Subgroups — C₄×C₂₃×#446 #19168143

Order 5888.

958 conjugacy classes of subgroups.

The subgroup lattice has 276 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	2	yes	maximal
2	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1771	C₂	1	4	yes	maximal
4	C₂₃×#1770	C₂	1	2	yes	maximal
5	C₄×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal
6	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₃×#97	C₂	1	2	yes	maximal
8	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₂₃	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#446	C₄	1	16	yes	direct product factor
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₂₃×#97	C₂²	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₄³×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#336	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₄×Q₈×C₂₃	—	2	8	no	—
34	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
50	C₂₃×#97	—	2	8	no	—
51	C₂₃×#97	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₂₃	—	2	4	no	—
53	C₄×C₈×C₂₃	—	2	2	no	—
54	C₄×C₈×C₂₃	—	2	4	no	—
55	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₂₃	Q₈	1	4	yes	complemented
57	C₂₃×#95	C₂×C₄	1	2	yes	—
58	C₂₃×#95	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₂₃	—	2	4	no	—
60	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₂₃	#41	1	4	yes	—
78	C₄⋊C₄×C₂₃	—	2	8	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	C₄²	1	8	yes	—
81	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
82	C₄²×C₂₃	—	2	8	no	—
83	C₄²×C₂₃	—	2	4	no	—
84	C₄²×C₂₃	—	2	8	no	—
85	C₄²×C₂₃	—	2	2	no	—
86	C₄²×C₂₃	—	2	4	no	—
87	C₄²×C₂₃	M₁₆	1	8	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
90	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₂₃	#41	1	2	yes	—
92	C₄×#446	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₂₃	—	2	8	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
110	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₂₃	#131	1	4	yes	—
112	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₈×C₂₃	—	2	4	no	—
117	C₈×C₂₃	—	2	8	no	—
118	C₈×C₂₃	C₄×Q₈	1	8	yes	complemented
119	C₂×#446	C₂×C₂₃	1	2	yes	—
120	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
121	#1771	C₂×C₂₃	1	4	yes	—
122	#1770	C₂×C₂₃	1	2	yes	—
123	C₄×C₄⋊C₈	C₂×C₂₃	1	2	yes	—
124	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
125	C₄×#97	C₂×C₂₃	1	2	yes	—
126	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
127	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
130	C₂²×C₂₃	C₂²×M₁₆	1	2	yes	—
131	C₂²×C₂₃	C₂×#131	1	2	yes	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	4	no	—
135	C₄×C₂₃	—	2	4	no	—
136	C₄×C₂₃	—	4	4	no	—
137	C₄×C₂₃	C₄×M₁₆	1	4	yes	—
138	C₄×C₂₃	#446	1	8	yes	direct product factor
139	C₄×C₂₃	C₂×C₄×Q₈	1	2	yes	—
140	C₄×C₂₃	C₄×#41	1	2	yes	—
141	C₂×C₄×Q₈	C₄×C₂₃	1	2	yes	—
142	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
143	#446	C₄×C₂₃	1	16	yes	direct product factor
144	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
146	C₂×#97	C₂²×C₂₃	1	4	yes	—
147	C₂×#97	C₂²×C₂₃	1	4	yes	—
148	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
154	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
155	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
157	C₄³	C₂²×C₂₃	1	1	yes	characteristic
158	C₄³	C₂²×C₂₃	1	2	yes	—
159	#336	C₂²×C₂₃	1	4	yes	—
160	#336	C₂²×C₂₃	1	2	yes	—
161	C₂×C₂₃	C₄²×Q₈	1	1	yes	characteristic
162	C₂×C₂₃	C₂×C₄×M₁₆	1	1	yes	characteristic
163	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
164	C₂×C₂₃	C₂×#446	1	2	yes	—
165	C₂×C₂₃	#2965	1	2	yes	—
166	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
169	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
170	C₄×Q₈	—	2	8	no	—
171	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
186	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
187	#97	—	2	8	no	—
188	#97	C₂×C₄×C₂₃	1	16	yes	—
189	C₄×C₈	—	2	4	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
193	C₄×C₈	Q₈×C₂₃	1	4	yes	complemented
194	#95	C₂×C₄×C₂₃	1	2	yes	—
195	#95	C₂×C₄×C₂₃	1	4	yes	—
196	C₂₃	C₄×#446	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
213	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
215	C₂×C₈	#41×C₂₃	1	4	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₄²×C₂₃	1	8	yes	—
219	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	4	no	—
225	C₄²	M₁₆×C₂₃	1	8	yes	—
226	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
227	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
228	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
229	C₄²	#41×C₂₃	1	2	yes	—
230	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
241	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
242	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
243	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
244	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
245	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
246	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
247	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
248	C₂×C₄	C₂₃×#131	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
250	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
252	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₄×Q₈×C₂₃	1	8	yes	complemented
256	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
257	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
258	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
259	C₂²	C₂²×M₁₆×C₂₃	1	2	yes	—
260	C₂²	C₂×C₂₃×#131	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	4	no	—
266	C₄	C₄×M₁₆×C₂₃	1	4	yes	—
267	C₄	C₂₃×#446	1	8	yes	direct product factor
268	C₄	C₂×C₄×Q₈×C₂₃	1	2	yes	—
269	C₄	C₄×#41×C₂₃	1	2	yes	—
270	C₂	C₄²×Q₈×C₂₃	1	1	yes	characteristic
271	C₂	C₂×C₄×M₁₆×C₂₃	1	1	yes	characteristic
272	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
273	C₂	C₂×C₂₃×#446	1	2	yes	—
274	C₂	C₂₃×#2965	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂₃×#446	C₄	direct (N × H)
56. C₄×C₈×C₂₃	Q₈	semidirect
92. C₄×#446	C₂₃	direct (N × H)
118. C₈×C₂₃	C₄×Q₈	semidirect
138. C₄×C₂₃	#446	direct (N × H)
143. #446	C₄×C₂₃	direct (N × H)
193. C₄×C₈	Q₈×C₂₃	semidirect
196. C₂₃	C₄×#446	direct (N × H)
255. C₈	C₄×Q₈×C₂₃	semidirect
267. C₄	C₂₃×#446	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
C₄²×Q₈×C₂₃	2
C₂₃×#1771	2
C₂₃×#1770	2
C₄×C₂₃×C₄⋊C₈	2
C₄×C₂₃×#97	2
C₄²×C₈×C₂₃	2
C₄×#446	23

Subgroups — C4×C23×#446 #19168143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#446 #19168143