Subgroups — C₂×C₂₃×#1338 #19159235

Order 5888.

346 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	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1338	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₂₃	—	2	1	no	—
7	C₂²×C₁₆×C₂₃	—	2	1	no	—
8	C₂₃×#446	—	2	1	no	—
9	C₂₃×#446	—	2	1	no	—
10	C₂₃×#446	C₄	1	2	yes	—
11	C₂₃×#446	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₂₃×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₁₆	—	2	1	no	—
17	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂₃×#346	—	2	1	no	—
19	C₂₃×#346	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	—	2	1	no	—
25	C₄×Q₈×C₂₃	—	2	1	no	—
26	C₄×Q₈×C₂₃	C₈	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	—	2	4	no	—
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂×C₁₆×C₂₃	—	2	2	no	—
35	C₂×C₁₆×C₂₃	—	4	2	no	—
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	1	no	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂₃×#97	—	2	2	no	—
41	C₂₃×#97	—	2	2	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	—	2	1	no	—
47	C₂×Q₈×C₂₃	—	4	2	no	—
48	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
60	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
61	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
72	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₁₆×C₂₃	—	2	4	no	—
74	C₁₆×C₂₃	—	4	2	no	—
75	C₂×#1338	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
77	Q₈×C₂₃	—	4	2	no	—
78	Q₈×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#103	1	2	yes	—
89	C₂×C₄×C₂₃	#104	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	4	2	no	—
96	C₈×C₂₃	#103	1	2	yes	—
97	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
100	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
101	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
102	#1338	C₂×C₂₃	1	4	yes	direct product factor
103	C₂²×C₂₃	Q₈⋊C₈	1	2	yes	—
104	C₂²×C₂₃	#349	1	2	yes	—
105	C₂²×C₂₃	#350	1	2	yes	—
106	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	Q₈⋊C₈	1	2	yes	—
114	C₄×C₂₃	C₂×#103	1	1	yes	characteristic
115	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
116	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₁₆	—	2	1	no	—
118	C₂²×C₁₆	—	2	1	no	—
119	#446	—	2	1	no	—
120	#446	—	2	1	no	—
121	#446	C₄×C₂₃	1	2	yes	—
122	#446	C₂²×C₂₃	1	2	yes	—
123	C₂×C₄⋊C₈	—	2	1	no	—
124	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
125	C₂×#97	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
129	#346	—	2	1	no	—
130	#346	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂₃	#1338	1	4	yes	direct product factor
132	C₂×C₂₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
133	C₂×C₂₃	C₂×#349	1	1	yes	characteristic
134	C₂×C₂₃	C₂×#350	1	1	yes	characteristic
135	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₈×C₂₃	1	2	yes	—
142	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₁₆	—	2	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	4	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
155	#97	—	2	2	no	—
156	#97	—	2	2	no	—
157	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₂₃	C₂×#1338	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
176	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
177	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
186	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
187	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
188	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₁₆	—	2	4	no	—
190	C₁₆	—	4	2	no	—
191	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂₃×#103	1	2	yes	—
204	C₂×C₄	C₂₃×#104	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	C₂₃×#103	1	2	yes	—
212	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
213	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
214	C₂²	C₂₃×Q₈⋊C₈	1	2	yes	—
215	C₂²	C₂₃×#349	1	2	yes	—
216	C₂²	C₂₃×#350	1	2	yes	—
217	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₂₃×Q₈⋊C₈	1	2	yes	—
225	C₄	C₂×C₂₃×#103	1	1	yes	characteristic
226	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
227	C₂	C₂₃×#1338	1	4	yes	direct product factor
228	C₂	C₂×C₂₃×Q₈⋊C₈	1	1	yes	characteristic
229	C₂	C₂×C₂₃×#349	1	1	yes	characteristic
230	C₂	C₂×C₂₃×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1338	C₂	direct (N × H)
75. C₂×#1338	C₂₃	direct (N × H)
102. #1338	C₂×C₂₃	direct (N × H)
131. C₂×C₂₃	#1338	direct (N × H)
161. C₂₃	C₂×#1338	direct (N × H)
227. C₂	C₂₃×#1338	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#446	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₂₃×#346	2
C₂₃×#1338	2
C₂×#1338	23

Subgroups — C2×C23×#1338 #19159235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1338 #19159235