Subgroups — C₈×#67288 #15682354

Order 2944.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₄₆	C₂	1	2	yes	maximal, complemented
3	C₈×D₉₂	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊D₂₃	C₂	1	4	yes	maximal, complemented
5	#13092470	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092468	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×D₂₃	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#67288	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₂₃	C₄	1	2	yes	—
13	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
14	#10625097	C₄	1	1	yes	characteristic
15	C₂×C₈×D₂₃	C₂²	1	4	yes	—
16	C₄×D₉₂	C₄	1	1	yes	characteristic
17	C₄×D₉₂	C₂²	1	1	yes	characteristic
18	#10625043	C₄	1	2	yes	—
19	C₄²×D₂₃	C₂²	1	2	yes	—
20	C₄×Dic₄₆	C₄	1	1	yes	characteristic
21	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
26	(C₂×D₂₃)⋊C₈	C₂²	1	4	yes	—
27	Dic₂₃⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₂₃	C₂²	1	4	yes	—
29	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
30	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
35	#67288	C₈	1	8	yes	direct product factor
36	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₃	C₂³	1	2	yes	—
40	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	—	2	4	no	—
42	C₂×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂×C₈×C₂₃	C₂³	1	2	yes	—
44	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
49	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
53	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₃	C₂³	1	2	yes	—
58	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
59	C₈×D₂₃	—	2	8	no	—
60	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	complemented
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₂₃	C₂×C₈	1	8	yes	complemented
72	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
75	D₉₂	C₂×C₈	1	4	yes	complemented
76	C₄×D₂₃	—	2	4	no	—
77	C₄×D₂₃	—	2	4	no	—
78	C₄×D₂₃	C₂×C₈	1	8	yes	complemented
79	Dic₄₆	C₂×C₈	1	4	yes	complemented
80	C₈×C₂₃	—	2	2	no	—
81	C₈×C₂₃	#41	1	4	yes	complemented
82	C₂₃⋊C₈	—	2	4	no	—
83	C₈×#41	—	23	1	no	maximal, Sylow
84	C₂²×C₂₃	—	2	1	no	—
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
87	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₂₃	—	2	2	no	—
89	C₂×D₂₃	—	2	2	no	—
90	C₂×D₂₃	C₂²×C₈	1	4	yes	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	C₂²×C₈	1	2	yes	—
94	C₄×C₂₃	C₂²×C₈	1	2	yes	—
95	C₄×C₂₃	#131	1	2	yes	—
96	C₄×C₂₃	C₂×#41	1	2	yes	—
97	Dic₂₃	—	2	2	no	—
98	Dic₂₃	—	2	2	no	—
99	Dic₂₃	C₂²×C₈	1	4	yes	—
100	C₄×#41	—	23	1	no	—
101	C₈×Q₈	—	23	2	no	—
102	C₈×D₄	—	23	4	no	—
103	C₈×D₄	—	23	2	no	—
104	#431	—	23	1	no	—
105	#431	—	23	2	no	—
106	C₂×C₄×C₈	—	23	2	no	—
107	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic, complemented
108	C₂×C₂₃	—	2	2	no	—
109	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
110	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
111	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
112	D₂₃	—	2	4	no	—
113	C₂×#41	—	23	1	no	—
114	C₂²×C₈	—	23	4	no	—
115	C₂²×C₈	C₂×D₂₃	1	2	yes	—
116	C₄×Q₈	—	23	1	no	—
117	C₄×Q₈	—	23	1	no	—
118	C₄×D₄	—	23	2	no	—
119	C₄×D₄	—	23	2	no	—
120	C₄×D₄	—	23	1	no	—
121	C₄×D₄	—	23	1	no	—
122	#117	—	23	1	no	—
123	#117	—	23	2	no	—
124	C₂×C₄²	—	23	2	no	—
125	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
126	C₄⋊C₈	—	23	4	no	—
127	C₄⋊C₈	—	23	2	no	—
128	C₂²⋊C₈	—	23	2	no	—
129	C₂²⋊C₈	—	23	4	no	—
130	C₄×C₈	—	23	4	no	—
131	C₄×C₈	—	23	2	no	—
132	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
133	C₄×C₈	C₂×D₂₃	1	2	yes	—
134	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
135	C₂₃	C₈×#41	1	1	yes	characteristic, complemented, Sylow
136	#41	—	23	8	no	—
137	C₂×Q₈	—	23	1	no	—
138	C₂×D₄	—	23	2	no	—
139	C₂×D₄	—	23	1	no	—
140	C₂²×C₄	—	23	2	no	—
141	C₂²×C₄	—	23	2	no	—
142	C₂²×C₄	—	23	2	no	—
143	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	23	4	no	—
148	C₂×C₈	—	46	8	no	—
149	C₂×C₈	C₂²×D₂₃	1	2	yes	—
150	C₂×C₈	C₂²×D₂₃	1	2	yes	—
151	C₄⋊C₄	—	23	2	no	—
152	C₄⋊C₄	—	23	2	no	—
153	C₄⋊C₄	—	23	1	no	—
154	C₄⋊C₄	—	23	1	no	—
155	C₂²⋊C₄	—	23	2	no	—
156	C₂²⋊C₄	—	23	2	no	—
157	C₂²⋊C₄	—	23	2	no	—
158	C₄²	—	23	2	no	—
159	C₄²	—	23	2	no	—
160	C₄²	—	23	2	no	—
161	C₄²	C₄×D₂₃	1	1	yes	characteristic
162	C₄²	C₄×D₂₃	1	1	yes	characteristic
163	C₄²	C₂²×D₂₃	1	1	yes	characteristic
164	C₄²	C₂²×D₂₃	1	1	yes	characteristic
165	C₂³	—	23	2	no	—
166	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
167	Q₈	—	23	4	no	—
168	D₄	—	23	8	no	—
169	D₄	—	23	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	23	2	no	—
173	C₂×C₄	—	23	8	no	—
174	C₂×C₄	—	23	2	no	—
175	C₂×C₄	—	23	2	no	—
176	C₂×C₄	—	46	4	no	—
177	C₂×C₄	—	46	4	no	—
178	C₂×C₄	C₈×D₂₃	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	46	4	no	—
187	C₈	#67288	1	4	yes	direct product factor
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	23	4	no	—
191	C₂²	—	46	2	no	—
192	C₂²	—	46	2	no	—
193	C₂²	C₂×C₈×D₂₃	1	2	yes	—
194	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	23	4	no	—
198	C₄	—	46	2	no	—
199	C₄	—	46	2	no	—
200	C₄	C₂×C₈×D₂₃	1	2	yes	—
201	C₄	C₂×C₈×D₂₃	1	2	yes	—
202	C₄	#10625072	1	2	yes	—
203	C₄	C₂×#67288	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	46	4	no	—
206	C₂	C₄×#67288	1	1	yes	characteristic
207	C₂	C₂²×C₈×D₂₃	1	1	yes	characteristic
208	C₂	C₂×#10625072	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₂₃
commutator	C₂×C₂₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂×#67288	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	C₈×#41
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₄₆	C₂	semidirect
3. C₈×D₉₂	C₂	semidirect
4. C₈×C₂²⋊D₂₃	C₂	semidirect
5. #13092470	C₂	semidirect
6. #13092468	C₂	semidirect
7. C₄×C₈×D₂₃	C₂	semidirect
8. C₂×C₄×C₈×C₂₃	C₂	semidirect
35. #67288	C₈	direct (N × H)
64. C₂×C₄×C₂₃	C₂×C₈	semidirect
71. C₂²⋊D₂₃	C₂×C₈	semidirect
75. D₉₂	C₂×C₈	semidirect
78. C₄×D₂₃	C₂×C₈	semidirect
79. Dic₄₆	C₂×C₈	semidirect
81. C₈×C₂₃	#41	semidirect
107. C₂×C₄×C₈	D₂₃	semidirect
135. C₂₃	C₈×#41	semidirect
178. C₂×C₄	C₈×D₂₃	semidirect
187. C₈	#67288	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×#67288	2
C₈×Dic₄₆	2
C₈×D₉₂	2
C₈×C₂²⋊D₂₃	2
#13092470	2
#13092468	2
C₄×C₈×D₂₃	2
C₂×C₄×C₈×C₂₃	2
C₈×#41	23

Subgroups — C8×#67288 #15682354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#67288 #15682354