Subgroups — C₂×C₂₃×#1716 #19167233

Order 5888.

798 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#498	C₂	1	2	yes	maximal
2	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#487	C₂	1	2	yes	maximal
5	C₂₃×#1716	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₂₃	—	2	2	no	—
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₃×#498	—	2	4	no	—
12	C₂₃×#498	—	2	2	no	—
13	C₂₃×#498	C₂²	1	4	yes	—
14	C₂₃×#498	C₂²	1	8	yes	—
15	C₂₃×#494	—	2	2	no	—
16	C₂₃×#494	C₂²	1	4	yes	—
17	C₂₃×#487	—	2	4	no	—
18	C₂₃×#487	C₂²	1	8	yes	—
19	C₂×C₂₃×#106	—	2	4	no	—
20	C₂×C₂₃×#106	—	2	2	no	—
21	C₂×C₂₃×#103	—	2	4	no	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#322	C₂²	1	4	yes	—
25	C₂²×Q₈×C₂₃	—	2	1	no	—
26	C₂²×Q₈×C₂₃	—	2	1	no	—
27	C₂²×Q₈×C₂₃	—	2	2	no	—
28	C₂×Q₁₆×C₂₃	—	2	8	no	—
29	C₂×Q₁₆×C₂₃	—	4	8	no	—
30	C₂²×C₈×C₂₃	D₄	1	4	yes	—
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₂₃×C₄⋊Q₈	—	2	2	no	—
33	C₂₃×C₄⋊Q₈	—	2	2	no	—
34	C₂₃×C₄⋊Q₈	—	2	2	no	—
35	C₂₃×C₄⋊Q₈	—	2	2	no	—
36	C₂₃×C₄⋊Q₈	—	2	2	no	—
37	C₂₃×C₄⋊Q₈	—	2	1	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#106	—	2	16	no	—
48	C₂₃×#106	—	2	8	no	—
49	C₂₃×#103	—	2	16	no	—
50	C₄×C₈×C₂₃	D₄	1	4	yes	—
51	C₄×C₈×C₂₃	D₄	1	2	yes	—
52	C₄×C₈×C₂₃	C₂³	1	4	yes	—
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×Q₈×C₂₃	—	2	4	no	—
55	C₂×Q₈×C₂₃	—	2	2	no	—
56	C₂×Q₈×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	—	4	1	no	—
58	C₂×Q₈×C₂₃	—	4	1	no	—
59	C₂×Q₈×C₂₃	—	4	2	no	—
60	C₂×Q₈×C₂₃	—	4	4	no	—
61	C₂×Q₈×C₂₃	—	4	4	no	—
62	C₂×Q₈×C₂₃	—	4	8	no	—
63	C₂²×C₄×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	—	4	2	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	Q₁₆×C₂₃	—	4	16	no	—
69	C₂×C₈×C₂₃	—	2	8	no	—
70	C₂×C₈×C₂₃	SD₁₆	1	8	yes	—
71	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
72	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	2	4	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₄²×C₂₃	C₂×D₄	1	2	yes	—
82	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
84	C₂×#1716	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₃	#127	1	1	yes	characteristic
86	Q₈×C₂₃	—	4	4	no	—
87	Q₈×C₂₃	—	4	4	no	—
88	Q₈×C₂₃	—	4	8	no	—
89	Q₈×C₂₃	—	8	2	no	—
90	Q₈×C₂₃	—	8	2	no	—
91	Q₈×C₂₃	—	8	4	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	—	8	2	no	—
97	C₂×C₄×C₂₃	—	8	4	no	—
98	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#127	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
101	C₂×C₄×C₂₃	#137	1	8	yes	—
102	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
104	C₈×C₂₃	—	2	8	no	—
105	C₈×C₂₃	C₂×SD₁₆	1	8	yes	—
106	C₂×#498	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#498	C₂×C₂₃	1	2	yes	—
108	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
109	C₂×#487	C₂×C₂₃	1	2	yes	—
110	#1716	C₂×C₂₃	1	8	yes	direct product factor
111	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
112	C₂²×C₂₃	#492	1	2	yes	—
113	C₂²×C₂₃	#497	1	4	yes	—
114	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
115	C₄×C₂₃	—	8	2	no	—
116	C₄×C₂₃	—	8	2	no	—
117	C₄×C₂₃	—	8	4	no	—
118	C₄×C₂₃	#492	1	2	yes	—
119	C₄×C₂₃	#497	1	4	yes	—
120	C₄×C₂₃	C₂²×SD₁₆	1	2	yes	—
121	C₄×C₂₃	C₂×#137	1	4	yes	—
122	C₂²×Q₁₆	—	2	2	no	—
123	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
125	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
126	#498	—	2	4	no	—
127	#498	—	2	2	no	—
128	#498	C₂²×C₂₃	1	8	yes	—
129	#498	C₂²×C₂₃	1	4	yes	—
130	#494	—	2	2	no	—
131	#494	C₂²×C₂₃	1	4	yes	—
132	#487	—	2	4	no	—
133	#487	C₂²×C₂₃	1	8	yes	—
134	C₂×#106	—	2	4	no	—
135	C₂×#106	—	2	2	no	—
136	C₂×#103	—	2	4	no	—
137	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
139	#322	C₂²×C₂₃	1	4	yes	—
140	C₂×C₂₃	#1716	1	4	yes	direct product factor
141	C₂×C₂₃	C₂×#492	1	1	yes	characteristic
142	C₂×C₂₃	C₂×#497	1	2	yes	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂×Q₁₆	—	2	8	no	—
147	C₂×Q₁₆	—	4	8	no	—
148	C₂²×C₈	D₄×C₂₃	1	4	yes	—
149	C₂²×C₈	D₄×C₂₃	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	1	no	—
156	C₄⋊Q₈	—	2	1	no	—
157	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
158	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
159	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
165	#106	—	2	16	no	—
166	#106	—	2	8	no	—
167	#103	—	2	16	no	—
168	C₄×C₈	D₄×C₂₃	1	4	yes	—
169	C₄×C₈	D₄×C₂₃	1	2	yes	—
170	C₄×C₈	C₂³×C₂₃	1	4	yes	—
171	C₄×C₈	C₂³×C₂₃	1	2	yes	—
172	C₂₃	C₂×#1716	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	4	1	no	—
177	C₂×Q₈	—	4	1	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
187	Q₁₆	—	4	16	no	—
188	C₂×C₈	—	2	8	no	—
189	C₂×C₈	SD₁₆×C₂₃	1	8	yes	—
190	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
191	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄²	C₂×D₄×C₂₃	1	2	yes	—
201	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
203	C₂³	C₂₃×#127	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	4	8	no	—
207	Q₈	—	8	2	no	—
208	Q₈	—	8	2	no	—
209	Q₈	—	8	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#127	1	2	yes	—
218	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
219	C₂×C₄	C₂₃×#137	1	8	yes	—
220	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
222	C₈	—	2	8	no	—
223	C₈	C₂×SD₁₆×C₂₃	1	8	yes	—
224	C₂²	C₂₃×#492	1	2	yes	—
225	C₂²	C₂₃×#497	1	4	yes	—
226	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
227	C₄	—	8	2	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	4	no	—
230	C₄	C₂₃×#492	1	2	yes	—
231	C₄	C₂₃×#497	1	4	yes	—
232	C₄	C₂²×SD₁₆×C₂₃	1	2	yes	—
233	C₄	C₂×C₂₃×#137	1	4	yes	—
234	C₂	C₂₃×#1716	1	4	yes	direct product factor
235	C₂	C₂×C₂₃×#492	1	1	yes	characteristic
236	C₂	C₂×C₂₃×#497	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1716	C₂	direct (N × H)
84. C₂×#1716	C₂₃	direct (N × H)
110. #1716	C₂×C₂₃	direct (N × H)
140. C₂×C₂₃	#1716	direct (N × H)
172. C₂₃	C₂×#1716	direct (N × H)
234. C₂	C₂₃×#1716	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×#498	2
C₂×C₂₃×#494	2
C₂×C₂₃×#487	2
C₂₃×#1716	2
C₂×C₂₃×#322	2
C₂×#1716	23

Subgroups — C2×C23×#1716 #19167233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1716 #19167233