Subgroups — C₂×C₂₃×#1362 #19159554

Order 5888.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1362	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₂₃	—	2	1	no	—
6	C₂²×Q₁₆×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₁₆×C₂₃	—	2	1	no	—
9	C₂₃×#494	—	2	1	no	—
10	C₂₃×#494	—	2	1	no	—
11	C₂₃×#494	C₄	1	2	yes	—
12	C₂₃×#494	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₁₆	—	2	1	no	—
16	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂₃×#335	—	2	1	no	—
18	C₂₃×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×Q₁₆×C₂₃	—	2	2	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	—	4	4	no	—
24	C₂×Q₁₆×C₂₃	—	4	2	no	—
25	C₂×Q₁₆×C₂₃	—	4	2	no	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₂₃	—	4	1	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×C₄⋊Q₈	—	2	2	no	—
32	C₂₃×C₄⋊Q₈	—	4	1	no	—
33	C₂₃×C₄⋊Q₈	—	4	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
35	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂×C₁₆×C₂₃	—	4	2	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	—	4	1	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	—	4	4	no	—
48	C₂×Q₈×C₂₃	—	4	4	no	—
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
53	Q₁₆×C₂₃	—	4	4	no	—
54	Q₁₆×C₂₃	—	4	4	no	—
55	Q₁₆×C₂₃	—	4	4	no	—
56	Q₁₆×C₂₃	—	4	2	no	—
57	Q₁₆×C₂₃	—	4	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	8	2	no	—
61	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	D₈	1	2	yes	—
64	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₂₃	D₈	1	1	yes	characteristic
73	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₂₃	—	4	2	no	—
76	C₂×#1362	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	#102	1	1	yes	characteristic
78	Q₈×C₂₃	—	4	4	no	—
79	Q₈×C₂₃	—	4	4	no	—
80	Q₈×C₂₃	—	8	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	#100	1	2	yes	—
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#102	1	2	yes	—
88	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
90	C₂×C₄×C₂₃	Q₃₂	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	8	2	no	—
97	C₈×C₂₃	#102	1	2	yes	—
98	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
101	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
102	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
103	#1362	C₂×C₂₃	1	4	yes	direct product factor
104	C₂²×C₂₃	#331	1	2	yes	—
105	C₂²×C₂₃	#358	1	2	yes	—
106	C₂²×C₂₃	#362	1	2	yes	—
107	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	8	4	no	—
111	C₄×C₂₃	#331	1	2	yes	—
112	C₄×C₂₃	#358	1	2	yes	—
113	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
114	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
115	C₄×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
116	C₄×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
117	C₂²×Q₁₆	—	2	1	no	—
118	C₂²×Q₁₆	C₄×C₂₃	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂²×C₁₆	—	2	1	no	—
121	#494	—	2	1	no	—
122	#494	—	2	1	no	—
123	#494	C₄×C₂₃	1	2	yes	—
124	#494	C₂²×C₂₃	1	2	yes	—
125	C₂×C₄⋊C₈	—	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	#335	—	2	1	no	—
130	#335	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂₃	#1362	1	4	yes	direct product factor
132	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
133	C₂×C₂₃	C₂×#358	1	1	yes	characteristic
134	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	4	4	no	—
140	C₂×Q₁₆	—	4	2	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
143	C₂²×C₈	—	4	1	no	—
144	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	4	1	no	—
149	C₄⋊Q₈	—	4	1	no	—
150	C₂×C₄⋊C₄	—	4	1	no	—
151	C₂×C₄⋊C₄	—	4	1	no	—
152	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	4	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	4	1	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×#1362	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
170	Q₁₆	—	4	4	no	—
171	Q₁₆	—	4	4	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	2	no	—
174	Q₁₆	—	4	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	8	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₈	D₈×C₂₃	1	2	yes	—
181	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₄²	D₈×C₂₃	1	1	yes	characteristic
190	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₁₆	—	4	2	no	—
193	C₂³	C₂₃×#102	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₂₃×#100	1	2	yes	—
201	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
202	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
203	C₂×C₄	C₂₃×#102	1	2	yes	—
204	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
205	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
206	C₂×C₄	C₂₃×Q₃₂	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	8	2	no	—
213	C₈	C₂₃×#102	1	2	yes	—
214	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
215	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
216	C₂²	C₂₃×#331	1	2	yes	—
217	C₂²	C₂₃×#358	1	2	yes	—
218	C₂²	C₂₃×#362	1	2	yes	—
219	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	C₂₃×#331	1	2	yes	—
224	C₄	C₂₃×#358	1	2	yes	—
225	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
226	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
227	C₄	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
228	C₄	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
229	C₂	C₂₃×#1362	1	4	yes	direct product factor
230	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
231	C₂	C₂×C₂₃×#358	1	1	yes	characteristic
232	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1362	C₂	direct (N × H)
76. C₂×#1362	C₂₃	direct (N × H)
103. #1362	C₂×C₂₃	direct (N × H)
131. C₂×C₂₃	#1362	direct (N × H)
161. C₂₃	C₂×#1362	direct (N × H)
229. C₂	C₂₃×#1362	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×#494	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₂₃×#335	2
C₂₃×#1362	2
C₂×#1362	23

Subgroups — C2×C23×#1362 #19159554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1362 #19159554