₄×Dic₁₅₇	—	2	1	no	—
68	C₂×Q₈×C₁₅₇	—	2	1	no	—
69	C₂×Q₈×C₁₅₇	—	2	2	no	—
70	C₂×Q₈×C₁₅₇	C₂×C₄	1	4	yes	complemented
71	C₂×Q₈×C₁₅₇	C₂³	1	2	yes	—
72	C₂²×C₄×C₁₅₇	—	2	1	no	—
73	C₂²×C₄×C₁₅₇	—	2	1	no	—
74	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₅₇	—	2	2	no	—
78	C₄⋊C₄×C₁₅₇	—	2	2	no	—
79	C₄⋊C₄×C₁₅₇	—	2	1	no	—
80	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
81	C₄²×C₁₅₇	—	2	2	no	—
82	C₄²×C₁₅₇	—	2	2	no	—
83	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
85	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
87	C₂²×D₁₅₇	—	2	1	no	—
88	Dic₃₁₄	—	2	1	no	—
89	Dic₃₁₄	C₂²×C₄	1	2	yes	—
90	D₆₂₈	—	2	1	no	—
91	D₆₂₈	C₂²×C₄	1	2	yes	—
92	C₂²⋊D₁₅₇	—	2	4	no	—
93	C₂×Dic₁₅₇	—	2	1	no	—
94	C₂×Dic₁₅₇	—	2	1	no	—
95	C₂×Dic₁₅₇	—	2	1	no	—
96	C₄×D₁₅₇	—	2	4	no	—
97	C₄×D₁₅₇	—	4	2	no	—
98	C₄×D₁₅₇	—	4	2	no	—
99	#11929434	—	2	2	no	—
100	#11929434	C₂²×C₄	1	4	yes	—
101	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₁₅₇	—	2	4	no	—
103	Q₈×C₁₅₇	—	2	2	no	—
104	Q₈×C₁₅₇	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₅₇	—	2	4	no	—
106	C₂×C₄×C₁₅₇	—	2	2	no	—
107	C₂×C₄×C₁₅₇	—	2	2	no	—
108	C₂×C₄×C₁₅₇	—	2	2	no	—
109	C₂×C₄×C₁₅₇	—	4	1	no	—
110	C₂×C₄×C₁₅₇	—	4	2	no	—
111	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₅₇	#41	1	2	yes	—
118	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
119	C₂²×C₁₅₇	—	2	1	no	—
120	C₂²×C₁₅₇	—	2	1	no	—
121	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
122	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₅₇	—	2	2	no	—
124	C₂×D₁₅₇	—	4	1	no	—
125	C₂×D₁₅₇	—	4	1	no	—
126	C₄×C₁₅₇	—	2	4	no	—
127	C₄×C₁₅₇	—	2	1	no	—
128	C₄×C₁₅₇	—	2	1	no	—
129	C₄×C₁₅₇	—	2	1	no	—
130	C₄×C₁₅₇	—	4	2	no	—
131	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
132	C₄×C₁₅₇	#137	1	2	yes	complemented
133	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
135	Dic₁₅₇	—	2	2	no	—
136	Dic₁₅₇	—	4	1	no	—
137	Dic₁₅₇	—	4	1	no	—
138	C₂×C₁₅₇	—	2	2	no	—
139	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₅₇	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₅₇	#575	1	1	yes	characteristic
142	D₁₅₇	—	4	2	no	—
143	C₁₅₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	157	1	no	maximal, Sylow
145	C₂×#137	—	157	1	no	—
146	C₄×#41	—	157	1	no	—
147	C₂×C₄×Q₈	D₁₅₇	1	1	yes	characteristic, complemented
148	#441	—	157	2	no	—
149	#440	—	157	2	no	—
150	C₄×Q₁₆	—	157	2	no	—
151	C₄×SD₁₆	—	157	2	no	—
152	#428	—	157	1	no	—
153	#419	—	157	1	no	—
154	#417	—	157	1	no	—
155	C₄×M₁₆	—	157	1	no	—
156	C₂×#41	—	157	1	no	—
157	C₂²×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
158	#137	—	157	8	no	—
159	C₂×Q₁₆	—	157	2	no	—
160	C₂×SD₁₆	—	157	2	no	—
161	C₂×M₁₆	—	157	1	no	—
162	C₂×M₁₆	—	157	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	—	2	1	no	—
165	C₄×Q₈	—	157	1	no	—
166	C₄×Q₈	—	157	1	no	—
167	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
168	C₄×Q₈	C₂×D₁₅₇	1	2	yes	—
169	C₄×D₄	—	314	1	no	—
170	C₄×D₄	—	314	1	no	—
171	C₄×D₄	—	157	1	no	—
172	C₄×D₄	—	157	1	no	—
173	#117	—	314	1	no	—
174	#117	—	157	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	314	1	no	—
179	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
180	#107	—	157	2	no	—
181	#106	—	157	2	no	—
182	#103	—	157	2	no	—
183	#103	—	157	2	no	—
184	#103	—	157	2	no	—
185	#102	—	157	2	no	—
186	#97	—	157	2	no	—
187	C₄×C₈	—	157	2	no	—
188	#41	—	314	2	no	—
189	#41	—	157	4	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	157	1	no	—
193	C₂×Q₈	C₄×D₁₅₇	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₁₅₇	1	2	yes	—
195	C₂×D₄	—	314	1	no	—
196	C₂×D₄	—	157	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	314	1	no	—
200	C₂²×C₄	—	314	1	no	—
201	C₂²×C₄	—	314	1	no	—
202	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
205	Q₁₆	—	157	8	no	—
206	SD₁₆	—	157	8	no	—
207	M₁₆	—	314	2	no	—
208	M₁₆	—	157	4	no	—
209	C₂×C₈	—	157	2	no	—
210	C₂×C₈	—	157	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	314	1	no	—
215	C₄⋊C₄	—	314	1	no	—
216	C₄⋊C₄	—	157	1	no	—
217	C₄⋊C₄	—	157	1	no	—
218	C₄⋊C₄	C₂²×D₁₅₇	1	2	yes	—
219	C₂²⋊C₄	—	314	1	no	—
220	C₂²⋊C₄	—	314	1	no	—
221	C₂²⋊C₄	—	314	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	314	1	no	—
225	C₄²	—	314	1	no	—
226	C₄²	—	314	1	no	—
227	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
228	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
229	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
230	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
231	C₂³	—	314	1	no	—
232	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	314	1	no	—
236	Q₈	—	157	2	no	—
237	Q₈	C₂×C₄×D₁₅₇	1	4	yes	—
238	D₄	—	314	4	no	—
239	D₄	—	314	1	no	—
240	D₄	—	157	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	314	1	no	—
248	C₂×C₄	—	314	4	no	—
249	C₂×C₄	—	314	1	no	—
250	C₂×C₄	—	314	1	no	—
251	C₂×C₄	—	628	2	no	—
252	C₂×C₄	—	628	2	no	—
253	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
259	C₂×C₄	#14578954	1	2	yes	—
260	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
261	C₈	—	314	2	no	—
262	C₈	—	157	4	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	314	2	no	—
266	C₂²	—	628	1	no	—
267	C₂²	—	628	1	no	—
268	C₂²	C₄×C₂²⋊D₁₅₇	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	314	2	no	—
276	C₄	—	628	1	no	—
277	C₄	—	628	1	no	—
278	C₄	C₄×C₂²⋊D₁₅₇	1	2	yes	—
279	C₄	#19097802	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
281	C₄	C₂×#14578954	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	628	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₅₇	1	1	yes	characteristic
285	C₂	C₂×#19097802	1	1	yes	characteristic
286	C₂	#23758071	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄×Q₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂×#14578954	C₂²
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₅₇
hypocenter	C₁₅₇	C₄×#137
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #23757378	C₂	semidirect
4. #23757371	C₂	semidirect
5. C₄×#14578944	C₂	semidirect
6. C₄×#14578941	C₂	semidirect
7. #23757313	C₂	semidirect
8. #23757282	C₂	semidirect
10. C₄×#14578938	C₂	semidirect
11. C₂×C₄×Q₈×C₁₅₇	C₂	semidirect
13. #19097802	C₄	direct (N × H)
52. #14578944	C₂×C₄	semidirect
53. #14578941	C₂×C₄	semidirect
55. #14578938	C₂×C₄	semidirect
70. C₂×Q₈×C₁₅₇	C₂×C₄	semidirect
132. C₄×C₁₅₇	#137	semidirect
143. C₁₅₇	C₄×#137	semidirect
147. C₂×C₄×Q₈	D₁₅₇	semidirect
193. C₂×Q₈	C₄×D₁₅₇	semidirect
279. C₄	#19097802	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19097802	2
C₄×#14578954	2
#23757378	2
#23757371	2
C₄×#14578944	2
C₄×#14578941	2
#23757313	2
#23757282	2
#23757272	2
C₄×#14578938	2
C₂×C₄×Q₈×C₁₅₇	2
C₄×#137	157

Subgroups — C4×#19097802 #34108955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19097802 #34108955