Subgroups — C₂×#13092418 #15681909

Order 2944.

816 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	C₂²×#67267	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092418	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10625017	C₂	1	2	yes	maximal, complemented
5	C₂×#10625014	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#41×C₂₃	—	2	4	no	—
8	C₂×#41×C₂₃	—	2	2	no	—
9	C₂×#41×C₂₃	C₄	1	8	yes	complemented
10	C₂×#41×C₂₃	C₂²	1	4	yes	—
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	—	2	1	no	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	—	2	1	no	—
15	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×Dic₂₃	—	2	1	no	—
17	C₂×C₄⋊Dic₂₃	—	2	2	no	—
18	C₂×C₄⋊Dic₂₃	—	2	1	no	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
21	C₂×#67267	—	2	4	no	—
22	C₂×#67267	C₂²	1	4	yes	complemented
23	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
24	#10625017	C₂²	1	8	yes	complemented
25	#10625014	C₂²	1	8	yes	complemented
26	C₂⁴×C₂₃	—	2	1	no	—
27	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
28	#41×C₂₃	—	2	16	no	—
29	#41×C₂₃	—	2	8	no	—
30	#41×C₂₃	C₂×C₄	1	16	yes	complemented
31	C₂×Q₈×C₂₃	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	2	1	no	—
33	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
34	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
35	C₂×D₄×C₂₃	—	2	8	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	—	2	1	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	2	8	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	D₄	1	4	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×Dic₂₃	—	2	2	no	—
51	C₂²×Dic₂₃	—	2	4	no	—
52	C₂²×Dic₂₃	—	4	4	no	—
53	C₄⋊Dic₂₃	—	2	2	no	—
54	C₄⋊Dic₂₃	—	2	4	no	—
55	C₄⋊Dic₂₃	C₂³	1	4	yes	—
56	#67267	—	2	8	no	—
57	C₂×C₂₃⋊C₈	—	2	4	no	—
58	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	4	2	no	—
64	C₂³×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	C₂²⋊C₄	1	4	yes	—
66	C₂³×C₂₃	C₂×D₄	1	2	yes	—
67	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₂₃	—	2	4	no	—
69	Q₈×C₂₃	—	2	2	no	—
70	Q₈×C₂₃	C₂²×C₄	1	4	yes	complemented
71	D₄×C₂₃	—	2	16	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	C₂²×C₄	1	4	yes	complemented
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	16	no	—
78	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
80	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₂₃	—	2	4	no	—
88	C₂×Dic₂₃	—	4	8	no	—
89	C₂×Dic₂₃	—	4	4	no	—
90	C₂₃⋊C₈	—	2	4	no	—
91	C₂×#417	—	23	1	no	maximal, Sylow
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	8	no	—
99	C₂²×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂²×C₂₃	#137	1	2	yes	—
103	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
106	C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	Dic₂₃	—	4	4	no	—
110	C₂²×#41	D₂₃	1	1	yes	characteristic
111	C₂²×M₁₆	—	23	1	no	—
112	C₂²×C₄⋊C₄	—	23	1	no	—
113	#417	—	23	8	no	—
114	C₂×#103	—	23	2	no	—
115	C₂×#102	—	23	2	no	—
116	C₂×C₂₃	—	2	4	no	—
117	C₂×C₂₃	—	4	4	no	—
118	C₂×C₂₃	#417	1	4	yes	complemented
119	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
122	C₂×#41	—	2	4	no	—
123	C₂×#41	—	2	2	no	—
124	C₂×#41	Dic₂₃	1	8	yes	complemented
125	C₂×#41	C₂×D₂₃	1	4	yes	—
126	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
127	C₂²×D₄	—	2	1	no	—
128	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
129	C₂³×C₄	—	2	1	no	—
130	C₂³×C₄	—	46	1	no	—
131	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
132	C₂×M₁₆	—	23	4	no	—
133	C₂×M₁₆	—	46	4	no	—
134	C₂²×C₈	—	23	2	no	—
135	C₂×C₄⋊C₄	—	23	2	no	—
136	C₂×C₄⋊C₄	—	23	4	no	—
137	C₂×C₄⋊C₄	—	46	2	no	—
138	C₂×C₄⋊C₄	—	46	1	no	—
139	#103	—	23	8	no	—
140	#102	—	23	8	no	—
141	C₂₃	C₂×#417	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	—	2	1	no	—
143	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
144	#41	—	2	16	no	—
145	#41	—	2	8	no	—
146	#41	C₂×Dic₂₃	1	16	yes	complemented
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂×Dic₂₃	1	4	yes	complemented
150	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
151	C₂×D₄	—	2	8	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₂×Dic₂₃	1	4	yes	complemented
156	C₂×D₄	C₂²×D₂₃	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	46	4	no	—
161	C₂²×C₄	—	46	2	no	—
162	C₂²×C₄	—	92	4	no	—
163	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
164	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
165	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
167	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
168	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
169	M₁₆	—	46	8	no	—
170	C₂×C₈	—	23	4	no	—
171	C₂×C₈	—	46	4	no	—
172	C₄⋊C₄	—	23	4	no	—
173	C₄⋊C₄	—	46	4	no	—
174	C₄⋊C₄	—	46	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂²⋊Dic₂₃	1	4	yes	—
182	C₂³	C₂×C₂²⋊D₂₃	1	2	yes	—
183	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
184	Q₈	—	2	4	no	—
185	Q₈	—	2	2	no	—
186	Q₈	C₂²×Dic₂₃	1	4	yes	complemented
187	D₄	—	2	16	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	C₂²×Dic₂₃	1	4	yes	complemented
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	46	4	no	—
195	C₂×C₄	—	92	8	no	—
196	C₂×C₄	—	92	4	no	—
197	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
198	C₂×C₄	C₂²⋊Dic₂₃	1	8	yes	—
199	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
201	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
205	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
206	C₈	—	46	4	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#10625120	1	2	yes	—
215	C₂²	#10625122	1	2	yes	—
216	C₂²	C₂×C₂²⋊Dic₂₃	1	4	yes	—
217	C₂²	C₂×C₂²⋊Dic₂₃	1	2	yes	—
218	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	92	4	no	—
221	C₄	C₂×C₂²⋊Dic₂₃	1	4	yes	—
222	C₄	C₂×C₂²⋊Dic₂₃	1	2	yes	—
223	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
224	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	#13092418	1	4	yes	direct product factor
228	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
229	C₂	C₂×#10625120	1	1	yes	characteristic
230	C₂	C₂×#10625122	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂²×#41×C₂₃	C₂
socle	C₂³×C₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×#41	D₂₃
hypocenter	C₂₃	C₂×#417
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#67267	C₂	semidirect
2. C₂²×C₄⋊Dic₂₃	C₂	semidirect
3. #13092418	C₂	direct (N × H)
4. C₂×#10625017	C₂	semidirect
5. C₂×#10625014	C₂	semidirect
9. C₂×#41×C₂₃	C₄	semidirect
20. C₂×C₄⋊Dic₂₃	C₂²	semidirect
22. C₂×#67267	C₂²	semidirect
24. #10625017	C₂²	semidirect
25. #10625014	C₂²	semidirect
30. #41×C₂₃	C₂×C₄	semidirect
33. C₂×Q₈×C₂₃	C₂×C₄	semidirect
39. C₂×D₄×C₂₃	C₂×C₄	semidirect
70. Q₈×C₂₃	C₂²×C₄	semidirect
74. D₄×C₂₃	C₂²×C₄	semidirect
118. C₂×C₂₃	#417	semidirect
124. C₂×#41	Dic₂₃	semidirect
141. C₂₃	C₂×#417	semidirect
146. #41	C₂×Dic₂₃	semidirect
149. C₂×Q₈	C₂×Dic₂₃	semidirect
155. C₂×D₄	C₂×Dic₂₃	semidirect
186. Q₈	C₂²×Dic₂₃	semidirect
190. D₄	C₂²×Dic₂₃	semidirect
227. C₂	#13092418	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#67267	2
C₂²×C₄⋊Dic₂₃	2
#13092418	2
C₂×#10625017	2
C₂×#10625014	2
C₂²×#41×C₂₃	2
C₂×#417	23

Subgroups — C2×#13092418 #15681909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092418 #15681909