Subgroups — C₈×#14556658 #34003920

Order 18688.

550 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556658	C₂	1	2	yes	maximal
2	C₂×C₈×#10596833	C₂	1	2	yes	maximal
3	#22624753	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556658	C₄	1	4	yes	—
6	C₂×C₄×#10596833	C₂²	1	4	yes	—
7	#17989453	C₄	1	4	yes	—
8	C₈×#10596833	—	2	4	no	—
9	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
16	C₂²×#10596833	C₂×C₄	1	8	yes	—
17	#14556658	C₈	1	16	yes	direct product factor
18	C₄×#10596833	—	2	4	no	—
19	C₄×#10596833	—	2	4	no	—
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₄×#10596831	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₈×Dic₇₃	C₈	1	4	yes	complemented
29	C₈×Dic₇₃	C₈	1	4	yes	complemented
30	C₈×Dic₇₃	D₄	1	2	yes	—
31	C₈×Dic₇₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	—	2	2	no	—
36	C₂×#10596833	—	2	8	no	—
37	C₂×#10596833	—	2	8	no	—
38	C₂×#10596833	C₂×C₈	1	16	yes	—
39	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
40	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
41	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	C₄⋊C₄	1	2	yes	—
45	C₂×#10596831	C₂×C₈	1	2	yes	—
46	C₂×#10596831	C₂×C₈	1	2	yes	—
47	C₂×#10596831	M₁₆	1	2	yes	—
48	C₄×Dic₇₃	—	2	4	no	—
49	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
50	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
51	C₄×Dic₇₃	C₂×C₈	1	8	yes	—
52	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₄×Dic₇₃	#41	1	2	yes	—
55	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
63	C₄²×C₇₃	—	2	1	no	—
64	C₄²×C₇₃	—	2	1	no	—
65	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
77	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
78	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
79	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
80	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
81	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
82	C₂×Dic₇₃	#131	1	4	yes	—
83	#10596833	—	2	16	no	—
84	C₈×C₇₃	—	2	2	no	—
85	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
86	#10596831	—	2	2	no	—
87	#10596831	C₄⋊C₈	1	4	yes	complemented
88	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
90	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
92	C₂²×C₇₃	#405	1	2	yes	—
93	C₄×C₇₃	—	2	4	no	—
94	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₇₃	#431	1	1	yes	characteristic
97	C₄×C₇₃	#431	1	1	yes	characteristic
98	Dic₇₃	—	2	2	no	—
99	Dic₇₃	C₈²	1	4	yes	complemented
100	Dic₇₃	C₈×D₄	1	4	yes	—
101	Dic₇₃	C₈×Q₈	1	4	yes	—
102	C₈×C₄⋊C₈	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
104	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
105	C₂×C₇₃	#1450	1	1	yes	characteristic
106	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₇₃	C₈×C₄⋊C₄	1	2	yes	—
108	C₄×C₄⋊C₈	—	73	2	no	—
109	C₄²×C₈	—	73	1	no	—
110	C₂×C₈²	—	73	2	no	—
111	#1273	—	73	2	no	—
112	C₇₃	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊C₈	—	73	4	no	—
114	C₂×C₄×C₈	—	73	1	no	—
115	C₂×C₄×C₈	—	73	1	no	—
116	C₂×C₄×C₈	—	73	1	no	—
117	C₂×C₄×C₈	—	73	4	no	—
118	C₂×C₄×C₈	—	73	1	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	#64408	1	1	yes	characteristic
121	C₄³	—	73	1	no	—
122	#336	—	73	4	no	—
123	C₈²	—	146	4	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	8	no	—
126	C₂²×C₈	—	73	1	no	—
127	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
128	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
129	C₂×C₄²	—	73	2	no	—
130	C₂×C₄²	—	73	1	no	—
131	C₂×C₄²	—	73	1	no	—
132	C₂×C₄²	—	73	2	no	—
133	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
134	C₄⋊C₈	—	73	16	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	73	2	no	—
137	C₄×C₈	—	73	2	no	—
138	C₄×C₈	—	73	4	no	—
139	C₄×C₈	—	73	4	no	—
140	C₄×C₈	—	146	2	no	—
141	C₄×C₈	—	146	2	no	—
142	C₄×C₈	—	146	4	no	—
143	C₄×C₈	—	146	4	no	—
144	C₂²×C₄	—	73	1	no	—
145	C₂²×C₄	—	73	2	no	—
146	C₂²×C₄	—	73	1	no	—
147	C₂²×C₄	C₄×#64408	1	2	yes	—
148	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	73	2	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	—	73	2	no	—
154	C₂×C₈	—	73	16	no	—
155	C₂×C₈	—	73	2	no	—
156	C₂×C₈	—	146	8	no	—
157	C₂×C₈	—	146	8	no	—
158	C₂×C₈	—	146	1	no	—
159	C₂×C₈	—	146	1	no	—
160	C₂×C₈	C₄⋊#64408	1	2	yes	—
161	C₂×C₈	#11924036	1	2	yes	—
162	C₂×C₈	#11924036	1	2	yes	—
163	C₂×C₈	#11924037	1	2	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	73	2	no	—
167	C₄²	—	73	2	no	—
168	C₄²	—	73	8	no	—
169	C₄²	—	73	1	no	—
170	C₄²	—	73	1	no	—
171	C₄²	—	73	2	no	—
172	C₄²	—	146	4	no	—
173	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	73	2	no	—
177	C₂×C₄	—	73	2	no	—
178	C₂×C₄	—	73	4	no	—
179	C₂×C₄	—	73	4	no	—
180	C₂×C₄	—	73	2	no	—
181	C₂×C₄	—	73	4	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	C₈×#64408	1	4	yes	—
185	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
186	C₂×C₄	#14556670	1	1	yes	characteristic
187	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
188	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
189	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
190	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	73	4	no	—
193	C₈	—	146	16	no	—
194	C₈	—	146	2	no	—
195	C₈	#14556658	1	4	yes	direct product factor
196	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
197	C₂²	C₂×C₈×#64408	1	2	yes	—
198	C₂²	C₄×#11924036	1	1	yes	characteristic
199	C₂²	C₄×#11924037	1	1	yes	characteristic
200	C₂²	#17989550	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	73	4	no	—
203	C₄	—	73	4	no	—
204	C₄	—	73	4	no	—
205	C₄	—	146	2	no	—
206	C₄	#17989588	1	1	yes	characteristic
207	C₄	C₂×#14556658	1	1	yes	characteristic
208	C₄	#17989604	1	1	yes	characteristic
209	C₄	#17989605	1	1	yes	characteristic
210	C₂	C₈×#11924036	1	1	yes	characteristic
211	C₂	C₈×#11924037	1	2	yes	—
212	C₂	#22625135	1	1	yes	characteristic
213	C₂	C₄×#14556658	1	1	yes	characteristic
214	C₂	C₈×C₄⋊#64408	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
17. #14556658	C₈	direct (N × H)
28. C₈×Dic₇₃	C₈	semidirect
29. C₈×Dic₇₃	C₈	semidirect
85. C₈×C₇₃	C₄⋊C₈	semidirect
87. #10596831	C₄⋊C₈	semidirect
99. Dic₇₃	C₈²	semidirect
112. C₇₃	C₈×C₄⋊C₈	semidirect
195. C₈	#14556658	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14556658	2
C₂×C₈×#10596833	2
#22624753	2
C₄×C₈×Dic₇₃	2
C₈×C₄⋊C₈	73

Subgroups — C8×#14556658 #34003920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14556658 #34003920