Subgroups — C₂²×C₂₃×#362 #19181016

Order 5888.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#362	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₂₃	—	2	1	no	—
6	C₂²×Q₁₆×C₂₃	—	2	3	no	—
7	C₂²×Q₁₆×C₂₃	—	2	1	no	—
8	C₂²×Q₁₆×C₂₃	—	2	3	no	—
9	C₂²×Q₁₆×C₂₃	C₄	1	6	yes	—
10	C₂²×Q₁₆×C₂₃	C₄	1	2	yes	—
11	C₂²×Q₁₆×C₂₃	C₂²	1	6	yes	—
12	C₂²×M₁₆×C₂₃	—	2	1	no	—
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×M₃₂	—	2	3	no	—
15	C₂×C₂₃×M₃₂	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₂₃	—	2	1	no	—
17	C₂×C₂₃×#108	—	2	3	no	—
18	C₂×C₂₃×#108	C₂²	1	6	yes	—
19	C₂₃×#362	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₂₃	—	2	6	no	—
21	C₂²×Q₈×C₂₃	—	2	2	no	—
22	C₂²×Q₈×C₂₃	—	2	6	no	—
23	C₂²×Q₈×C₂₃	—	4	3	no	—
24	C₂²×Q₈×C₂₃	—	4	1	no	—
25	C₂²×Q₈×C₂₃	—	4	3	no	—
26	C₂³×C₄×C₂₃	—	4	1	no	—
27	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₂₃	—	2	6	no	—
29	C₂×Q₁₆×C₂₃	—	2	12	no	—
30	C₂×Q₁₆×C₂₃	—	2	6	no	—
31	C₂×Q₁₆×C₂₃	—	2	6	no	—
32	C₂×Q₁₆×C₂₃	—	2	6	no	—
33	C₂×Q₁₆×C₂₃	—	2	6	no	—
34	C₂×Q₁₆×C₂₃	C₂×C₄	1	12	yes	—
35	C₂×Q₁₆×C₂₃	C₂×C₄	1	12	yes	—
36	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
37	C₂×M₁₆×C₂₃	—	2	6	no	—
38	C₂×M₁₆×C₂₃	—	4	3	no	—
39	C₂²×C₈×C₂₃	—	4	1	no	—
40	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	D₄	1	3	yes	—
43	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	D₄	1	3	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
46	C₂₃×M₃₂	—	2	6	no	—
47	C₂₃×M₃₂	C₂³	1	4	yes	—
48	C₂×C₁₆×C₂₃	—	2	6	no	—
49	C₂₃×#108	—	2	6	no	—
50	C₂₃×#108	C₂³	1	4	yes	—
51	C₂⁴×C₂₃	SD₁₆	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	—	2	12	no	—
53	C₂×Q₈×C₂₃	—	2	12	no	—
54	C₂×Q₈×C₂₃	—	2	4	no	—
55	C₂×Q₈×C₂₃	—	4	12	no	—
56	C₂×Q₈×C₂₃	—	4	6	no	—
57	C₂×Q₈×C₂₃	—	4	6	no	—
58	C₂×Q₈×C₂₃	—	4	6	no	—
59	C₂×Q₈×C₂₃	—	4	6	no	—
60	C₂×Q₈×C₂₃	—	4	6	no	—
61	C₂²×C₄×C₂₃	—	4	6	no	—
62	C₂²×C₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	4	6	no	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	D₈	1	3	yes	—
67	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	SD₁₆	1	3	yes	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
70	Q₁₆×C₂₃	—	2	12	no	—
71	Q₁₆×C₂₃	—	2	12	no	—
72	Q₁₆×C₂₃	—	2	4	no	—
73	Q₁₆×C₂₃	C₂²×C₄	1	8	yes	—
74	M₁₆×C₂₃	—	2	4	no	—
75	M₁₆×C₂₃	—	4	6	no	—
76	C₂×C₈×C₂₃	—	4	6	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	3	yes	—
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	6	yes	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	3	yes	—
80	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₁₆×C₂₃	—	2	4	no	—
87	C₂²×#362	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	4	no	—
89	C₂³×C₂₃	#102	1	3	yes	—
90	C₂³×C₂₃	#102	1	1	yes	characteristic
91	C₂³×C₂₃	C₂×SD₁₆	1	3	yes	—
92	Q₈×C₂₃	—	2	8	no	—
93	Q₈×C₂₃	—	4	4	no	—
94	Q₈×C₂₃	—	4	12	no	—
95	Q₈×C₂₃	—	4	12	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	12	no	—
98	C₂×C₄×C₂₃	—	4	12	no	—
99	C₂×C₄×C₂₃	#102	1	6	yes	—
100	C₂×C₄×C₂₃	#102	1	3	yes	—
101	C₂×C₄×C₂₃	#102	1	3	yes	—
102	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
104	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
105	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
106	C₂×C₄×C₂₃	C₂×SD₁₆	1	3	yes	—
107	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₂₃	—	4	4	no	—
109	C₈×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
110	C₈×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
111	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
114	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
115	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
116	C₂×#362	C₂×C₂₃	1	12	yes	direct product factor
117	C₂²×C₂₃	—	2	6	no	—
118	C₂²×C₂₃	—	2	6	no	—
119	C₂²×C₂₃	#362	1	4	yes	direct product factor
120	C₂²×C₂₃	C₂×#102	1	3	yes	—
121	C₂²×C₂₃	C₂×#102	1	3	yes	—
122	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
123	C₄×C₂₃	—	4	8	no	—
124	C₄×C₂₃	C₂×#102	1	3	yes	—
125	C₄×C₂₃	C₂×#102	1	3	yes	—
126	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
128	C₂³×Q₈	—	2	1	no	—
129	C₂²×Q₁₆	—	2	3	no	—
130	C₂²×Q₁₆	—	2	1	no	—
131	C₂²×Q₁₆	—	2	3	no	—
132	C₂²×Q₁₆	C₄×C₂₃	1	6	yes	—
133	C₂²×Q₁₆	C₄×C₂₃	1	2	yes	—
134	C₂²×Q₁₆	C₂²×C₂₃	1	6	yes	—
135	C₂²×M₁₆	—	2	1	no	—
136	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×M₃₂	—	2	3	no	—
138	C₂×M₃₂	C₂²×C₂₃	1	6	yes	—
139	C₂²×C₁₆	—	2	1	no	—
140	C₂×#108	—	2	3	no	—
141	C₂×#108	C₂²×C₂₃	1	6	yes	—
142	#362	C₂²×C₂₃	1	16	yes	direct product factor
143	C₂×C₂₃	—	2	1	no	—
144	C₂×C₂₃	—	2	3	no	—
145	C₂×C₂₃	C₂×#362	1	6	yes	direct product factor
146	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
147	C₂²×Q₈	—	2	6	no	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×Q₈	—	2	6	no	—
150	C₂²×Q₈	—	4	1	no	—
151	C₂²×Q₈	—	4	3	no	—
152	C₂²×Q₈	—	4	3	no	—
153	C₂³×C₄	—	4	1	no	—
154	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
155	C₂×Q₁₆	—	2	6	no	—
156	C₂×Q₁₆	—	2	12	no	—
157	C₂×Q₁₆	—	2	6	no	—
158	C₂×Q₁₆	—	2	6	no	—
159	C₂×Q₁₆	—	2	6	no	—
160	C₂×Q₁₆	—	2	6	no	—
161	C₂×Q₁₆	C₂×C₄×C₂₃	1	12	yes	—
162	C₂×Q₁₆	C₂×C₄×C₂₃	1	12	yes	—
163	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
164	C₂×M₁₆	—	2	6	no	—
165	C₂×M₁₆	—	4	3	no	—
166	C₂²×C₈	—	4	1	no	—
167	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
168	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	D₄×C₂₃	1	3	yes	—
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₂₃	1	3	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
173	M₃₂	—	2	6	no	—
174	M₃₂	C₂³×C₂₃	1	4	yes	—
175	C₂×C₁₆	—	2	6	no	—
176	#108	—	2	6	no	—
177	#108	C₂³×C₂₃	1	4	yes	—
178	C₂₃	C₂²×#362	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁴	SD₁₆×C₂₃	1	1	yes	characteristic
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	12	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	12	no	—
184	C₂×Q₈	—	4	6	no	—
185	C₂×Q₈	—	4	6	no	—
186	C₂×Q₈	—	4	6	no	—
187	C₂×Q₈	—	4	6	no	—
188	C₂×Q₈	—	4	6	no	—
189	C₂²×C₄	—	4	6	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	6	no	—
192	C₂²×C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	D₈×C₂₃	1	3	yes	—
196	C₂²×C₄	SD₁₆×C₂₃	1	3	yes	—
197	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
198	Q₁₆	—	2	12	no	—
199	Q₁₆	—	2	12	no	—
200	Q₁₆	—	2	4	no	—
201	Q₁₆	C₂²×C₄×C₂₃	1	8	yes	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	6	no	—
204	C₂×C₈	—	4	6	no	—
205	C₂×C₈	C₂²⋊C₄×C₂₃	1	3	yes	—
206	C₂×C₈	C₂²⋊C₄×C₂₃	1	6	yes	—
207	C₂×C₈	C₂²⋊C₄×C₂₃	1	3	yes	—
208	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
209	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
210	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
211	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
212	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
213	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
214	C₁₆	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	C₂₃×#102	1	3	yes	—
217	C₂³	C₂₃×#102	1	1	yes	characteristic
218	C₂³	C₂×SD₁₆×C₂₃	1	3	yes	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	12	no	—
226	C₂×C₄	C₂₃×#102	1	6	yes	—
227	C₂×C₄	C₂₃×#102	1	3	yes	—
228	C₂×C₄	C₂₃×#102	1	3	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
231	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
232	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
233	C₂×C₄	C₂×SD₁₆×C₂₃	1	3	yes	—
234	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
235	C₈	—	4	4	no	—
236	C₈	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
237	C₈	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
238	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
239	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	C₂₃×#362	1	4	yes	direct product factor
243	C₂²	C₂×C₂₃×#102	1	3	yes	—
244	C₂²	C₂×C₂₃×#102	1	3	yes	—
245	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×C₂₃×#102	1	3	yes	—
248	C₄	C₂×C₂₃×#102	1	3	yes	—
249	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
250	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	3	no	—
253	C₂	C₂×C₂₃×#362	1	6	yes	direct product factor
254	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#102
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#362	C₂	direct (N × H)
19. C₂₃×#362	C₂²	direct (N × H)
87. C₂²×#362	C₂₃	direct (N × H)
116. C₂×#362	C₂×C₂₃	direct (N × H)
119. C₂²×C₂₃	#362	direct (N × H)
142. #362	C₂²×C₂₃	direct (N × H)
145. C₂×C₂₃	C₂×#362	direct (N × H)
178. C₂₃	C₂²×#362	direct (N × H)
242. C₂²	C₂₃×#362	direct (N × H)
253. C₂	C₂×C₂₃×#362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₂₃	2
C₂²×C₂₃×M₃₂	2
C₂²×C₂₃×#108	2
C₂×C₂₃×#362	2
C₂²×#362	23

Subgroups — C22×C23×#362 #19181016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#362 #19181016