Subgroups — C₂₃×#3079 #15670698

Order 2944.

416 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#477	C₂	1	2	yes	maximal, complemented
4	C₂₃×#476	C₂	1	2	yes	maximal, complemented
5	C₂₃×#475	C₂	1	2	yes	maximal, complemented
6	C₂₃×#474	C₂	1	2	yes	maximal, complemented
7	C₂₃×#428	C₂	1	2	yes	maximal, complemented
8	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#417	C₂	1	2	yes	maximal
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₃	—	2	1	no	—
12	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₂₃	—	2	1	no	—
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#117	—	2	1	no	—
24	C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₃	—	2	1	no	—
28	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	4	yes	—
30	C₂₃×#106	C₂²	1	4	yes	—
31	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₂₃×#103	C₂²	1	4	yes	—
34	C₂₃×#102	C₂²	1	4	yes	—
35	#41×C₂₃	—	2	2	no	—
36	#41×C₂₃	Q₈	1	4	yes	complemented
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	4	4	no	—
40	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₂₃	—	2	1	no	—
42	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	M₁₆×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	C₂³	1	4	yes	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
58	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₂²⋊C₄×C₂₃	—	2	1	no	—
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	1	no	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	D₄	1	1	yes	characteristic
67	C₄²×C₂₃	D₄	1	1	yes	characteristic
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₂³×C₂₃	—	2	1	no	—
71	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₂₃	—	2	1	no	—
73	Q₈×C₂₃	—	4	4	no	—
74	Q₈×C₂₃	—	4	4	no	—
75	Q₈×C₂₃	C₂×Q₈	1	2	yes	complemented
76	D₄×C₂₃	—	2	4	no	—
77	D₄×C₂₃	—	2	1	no	—
78	D₄×C₂₃	C₂×Q₈	1	2	yes	complemented
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₃	—	2	4	no	—
98	#3079	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	4	1	no	—
103	C₂²×C₂₃	—	4	1	no	—
104	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
105	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	4	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₄×C₂₃	#137	1	2	yes	—
115	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
116	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₄×#41	C₂×C₂₃	1	1	yes	characteristic
119	#477	C₂×C₂₃	1	2	yes	complemented
120	#476	C₂×C₂₃	1	2	yes	complemented
121	#475	C₂×C₂₃	1	2	yes	complemented
122	#474	C₂×C₂₃	1	2	yes	complemented
123	#428	C₂×C₂₃	1	2	yes	complemented
124	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#417	C₂×C₂₃	1	2	yes	—
126	C₂×C₂₃	—	2	2	no	—
127	C₂×C₂₃	—	4	2	no	—
128	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
131	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
132	C₂²×Q₈	—	2	1	no	—
133	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
134	C₄⋊Q₈	—	2	1	no	—
135	C₄⋊Q₈	—	2	1	no	—
136	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
138	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
139	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
144	#117	—	2	1	no	—
145	#117	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
150	#107	C₂²×C₂₃	1	4	yes	—
151	#106	C₂²×C₂₃	1	4	yes	—
152	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
154	#103	C₂²×C₂₃	1	4	yes	—
155	#102	C₂²×C₂₃	1	4	yes	—
156	C₂₃	#3079	1	1	yes	characteristic, direct product factor, Sylow
157	#41	—	2	2	no	—
158	#41	Q₈×C₂₃	1	4	yes	complemented
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	M₁₆	—	2	4	no	—
173	C₂×C₈	C₂³×C₂₃	1	4	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
180	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	D₄×C₂₃	1	1	yes	characteristic
189	C₄²	D₄×C₂₃	1	1	yes	characteristic
190	C₄²	C₂³×C₂₃	1	1	yes	characteristic
191	C₄²	C₂³×C₂₃	1	1	yes	characteristic
192	C₂³	—	2	1	no	—
193	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
194	Q₈	—	2	1	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	4	4	no	—
197	Q₈	C₂×Q₈×C₂₃	1	2	yes	complemented
198	D₄	—	2	4	no	—
199	D₄	—	2	1	no	—
200	D₄	C₂×Q₈×C₂₃	1	2	yes	complemented
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
217	C₂×C₄	#41×C₂₃	1	2	yes	—
218	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
226	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	4	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
234	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
235	C₄	C₂₃×#137	1	2	yes	—
236	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
237	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#477	C₂	semidirect
4. C₂₃×#476	C₂	semidirect
5. C₂₃×#475	C₂	semidirect
6. C₂₃×#474	C₂	semidirect
7. C₂₃×#428	C₂	semidirect
8. C₂₃×#423	C₂	semidirect
36. #41×C₂₃	Q₈	semidirect
75. Q₈×C₂₃	C₂×Q₈	semidirect
78. D₄×C₂₃	C₂×Q₈	semidirect
98. #3079	C₂₃	direct (N × H)
117. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
119. #477	C₂×C₂₃	semidirect
120. #476	C₂×C₂₃	semidirect
121. #475	C₂×C₂₃	semidirect
122. #474	C₂×C₂₃	semidirect
123. #428	C₂×C₂₃	semidirect
124. #423	C₂×C₂₃	semidirect
156. C₂₃	#3079	direct (N × H)
158. #41	Q₈×C₂₃	semidirect
197. Q₈	C₂×Q₈×C₂₃	semidirect
200. D₄	C₂×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₄×#41×C₂₃	2
C₂₃×#477	2
C₂₃×#476	2
C₂₃×#475	2
C₂₃×#474	2
C₂₃×#428	2
C₂₃×#423	2
C₂₃×#417	2
#3079	23

Subgroups — C23×#3079 #15670698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3079 #15670698