Subgroups — C₈×#14556645 #34001925

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625644	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556645	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	#22625127	C₂	1	1	yes	maximal, characteristic
5	#22625125	C₂	1	1	yes	maximal, characteristic
6	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
7	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556645	C₄	1	4	yes	—
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₄×#11924036	C₂²	1	1	yes	characteristic
11	C₄×#11924036	C₂²	1	1	yes	characteristic
12	C₂×C₈×#64408	C₄	1	4	yes	—
13	C₄²×#64408	C₂²	1	1	yes	characteristic
14	#17989452	C₄	1	4	yes	—
15	#17989429	C₄	1	4	yes	complemented
16	#17989428	C₄	1	4	yes	complemented
17	C₈×#10596833	C₄	1	4	yes	complemented
18	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
21	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
22	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
23	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
24	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
25	C₂×#11924036	C₂×C₄	1	2	yes	—
26	C₂×#11924036	C₂×C₄	1	2	yes	—
27	C₂×#11924036	C₂×C₄	1	2	yes	—
28	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
29	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
30	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
31	#14556645	C₈	1	16	yes	direct product factor
32	C₄×#10596833	C₂×C₄	1	2	yes	—
33	C₄×#10596833	C₂×C₄	1	2	yes	—
34	C₄×#10596833	C₂×C₄	1	2	yes	—
35	C₄×#10596833	C₂×C₄	1	2	yes	—
36	C₄×#10596833	C₂×C₄	1	2	yes	—
37	C₄×#10596833	C₂×C₄	1	2	yes	—
38	C₈×#64408	—	2	8	no	—
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
42	C₄²×D₇₃	C₂³	1	1	yes	characteristic
43	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
44	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
45	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
47	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×#64408	C₄²	1	4	yes	—
53	#11924036	—	2	4	no	—
54	#11924036	—	2	4	no	—
55	#11924036	C₂×C₈	1	8	yes	—
56	C₂×#10596833	C₄²	1	4	yes	—
57	C₂×#10596833	C₄²	1	4	yes	—
58	C₂×#10596833	C₄²	1	4	yes	—
59	C₄×#64408	—	2	4	no	—
60	C₄×#64408	—	2	4	no	—
61	C₄×#64408	C₂×C₈	1	8	yes	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×#10596831	C₄²	1	2	yes	—
66	C₂×#10596831	C₄²	1	2	yes	—
67	C₂×#10596831	C₄²	1	2	yes	—
68	C₈×D₇₃	—	2	4	no	—
69	C₈×D₇₃	C₂×C₈	1	8	yes	—
70	C₈×D₇₃	M₁₆	1	8	yes	—
71	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₄²	1	2	yes	—
75	C₂×C₈×C₇₃	C₄²	1	2	yes	—
76	C₂×C₈×C₇₃	C₄²	1	2	yes	—
77	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×#64408	—	2	4	no	—
80	C₂×#64408	—	2	4	no	—
81	C₂×#64408	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
88	C₄×D₇₃	C₂²×C₈	1	4	yes	—
89	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
90	C₄×D₇₃	#131	1	4	yes	—
91	#10596833	—	2	4	no	—
92	#10596833	—	2	4	no	—
93	#10596833	C₄×C₈	1	8	yes	complemented
94	C₈×C₇₃	—	2	2	no	—
95	C₈×C₇₃	C₄×C₈	1	4	yes	complemented
96	C₈×C₇₃	#97	1	4	yes	complemented
97	#10596831	—	2	2	no	—
98	#10596831	C₄×C₈	1	4	yes	complemented
99	#10596831	#97	1	4	yes	complemented
100	C₂²×C₇₃	C₄³	1	1	yes	characteristic
101	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
102	C₂×D₇₃	C₄×M₁₆	1	2	yes	—
103	C₂×D₇₃	#405	1	2	yes	—
104	#64408	—	2	8	no	—
105	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
106	C₄×C₇₃	C₂×#97	1	2	yes	—
107	C₄×C₇₃	#405	1	2	yes	—
108	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
109	Dic₇₃	C₄×M₁₆	1	2	yes	—
110	Dic₇₃	#405	1	2	yes	—
111	C₈×#97	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
113	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
114	C₂×C₇₃	#1726	1	1	yes	characteristic
115	D₇₃	C₈×M₁₆	1	4	yes	—
116	#1744	—	73	1	no	—
117	C₄×#97	—	73	1	no	—
118	C₄²×C₈	—	73	1	no	—
119	C₂×#322	—	73	1	no	—
120	C₂×#322	—	73	1	no	—
121	C₂×C₈²	—	73	1	no	—
122	C₂×C₈²	—	73	1	no	—
123	C₇₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
124	C₂×#97	—	73	4	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	4	no	—
132	C₄³	—	73	1	no	—
133	#336	—	73	4	no	—
134	#322	—	73	4	no	—
135	#322	—	73	4	no	—
136	C₈²	—	73	4	no	—
137	C₈²	—	73	1	no	—
138	C₈²	—	73	1	no	—
139	C₈²	—	73	1	no	—
140	C₈²	#64408	1	1	yes	characteristic, complemented
141	C₂²×C₈	—	73	2	no	—
142	C₂²×C₈	—	73	2	no	—
143	C₂²×C₈	—	73	2	no	—
144	C₂²×C₈	—	73	2	no	—
145	C₂²×C₈	—	73	2	no	—
146	C₂²×C₈	—	73	2	no	—
147	C₂×C₄²	—	73	2	no	—
148	C₂×C₄²	—	73	1	no	—
149	C₂×C₄²	—	73	2	no	—
150	C₂×C₄²	—	73	2	no	—
151	#97	—	73	16	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	73	2	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	—	73	2	no	—
158	C₄×C₈	—	73	2	no	—
159	C₄×C₈	—	73	2	no	—
160	C₄×C₈	—	73	2	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	146	8	no	—
165	C₄×C₈	C₂×#64408	1	1	yes	characteristic
166	C₄×C₈	C₂×#64408	1	1	yes	characteristic
167	C₄×C₈	C₂×#64408	1	1	yes	characteristic
168	C₂²×C₄	—	73	1	no	—
169	C₂²×C₄	—	73	1	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	—	73	4	no	—
172	C₂×C₈	—	73	4	no	—
173	C₂×C₈	—	73	8	no	—
174	C₂×C₈	—	73	2	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	8	no	—
177	C₂×C₈	—	73	8	no	—
178	C₂×C₈	—	73	4	no	—
179	C₂×C₈	—	73	4	no	—
180	C₂×C₈	—	73	2	no	—
181	C₂×C₈	—	146	4	no	—
182	C₂×C₈	—	146	4	no	—
183	C₂×C₈	—	146	4	no	—
184	C₂×C₈	C₄×#64408	1	2	yes	—
185	C₂×C₈	C₄×#64408	1	2	yes	—
186	C₂×C₈	C₄×#64408	1	2	yes	—
187	C₄²	—	73	8	no	—
188	C₄²	—	73	1	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	146	4	no	—
192	C₄²	—	146	4	no	—
193	C₄²	C₂²×#64408	1	1	yes	characteristic
194	C₂³	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	4	no	—
199	C₂×C₄	—	73	4	no	—
200	C₂×C₄	—	73	4	no	—
201	C₂×C₄	—	73	8	no	—
202	C₂×C₄	—	146	4	no	—
203	C₂×C₄	—	146	4	no	—
204	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	73	8	no	—
209	C₈	—	73	4	no	—
210	C₈	—	73	4	no	—
211	C₈	—	146	4	no	—
212	C₈	—	146	4	no	—
213	C₈	—	146	2	no	—
214	C₈	C₈×#64408	1	4	yes	complemented
215	C₈	#14556645	1	4	yes	direct product factor
216	C₂²	—	73	2	no	—
217	C₂²	—	73	2	no	—
218	C₂²	—	73	2	no	—
219	C₂²	C₄²×#64408	1	1	yes	characteristic
220	C₄	—	73	2	no	—
221	C₄	—	73	2	no	—
222	C₄	—	73	2	no	—
223	C₄	—	146	8	no	—
224	C₄	C₂×C₈×#64408	1	2	yes	—
225	C₄	C₂×#14556645	1	2	yes	—
226	C₄	#17989550	1	2	yes	—
227	C₂	—	73	4	no	—
228	C₂	C₄×C₈×#64408	1	1	yes	characteristic
229	C₂	C₄×#14556645	1	1	yes	characteristic
230	C₂	#22625587	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₄²×C₈
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₄³
prime radical	C₂²×D₇₃	C₂×C₄²
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	C₈×#97
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
15. #17989429	C₄	semidirect
16. #17989428	C₄	semidirect
17. C₈×#10596833	C₄	semidirect
21. C₈×#10596831	C₄	semidirect
22. C₈×#10596831	C₄	semidirect
23. C₈×#10596831	C₄	semidirect
24. C₈²×C₇₃	C₄	semidirect
31. #14556645	C₈	direct (N × H)
93. #10596833	C₄×C₈	semidirect
95. C₈×C₇₃	C₄×C₈	semidirect
96. C₈×C₇₃	#97	semidirect
98. #10596831	C₄×C₈	semidirect
99. #10596831	#97	semidirect
123. C₇₃	C₈×#97	semidirect
140. C₈²	#64408	semidirect
214. C₈	C₈×#64408	semidirect
215. C₈	#14556645	direct (N × H)

Maximal subgroups

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

subgroup	index
#22625644	2
C₄×#14556645	2
C₄×C₈×#64408	2
#22625127	2
#22625125	2
C₈×#11924036	2
C₈²×D₇₃	2
C₈×#97	73

Subgroups — C8×#14556645 #34001925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14556645 #34001925