Subgroups — C₂×C₄×C₈×#6923 #30613809

Order 15616.

2046 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×C₄×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#6923	C₂	1	4	yes	maximal
3	C₂×C₄²×#6923	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×#6923	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₆₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×#6923	C₄	1	4	yes	—
8	C₂²×C₄×#6923	C₂²	1	2	yes	—
9	C₄×D₆₁⋊C₈	C₄	1	4	yes	complemented
10	C₄×D₆₁⋊C₈	C₂²	1	4	yes	—
11	C₂×C₄×#92420	C₄	1	4	yes	complemented
12	C₂×C₈×#6923	C₄	1	32	yes	direct product factor
13	C₂×C₈×#6923	C₂²	1	16	yes	—
14	C₄²×#6923	C₄	1	4	yes	—
15	C₄²×#6923	C₂²	1	4	yes	—
16	C₂²×C₈×D₆₁	C₂²	1	2	yes	—
17	C₂×C₄²×D₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×#92418	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₆₁	C₄	1	2	yes	complemented
20	C₄×C₈×D₆₁	C₄	1	4	yes	complemented
21	C₄×C₈×D₆₁	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic, complemented
23	C₂³×#6923	C₂×C₄	1	4	yes	—
24	C₂×D₆₁⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×D₆₁⋊C₈	C₂×C₄	1	16	yes	—
26	C₂×D₆₁⋊C₈	C₂³	1	4	yes	—
27	C₂²×#92420	C₂×C₄	1	4	yes	—
28	C₂×C₄×#6923	C₈	1	32	yes	direct product factor
29	C₂×C₄×#6923	C₂×C₄	1	4	yes	—
30	C₂×C₄×#6923	C₂×C₄	1	16	yes	—
31	C₂×C₄×#6923	C₂×C₄	1	16	yes	—
32	C₂×C₄×#6923	C₂³	1	4	yes	—
33	C₄×#92420	C₂×C₄	1	16	yes	complemented
34	C₈×#6923	C₂×C₄	1	64	yes	direct product factor
35	C₂²×C₄×D₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×D₆₁	C₂³	1	1	yes	characteristic
37	C₂²×#92418	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₆₁	C₂×C₄	1	16	yes	—
40	C₂×C₈×D₆₁	C₂³	1	4	yes	—
41	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
43	C₄²×D₆₁	C₂×C₄	1	2	yes	—
44	C₄²×D₆₁	C₂×C₄	1	2	yes	—
45	C₄²×D₆₁	C₂×C₄	1	2	yes	—
46	C₄²×D₆₁	C₂³	1	2	yes	—
47	C₄×#92418	C₂×C₄	1	4	yes	complemented
48	C₈×Dic₆₁	C₂×C₄	1	8	yes	complemented
49	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	complemented
52	C₂³×D₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×#6923	C₄²	1	16	yes	—
54	C₂²×#6923	C₂×C₈	1	32	yes	—
55	C₂²×#6923	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₆₁	C₄²	1	2	yes	—
59	C₂²×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×D₆₁	C₄²	1	4	yes	—
61	C₂×C₄×D₆₁	C₂×C₈	1	16	yes	—
62	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×D₆₁	C₂²×C₄	1	8	yes	—
66	C₂×C₄×D₆₁	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
68	C₂×#92420	C₄²	1	16	yes	complemented
69	C₂×#92420	C₂²×C₄	1	8	yes	—
70	C₄×#6923	C₄²	1	16	yes	—
71	C₄×#6923	C₂×C₈	1	64	yes	direct product factor
72	C₄×#6923	C₂²×C₄	1	16	yes	—
73	D₆₁⋊C₈	C₄²	1	16	yes	complemented
74	D₆₁⋊C₈	C₂²×C₄	1	16	yes	—
75	C₂×C₈×C₆₁	C₄²	1	8	yes	complemented
76	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
77	C₄²×C₆₁	C₄²	1	2	yes	—
78	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
79	C₄×Dic₆₁	C₄²	1	4	yes	—
80	C₄×Dic₆₁	C₄²	1	2	yes	—
81	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
82	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
83	C₂×#92418	C₄²	1	8	yes	complemented
84	C₂×#92418	C₂²×C₄	1	4	yes	—
85	C₈×D₆₁	C₄²	1	16	yes	complemented
86	C₈×D₆₁	C₂²×C₄	1	16	yes	—
87	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
88	C₂²×D₆₁	C₂×C₄²	1	1	yes	characteristic
89	C₂²×D₆₁	C₂×C₄²	1	4	yes	—
90	C₂²×D₆₁	C₂²×C₈	1	8	yes	—
91	C₂²×D₆₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×#6923	C₄×C₈	1	64	yes	direct product factor
93	C₂×#6923	C₂×C₄²	1	16	yes	—
94	C₂×#6923	C₂²×C₈	1	32	yes	—
95	C₂×C₄×C₆₁	C₄×C₈	1	8	yes	complemented
96	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
98	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₆₁	C₄×C₈	1	8	yes	complemented
101	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
102	C₂×Dic₆₁	C₂×C₄²	1	4	yes	—
103	C₂×Dic₆₁	C₂×C₄²	1	4	yes	—
104	C₂×Dic₆₁	C₂³×C₄	1	1	yes	characteristic
105	C₄×D₆₁	C₄×C₈	1	16	yes	complemented
106	C₄×D₆₁	C₂×C₄²	1	4	yes	—
107	C₄×D₆₁	C₂×C₄²	1	4	yes	—
108	C₄×D₆₁	C₂×C₄²	1	4	yes	—
109	C₄×D₆₁	C₂²×C₈	1	16	yes	—
110	C₄×D₆₁	C₂³×C₄	1	4	yes	—
111	#92420	C₂×C₄²	1	16	yes	complemented
112	C₈×C₆₁	C₂×C₄²	1	8	yes	complemented
113	#92418	C₂×C₄²	1	8	yes	complemented
114	C₂×C₄²×C₈	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	C₄³	1	2	yes	—
116	C₂²×C₆₁	C₂×C₄×C₈	1	4	yes	—
117	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
118	C₂×D₆₁	C₄³	1	2	yes	—
119	C₂×D₆₁	C₂×C₄×C₈	1	16	yes	—
120	C₂×D₆₁	C₂×C₄×C₈	1	4	yes	—
121	C₂×D₆₁	C₂²×C₄²	1	2	yes	—
122	C₂×D₆₁	C₂³×C₈	1	4	yes	—
123	#6923	C₂×C₄×C₈	1	32	yes	direct product factor
124	C₄×C₆₁	C₄³	1	2	yes	—
125	C₄×C₆₁	C₂×C₄×C₈	1	8	yes	complemented
126	C₄×C₆₁	C₂²×C₄²	1	2	yes	—
127	Dic₆₁	C₄³	1	2	yes	—
128	Dic₆₁	C₂×C₄×C₈	1	8	yes	complemented
129	Dic₆₁	C₂²×C₄²	1	2	yes	—
130	C₂²×C₄×C₈	—	61	4	no	—
131	C₂²×C₄×C₈	—	61	1	no	—
132	C₂²×C₄×C₈	—	61	1	no	—
133	C₂×C₄³	—	61	1	no	—
134	C₄²×C₈	—	61	8	no	—
135	C₂×C₆₁	C₄²×C₈	1	4	yes	complemented
136	C₂×C₆₁	C₂×C₄³	1	1	yes	characteristic
137	C₂×C₆₁	C₂²×C₄×C₈	1	2	yes	—
138	D₆₁	C₄²×C₈	1	4	yes	complemented
139	D₆₁	C₂²×C₄×C₈	1	4	yes	—
140	C₂³×C₈	—	61	2	no	—
141	C₂³×C₈	—	61	2	no	—
142	C₂²×C₄²	—	61	2	no	—
143	C₂²×C₄²	—	61	4	no	—
144	C₂²×C₄²	—	61	1	no	—
145	C₂×C₄×C₈	—	61	32	no	—
146	C₂×C₄×C₈	—	61	16	no	—
147	C₂×C₄×C₈	—	61	4	no	—
148	C₂×C₄×C₈	—	61	4	no	—
149	C₂×C₄×C₈	—	61	4	no	—
150	C₂×C₄×C₈	—	61	2	no	—
151	C₂×C₄×C₈	—	61	1	no	—
152	C₂×C₄×C₈	—	61	4	no	—
153	C₂×C₄×C₈	—	61	4	no	—
154	C₂×C₄×C₈	#6923	1	1	yes	characteristic, direct product factor
155	C₄³	—	61	4	no	—
156	C₄³	—	61	4	no	—
157	C₆₁	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	61	4	no	—
159	C₂³×C₄	—	61	2	no	—
160	C₂³×C₄	—	61	1	no	—
161	C₂²×C₈	—	61	4	no	—
162	C₂²×C₈	—	61	16	no	—
163	C₂²×C₈	—	61	4	no	—
164	C₂²×C₈	—	61	4	no	—
165	C₂²×C₈	—	61	2	no	—
166	C₂²×C₈	—	61	16	no	—
167	C₂²×C₈	—	61	4	no	—
168	C₂²×C₈	—	61	4	no	—
169	C₂²×C₈	C₂×#6923	1	2	yes	—
170	C₂×C₄²	—	61	32	no	—
171	C₂×C₄²	—	61	4	no	—
172	C₂×C₄²	—	61	16	no	—
173	C₂×C₄²	—	61	16	no	—
174	C₂×C₄²	—	61	4	no	—
175	C₂×C₄²	—	61	2	no	—
176	C₂×C₄²	—	61	1	no	—
177	C₂×C₄²	—	61	2	no	—
178	C₂×C₄²	—	61	2	no	—
179	C₂×C₄²	—	61	2	no	—
180	C₂×C₄²	—	61	2	no	—
181	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
182	C₄×C₈	—	61	64	no	—
183	C₄×C₈	—	61	16	no	—
184	C₄×C₈	—	61	8	no	—
185	C₄×C₈	—	61	4	no	—
186	C₄×C₈	C₂×#6923	1	4	yes	direct product factor
187	C₂⁴	—	61	1	no	—
188	C₂²×C₄	—	61	8	no	—
189	C₂²×C₄	—	61	16	no	—
190	C₂²×C₄	—	61	32	no	—
191	C₂²×C₄	—	61	1	no	—
192	C₂²×C₄	—	61	2	no	—
193	C₂²×C₄	—	61	16	no	—
194	C₂²×C₄	—	61	4	no	—
195	C₂²×C₄	—	61	1	no	—
196	C₂²×C₄	—	61	1	no	—
197	C₂²×C₄	—	61	8	no	—
198	C₂²×C₄	—	61	4	no	—
199	C₂²×C₄	—	61	1	no	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	C₄×#6923	1	2	yes	—
202	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
203	C₂×C₈	—	61	16	no	—
204	C₂×C₈	—	61	8	no	—
205	C₂×C₈	—	61	16	no	—
206	C₂×C₈	—	61	16	no	—
207	C₂×C₈	—	61	8	no	—
208	C₂×C₈	—	61	4	no	—
209	C₂×C₈	—	61	16	no	—
210	C₂×C₈	—	61	16	no	—
211	C₂×C₈	C₄×#6923	1	8	yes	direct product factor
212	C₂×C₈	C₂²×#6923	1	4	yes	—
213	C₄²	—	61	16	no	—
214	C₄²	—	61	16	no	—
215	C₄²	—	61	64	no	—
216	C₄²	—	61	4	no	—
217	C₄²	—	61	4	no	—
218	C₄²	—	61	2	no	—
219	C₄²	—	61	2	no	—
220	C₄²	C₄×#6923	1	2	yes	—
221	C₄²	C₂²×#6923	1	2	yes	—
222	C₂³	—	61	8	no	—
223	C₂³	—	61	1	no	—
224	C₂³	—	61	1	no	—
225	C₂³	—	61	4	no	—
226	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
227	C₂×C₄	—	61	64	no	—
228	C₂×C₄	—	61	16	no	—
229	C₂×C₄	—	61	32	no	—
230	C₂×C₄	—	61	8	no	—
231	C₂×C₄	—	61	4	no	—
232	C₂×C₄	—	61	1	no	—
233	C₂×C₄	—	61	1	no	—
234	C₂×C₄	—	61	4	no	—
235	C₂×C₄	—	61	16	no	—
236	C₂×C₄	—	61	16	no	—
237	C₂×C₄	—	61	4	no	—
238	C₂×C₄	—	61	4	no	—
239	C₂×C₄	—	61	4	no	—
240	C₂×C₄	—	61	4	no	—
241	C₂×C₄	C₈×#6923	1	8	yes	direct product factor
242	C₂×C₄	C₂×C₄×#6923	1	4	yes	—
243	C₂×C₄	C₂×C₄×#6923	1	4	yes	—
244	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
245	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
246	C₈	—	61	16	no	—
247	C₈	—	61	8	no	—
248	C₈	C₂×C₄×#6923	1	8	yes	direct product factor
249	C₂²	—	61	4	no	—
250	C₂²	—	61	4	no	—
251	C₂²	—	61	16	no	—
252	C₂²	—	61	2	no	—
253	C₂²	—	61	2	no	—
254	C₂²	C₄²×#6923	1	2	yes	—
255	C₂²	C₂×C₈×#6923	1	4	yes	—
256	C₂²	C₂²×C₄×#6923	1	1	yes	characteristic
257	C₄	—	61	32	no	—
258	C₄	—	61	8	no	—
259	C₄	—	61	2	no	—
260	C₄	—	61	2	no	—
261	C₄	C₄²×#6923	1	2	yes	—
262	C₄	C₂×C₈×#6923	1	8	yes	direct product factor
263	C₄	C₂²×C₄×#6923	1	2	yes	—
264	C₂	—	61	4	no	—
265	C₂	—	61	4	no	—
266	C₂	C₄×C₈×#6923	1	4	yes	direct product factor
267	C₂	C₂×C₄²×#6923	1	1	yes	characteristic
268	C₂	C₂²×C₈×#6923	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#6923
commutator, hypocenter	C₆₁	C₂×C₄²×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂×C₄²
prime radical	C₂³×D₆₁	C₂²×C₄
cosocle	C₂×C₄×D₆₁	C₂⁴
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₈×#6923	C₂	direct (N × H)
9. C₄×D₆₁⋊C₈	C₄	semidirect
11. C₂×C₄×#92420	C₄	semidirect
12. C₂×C₈×#6923	C₄	direct (N × H)
18. C₂×C₄×#92418	C₄	semidirect
19. C₂×C₈×Dic₆₁	C₄	semidirect
20. C₄×C₈×D₆₁	C₄	semidirect
22. C₂×C₄×C₈×C₆₁	C₄	semidirect
28. C₂×C₄×#6923	C₈	direct (N × H)
33. C₄×#92420	C₂×C₄	semidirect
34. C₈×#6923	C₂×C₄	direct (N × H)
47. C₄×#92418	C₂×C₄	semidirect
48. C₈×Dic₆₁	C₂×C₄	semidirect
51. C₄×C₈×C₆₁	C₂×C₄	semidirect
68. C₂×#92420	C₄²	semidirect
71. C₄×#6923	C₂×C₈	direct (N × H)
73. D₆₁⋊C₈	C₄²	semidirect
75. C₂×C₈×C₆₁	C₄²	semidirect
83. C₂×#92418	C₄²	semidirect
85. C₈×D₆₁	C₄²	semidirect
92. C₂×#6923	C₄×C₈	direct (N × H)
95. C₂×C₄×C₆₁	C₄×C₈	semidirect
100. C₂×Dic₆₁	C₄×C₈	semidirect
105. C₄×D₆₁	C₄×C₈	semidirect
111. #92420	C₂×C₄²	semidirect
112. C₈×C₆₁	C₂×C₄²	semidirect
113. #92418	C₂×C₄²	semidirect
123. #6923	C₂×C₄×C₈	direct (N × H)
125. C₄×C₆₁	C₂×C₄×C₈	semidirect
128. Dic₆₁	C₂×C₄×C₈	semidirect
135. C₂×C₆₁	C₄²×C₈	semidirect
138. D₆₁	C₄²×C₈	semidirect
154. C₂×C₄×C₈	#6923	direct (N × H)
157. C₆₁	C₂×C₄²×C₈	semidirect
186. C₄×C₈	C₂×#6923	direct (N × H)
211. C₂×C₈	C₄×#6923	direct (N × H)
241. C₂×C₄	C₈×#6923	direct (N × H)
248. C₈	C₂×C₄×#6923	direct (N × H)
262. C₄	C₂×C₈×#6923	direct (N × H)
266. C₂	C₄×C₈×#6923	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×C₄×D₆₁⋊C₈	2
C₂²×C₈×#6923	2
C₂×C₄²×#6923	2
C₄×C₈×#6923	2
C₂×C₄×C₈×D₆₁	2
C₂×C₄²×C₈	61

Subgroups — C2×C4×C8×#6923 #30613809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×#6923 #30613809