Subgroups — C₂²×C₂₃×#475 #19208102

Order 5888.

2358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#475	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#106	C₂	1	2	yes	maximal
5	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₃×#103	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₂₃	—	2	1	no	—
8	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊Q₈	—	2	6	no	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×C₄×Q₈×C₂₃	—	2	6	no	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₂₃	—	2	1	no	—
19	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#475	C₂²	1	64	yes	direct product factor
21	C₂×C₂₃×#106	C₂²	1	24	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	12	yes	—
23	C₂×C₂₃×#103	C₂²	1	24	yes	—
24	C₂²×Q₈×C₂₃	—	2	3	no	—
25	C₂²×Q₈×C₂₃	—	2	12	no	—
26	C₂²×Q₈×C₂₃	—	2	4	no	—
27	C₂²×Q₈×C₂₃	—	4	8	no	—
28	C₂²×Q₈×C₂₃	Q₈	1	8	yes	complemented
29	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
30	C₂³×C₄×C₂₃	—	2	1	no	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂³×C₄×C₂₃	—	2	1	no	—
33	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	—	2	8	no	—
37	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
38	C₂₃×C₄⋊Q₈	—	2	24	no	—
39	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	—
40	C₄×Q₈×C₂₃	—	2	24	no	—
41	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
48	C₂×C₄²×C₂₃	—	2	12	no	—
49	C₂×C₄²×C₂₃	D₄	1	6	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
51	C₂₃×#106	C₂³	1	32	yes	—
52	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
53	C₂₃×#103	C₂³	1	32	yes	—
54	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₂₃	—	2	24	no	—
56	C₂×Q₈×C₂₃	—	2	6	no	—
57	C₂×Q₈×C₂₃	—	2	16	no	—
58	C₂×Q₈×C₂₃	—	2	12	no	—
59	C₂×Q₈×C₂₃	—	4	48	no	—
60	C₂×Q₈×C₂₃	C₂×Q₈	1	24	yes	complemented
61	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₂₃	—	2	8	no	—
63	C₂²×C₄×C₂₃	—	2	4	no	—
64	C₂²×C₄×C₂₃	—	2	6	no	—
65	C₂²×C₄×C₂₃	—	2	6	no	—
66	C₂²×C₄×C₂₃	—	2	12	no	—
67	C₂²×C₄×C₂₃	—	4	8	no	—
68	C₂²×C₄×C₂₃	—	4	4	no	—
69	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₂₃	#41	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
76	C₂×C₈×C₂₃	—	2	24	no	—
77	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
78	C₄⋊C₄×C₂₃	—	2	16	no	—
79	C₄⋊C₄×C₂₃	—	2	16	no	—
80	C₄⋊C₄×C₂₃	—	2	6	no	—
81	C₄⋊C₄×C₂₃	—	2	12	no	—
82	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
83	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
84	C₄²×C₂₃	—	2	16	no	—
85	C₄²×C₂₃	C₂×D₄	1	12	yes	—
86	C₄²×C₂₃	C₂⁴	1	4	yes	—
87	C₂²×#475	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	C₂²⋊Q₈	1	4	yes	—
89	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
90	C₂³×C₂₃	#137	1	4	yes	—
91	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
92	Q₈×C₂₃	—	2	24	no	—
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	4	32	no	—
95	Q₈×C₂₃	C₂²×Q₈	1	8	yes	complemented
96	C₂×C₄×C₂₃	—	2	24	no	—
97	C₂×C₄×C₂₃	—	2	12	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	8	no	—
101	C₂×C₄×C₂₃	—	4	24	no	—
102	C₂×C₄×C₂₃	—	4	12	no	—
103	C₂×C₄×C₂₃	C₂²⋊Q₈	1	12	yes	—
104	C₂×C₄×C₂₃	C₂×SD₁₆	1	24	yes	—
105	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
106	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
108	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
109	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
110	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
111	C₈×C₂₃	—	2	8	no	—
112	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
113	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
114	C₂×#475	C₂×C₂₃	1	24	yes	direct product factor
115	C₂²×#106	C₂×C₂₃	1	2	yes	—
116	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
117	C₂²×#103	C₂×C₂₃	1	2	yes	—
118	C₂²×C₂₃	#475	1	16	yes	direct product factor
119	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	6	yes	—
120	C₂²×C₂₃	C₂²×SD₁₆	1	6	yes	—
121	C₂²×C₂₃	C₂×#137	1	6	yes	—
122	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
123	C₄×C₂₃	—	2	8	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	4	8	no	—
126	C₄×C₂₃	—	4	4	no	—
127	C₄×C₂₃	C₂×C₂²⋊Q₈	1	6	yes	—
128	C₄×C₂₃	C₂²×SD₁₆	1	8	yes	—
129	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
130	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
131	C₂³×Q₈	—	2	1	no	—
132	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
134	C₂×C₄⋊Q₈	—	2	6	no	—
135	C₂×C₄⋊Q₈	C₂²×C₂₃	1	12	yes	—
136	C₂×C₄×Q₈	—	2	6	no	—
137	C₂×C₄×Q₈	C₂²×C₂₃	1	12	yes	—
138	C₂²×C₄⋊C₄	—	2	1	no	—
139	C₂²×C₄⋊C₄	—	2	1	no	—
140	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂²×C₄²	—	2	1	no	—
143	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
144	#475	C₂²×C₂₃	1	64	yes	direct product factor
145	C₂×#106	C₂²×C₂₃	1	24	yes	—
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	12	yes	—
147	C₂×#103	C₂²×C₂₃	1	24	yes	—
148	C₂×C₂₃	C₂×#475	1	12	yes	direct product factor
149	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
151	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
152	C₂²×Q₈	—	2	3	no	—
153	C₂²×Q₈	—	2	12	no	—
154	C₂²×Q₈	—	2	4	no	—
155	C₂²×Q₈	—	4	8	no	—
156	C₂²×Q₈	Q₈×C₂₃	1	8	yes	complemented
157	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
158	C₂³×C₄	—	2	1	no	—
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	—	2	1	no	—
161	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
162	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂²×C₈	—	2	8	no	—
165	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
166	C₄⋊Q₈	—	2	24	no	—
167	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	—
168	C₄×Q₈	—	2	24	no	—
169	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
170	C₂×C₄⋊C₄	—	2	12	no	—
171	C₂×C₄⋊C₄	—	2	12	no	—
172	C₂×C₄⋊C₄	—	2	3	no	—
173	C₂×C₄⋊C₄	—	2	6	no	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
176	C₂×C₄²	—	2	12	no	—
177	C₂×C₄²	D₄×C₂₃	1	6	yes	—
178	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
179	#106	C₂³×C₂₃	1	32	yes	—
180	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
181	#103	C₂³×C₂₃	1	32	yes	—
182	C₂₃	C₂²×#475	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂×Q₈	—	2	24	no	—
185	C₂×Q₈	—	2	6	no	—
186	C₂×Q₈	—	2	16	no	—
187	C₂×Q₈	—	2	12	no	—
188	C₂×Q₈	—	4	48	no	—
189	C₂×Q₈	C₂×Q₈×C₂₃	1	24	yes	complemented
190	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	2	12	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
199	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
201	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
202	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
203	C₂²×C₄	#41×C₂₃	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
205	C₂×C₈	—	2	24	no	—
206	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	16	no	—
209	C₄⋊C₄	—	2	6	no	—
210	C₄⋊C₄	—	2	12	no	—
211	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
212	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
213	C₄²	—	2	16	no	—
214	C₄²	C₂×D₄×C₂₃	1	12	yes	—
215	C₄²	C₂⁴×C₂₃	1	4	yes	—
216	C₂³	C₂₃×C₂²⋊Q₈	1	4	yes	—
217	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
218	C₂³	C₂₃×#137	1	4	yes	—
219	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
220	Q₈	—	2	4	no	—
221	Q₈	—	2	24	no	—
222	Q₈	—	4	32	no	—
223	Q₈	C₂²×Q₈×C₂₃	1	8	yes	complemented
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	24	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	4	24	no	—
230	C₂×C₄	—	4	12	no	—
231	C₂×C₄	C₂₃×C₂²⋊Q₈	1	12	yes	—
232	C₂×C₄	C₂×SD₁₆×C₂₃	1	24	yes	—
233	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
234	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
235	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
236	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
237	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
238	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
239	C₈	—	2	8	no	—
240	C₂²	C₂₃×#475	1	16	yes	direct product factor
241	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	6	yes	—
242	C₂²	C₂²×SD₁₆×C₂₃	1	6	yes	—
243	C₂²	C₂×C₂₃×#137	1	6	yes	—
244	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
245	C₄	—	2	8	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	8	no	—
248	C₄	—	4	4	no	—
249	C₄	C₂×C₂₃×C₂²⋊Q₈	1	6	yes	—
250	C₄	C₂²×SD₁₆×C₂₃	1	8	yes	—
251	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
252	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
253	C₂	C₂×C₂₃×#475	1	12	yes	direct product factor
254	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
256	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#475	C₂	direct (N × H)
20. C₂₃×#475	C₂²	direct (N × H)
28. C₂²×Q₈×C₂₃	Q₈	semidirect
60. C₂×Q₈×C₂₃	C₂×Q₈	semidirect
87. C₂²×#475	C₂₃	direct (N × H)
95. Q₈×C₂₃	C₂²×Q₈	semidirect
114. C₂×#475	C₂×C₂₃	direct (N × H)
118. C₂²×C₂₃	#475	direct (N × H)
144. #475	C₂²×C₂₃	direct (N × H)
148. C₂×C₂₃	C₂×#475	direct (N × H)
156. C₂²×Q₈	Q₈×C₂₃	semidirect
182. C₂₃	C₂²×#475	direct (N × H)
189. C₂×Q₈	C₂×Q₈×C₂₃	semidirect
223. Q₈	C₂²×Q₈×C₂₃	semidirect
240. C₂²	C₂₃×#475	direct (N × H)
253. C₂	C₂×C₂₃×#475	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×C₂₃×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₂₃	2
C₂×C₂₃×#475	2
C₂²×C₂₃×#106	2
C₂²×C₂₃×C₄⋊C₈	2
C₂²×C₂₃×#103	2
C₂²×#475	23

Subgroups — C22×C23×#475 #19208102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#475 #19208102