Subgroups — C₂×C₂₃×#410 #15670501

Order 2944.

788 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	2	yes	maximal, complemented
3	C₂₃×#410	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#99	C₂	1	4	yes	maximal, complemented
5	C₂×#41×C₂₃	—	2	2	no	—
6	C₂×#41×C₂₃	—	2	4	no	—
7	C₂×#41×C₂₃	—	2	2	no	—
8	C₂×#41×C₂₃	C₄	1	4	yes	complemented
9	C₂×#41×C₂₃	C₂²	1	4	yes	—
10	C₂²×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₄	1	2	yes	complemented
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	—	2	4	no	—
16	C₂₃×#117	C₂²	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₃	—	2	2	no	—
20	C₂₃×#99	C₂²	1	16	yes	complemented
21	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
22	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
23	#41×C₂₃	—	2	8	no	—
24	#41×C₂₃	—	2	8	no	—
25	#41×C₂₃	—	2	8	no	—
26	#41×C₂₃	—	2	8	no	—
27	C₂×Q₈×C₂₃	—	2	2	no	—
28	C₂×Q₈×C₂₃	—	2	2	no	—
29	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
30	C₂×D₄×C₂₃	—	2	4	no	—
31	C₂×D₄×C₂₃	—	2	4	no	—
32	C₂×D₄×C₂₃	—	2	2	no	—
33	C₂×D₄×C₂₃	—	2	2	no	—
34	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
36	C₂×D₄×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	complemented
46	C₂²×C₄×C₂₃	D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	D₄	1	4	yes	—
48	C₂²×C₄×C₂₃	D₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₂₃	—	2	8	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
53	C₄²×C₂₃	—	2	8	no	—
54	C₂³×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	2	no	—
56	C₂³×C₂₃	—	4	2	no	—
57	C₂³×C₂₃	—	4	4	no	—
58	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₂₃	C₂×D₄	1	2	yes	—
62	C₂³×C₂₃	C₂×D₄	1	2	yes	—
63	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	8	no	—
65	D₄×C₂₃	—	2	8	no	—
66	D₄×C₂₃	—	2	8	no	—
67	D₄×C₂₃	—	2	8	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	—	2	8	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	4	8	no	—
74	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
80	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
86	C₂×#410	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₂₃	—	2	4	no	—
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	4	1	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	—	4	1	no	—
93	C₂²×C₂₃	—	4	4	no	—
94	C₂²×C₂₃	—	4	4	no	—
95	C₂²×C₂₃	—	4	4	no	—
96	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
101	C₄×C₂₃	—	2	4	no	—
102	C₄×C₂₃	—	4	8	no	—
103	C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
105	C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂²×#41	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#117	C₂×C₂₃	1	2	yes	complemented
108	#410	C₂×C₂₃	1	8	yes	direct product factor
109	C₂×#99	C₂×C₂₃	1	4	yes	complemented
110	C₂×C₂₃	—	2	2	no	—
111	C₂×C₂₃	—	2	1	no	—
112	C₂×C₂₃	—	2	1	no	—
113	C₂×C₂₃	—	2	2	no	—
114	C₂×C₂₃	—	4	4	no	—
115	C₂×C₂₃	#410	1	2	yes	direct product factor
116	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×#41	—	2	2	no	—
118	C₂×#41	—	2	2	no	—
119	C₂×#41	—	2	4	no	—
120	C₂×#41	C₄×C₂₃	1	4	yes	complemented
121	C₂×#41	C₂²×C₂₃	1	4	yes	—
122	C₂²×Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
123	C₂²×D₄	C₄×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
125	C₂³×C₄	C₄×C₂₃	1	2	yes	complemented
126	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
127	#117	—	2	4	no	—
128	#117	C₂²×C₂₃	1	8	yes	complemented
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
131	C₂×C₄²	—	2	2	no	—
132	#99	C₂²×C₂₃	1	16	yes	complemented
133	C₂₃	C₂×#410	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	C₂×C₄×C₂₃	1	2	yes	—
135	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
136	#41	—	2	8	no	—
137	#41	—	2	8	no	—
138	#41	—	2	8	no	—
139	#41	—	2	8	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
148	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
149	C₂×D₄	C₂³×C₂₃	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
159	C₂²×C₄	D₄×C₂₃	1	2	yes	—
160	C₂²×C₄	D₄×C₂₃	1	2	yes	—
161	C₂²×C₄	D₄×C₂₃	1	4	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
166	C₄²	—	2	8	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
172	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
173	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
174	C₂³	C₂×D₄×C₂₃	1	2	yes	—
175	C₂³	C₂×D₄×C₂₃	1	2	yes	—
176	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	D₄	—	2	8	no	—
179	D₄	—	2	8	no	—
180	D₄	—	2	8	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
188	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
191	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
192	C₂×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
193	C₂×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
195	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
196	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
198	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂²	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
210	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
212	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	4	8	no	—
215	C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
216	C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
217	C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	2	2	no	—
222	C₂	—	4	4	no	—
223	C₂	C₂₃×#410	1	2	yes	direct product factor
224	C₂	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#117	C₂	semidirect
3. C₂₃×#410	C₂	direct (N × H)
4. C₂×C₂₃×#99	C₂	semidirect
8. C₂×#41×C₂₃	C₄	semidirect
10. C₂²×Q₈×C₂₃	C₄	semidirect
11. C₂²×D₄×C₂₃	C₄	semidirect
13. C₂³×C₄×C₂₃	C₄	semidirect
16. C₂₃×#117	C₂²	semidirect
20. C₂₃×#99	C₂²	semidirect
29. C₂×Q₈×C₂₃	C₂×C₄	semidirect
35. C₂×D₄×C₂₃	C₂×C₄	semidirect
45. C₂²×C₄×C₂₃	C₂×C₄	semidirect
86. C₂×#410	C₂₃	direct (N × H)
107. C₂×#117	C₂×C₂₃	semidirect
108. #410	C₂×C₂₃	direct (N × H)
109. C₂×#99	C₂×C₂₃	semidirect
115. C₂×C₂₃	#410	direct (N × H)
120. C₂×#41	C₄×C₂₃	semidirect
122. C₂²×Q₈	C₄×C₂₃	semidirect
123. C₂²×D₄	C₄×C₂₃	semidirect
125. C₂³×C₄	C₄×C₂₃	semidirect
128. #117	C₂²×C₂₃	semidirect
132. #99	C₂²×C₂₃	semidirect
133. C₂₃	C₂×#410	direct (N × H)
142. C₂×Q₈	C₂×C₄×C₂₃	semidirect
148. C₂×D₄	C₂×C₄×C₂₃	semidirect
158. C₂²×C₄	C₂×C₄×C₂₃	semidirect
223. C₂	C₂₃×#410	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×#41×C₂₃	2
C₂×C₂₃×#117	2
C₂₃×#410	2
C₂×C₂₃×#99	2
C₂×#410	23

Subgroups — C2×C23×#410 #15670501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#410 #15670501