Subgroups — C₂×C₂₃×#2327 #19181694

Order 5888.

2958 conjugacy classes of subgroups.

The subgroup lattice has 268 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	2	yes	maximal
3	C₂₃×#2327	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#391	C₂	1	2	yes	maximal
5	C₂×C₂₃×#387	C₂	1	8	yes	maximal
6	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₄	1	16	yes	complemented
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	—
11	C₂×C₄×Q₈×C₂₃	—	2	16	no	—
12	C₂×C₄×Q₈×C₂₃	C₂²	1	16	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#391	C₂²	1	16	yes	—
21	C₂₃×#387	C₂²	1	64	yes	—
22	C₂₃×#382	C₂²	1	8	yes	—
23	C₂₃×#376	C₂²	1	8	yes	—
24	C₂×C₂₃×#95	C₂²	1	4	yes	—
25	C₂×C₂₃×#95	C₂²	1	4	yes	—
26	C₂×C₂₃×#95	C₂²	1	8	yes	—
27	C₂²×Q₈×C₂₃	—	2	8	no	—
28	C₂²×Q₈×C₂₃	—	2	8	no	—
29	C₂²×Q₈×C₂₃	C₂×C₄	1	16	yes	—
30	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₄⋊Q₈	C₂×C₄	1	64	yes	complemented
38	C₄×Q₈×C₂₃	—	2	64	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	32	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
45	C₂×C₄²×C₂₃	—	2	16	no	—
46	C₂×C₄²×C₂₃	—	2	4	no	—
47	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
48	C₂×C₄²×C₂₃	D₄	1	8	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
53	C₂₃×#95	C₂³	1	16	yes	—
54	C₂₃×#95	C₂³	1	16	yes	—
55	C₂₃×#95	C₂³	1	32	yes	—
56	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₂₃	—	2	64	no	—
58	C₂×Q₈×C₂₃	—	2	16	no	—
59	C₂×Q₈×C₂₃	—	2	16	no	—
60	C₂×Q₈×C₂₃	C₂²×C₄	1	32	yes	—
61	C₂²×C₄×C₂₃	—	2	8	no	—
62	C₂²×C₄×C₂₃	—	2	8	no	—
63	C₂²×C₄×C₂₃	—	2	8	no	—
64	C₂²×C₄×C₂₃	—	2	8	no	—
65	C₂²×C₄×C₂₃	—	2	8	no	—
66	C₂²×C₄×C₂₃	—	2	4	no	—
67	C₂²×C₄×C₂₃	—	2	16	no	—
68	C₂²×C₄×C₂₃	—	2	8	no	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
73	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₂₃	#41	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
81	C₄⋊C₄×C₂₃	—	2	32	no	—
82	C₄⋊C₄×C₂₃	—	2	16	no	—
83	C₄⋊C₄×C₂₃	C₂²×C₄	1	64	yes	—
84	C₄²×C₂₃	—	2	32	no	—
85	C₄²×C₂₃	—	2	8	no	—
86	C₄²×C₂₃	C₂²×C₄	1	16	yes	—
87	C₄²×C₂₃	C₂×D₄	1	16	yes	—
88	C₂×#2327	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
90	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
91	C₂³×C₂₃	C₂×#41	1	2	yes	—
92	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
93	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
94	Q₈×C₂₃	—	2	64	no	—
95	C₂×C₄×C₂₃	—	2	32	no	—
96	C₂×C₄×C₂₃	—	2	16	no	—
97	C₂×C₄×C₂₃	—	2	16	no	—
98	C₂×C₄×C₂₃	—	2	8	no	—
99	C₂×C₄×C₂₃	—	2	8	no	—
100	C₂×C₄×C₂₃	—	2	8	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	—	2	8	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	16	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	4	16	no	—
107	C₂×C₄×C₂₃	—	4	8	no	—
108	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	—
109	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₂₃	C₂³×C₄	1	16	yes	—
111	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
113	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
114	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
115	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
116	C₂²×C₄×Q₈	C₂×C₂₃	1	2	yes	—
117	#2327	C₂×C₂₃	1	16	yes	direct product factor
118	C₂×#391	C₂×C₂₃	1	2	yes	—
119	C₂×#387	C₂×C₂₃	1	8	yes	—
120	C₂×#382	C₂×C₂₃	1	1	yes	characteristic
121	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
122	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
123	C₂²×C₂₃	#519	1	8	yes	—
124	C₂²×C₂₃	#536	1	8	yes	—
125	C₂²×C₂₃	#541	1	8	yes	—
126	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
127	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
128	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
130	C₄×C₂₃	—	2	16	no	—
131	C₄×C₂₃	—	2	8	no	—
132	C₄×C₂₃	—	2	8	no	—
133	C₄×C₂₃	—	4	8	no	—
134	C₄×C₂₃	—	4	4	no	—
135	C₄×C₂₃	C₂×C₄×D₄	1	8	yes	—
136	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
137	C₂×C₄⋊Q₈	C₄×C₂₃	1	16	yes	complemented
138	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	—
139	C₂×C₄×Q₈	—	2	16	no	—
140	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
144	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
145	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
147	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
148	#391	C₂²×C₂₃	1	16	yes	—
149	#387	C₂²×C₂₃	1	64	yes	—
150	#382	C₂²×C₂₃	1	8	yes	—
151	#376	C₂²×C₂₃	1	8	yes	—
152	C₂×#95	C₂²×C₂₃	1	4	yes	—
153	C₂×#95	C₂²×C₂₃	1	4	yes	—
154	C₂×#95	C₂²×C₂₃	1	8	yes	—
155	C₂×C₂₃	#2327	1	8	yes	direct product factor
156	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
157	C₂×C₂₃	C₂×#519	1	2	yes	—
158	C₂×C₂₃	C₂×#536	1	2	yes	—
159	C₂×C₂₃	C₂×#541	1	2	yes	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	C₂×C₄×C₂₃	1	16	yes	—
163	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
164	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
165	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
166	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
167	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
169	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₄⋊Q₈	C₂×C₄×C₂₃	1	64	yes	complemented
171	C₄×Q₈	—	2	64	no	—
172	C₂×C₄⋊C₄	—	2	16	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	32	yes	—
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
176	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
177	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
178	C₂×C₄²	—	2	16	no	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
181	C₂×C₄²	D₄×C₂₃	1	8	yes	—
182	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
183	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
184	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
186	#95	C₂³×C₂₃	1	16	yes	—
187	#95	C₂³×C₂₃	1	16	yes	—
188	#95	C₂³×C₂₃	1	32	yes	—
189	C₂₃	C₂×#2327	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
191	C₂×Q₈	—	2	64	no	—
192	C₂×Q₈	—	2	16	no	—
193	C₂×Q₈	—	2	16	no	—
194	C₂×Q₈	C₂²×C₄×C₂₃	1	32	yes	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
205	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
206	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
207	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
208	C₂²×C₄	#41×C₂₃	1	4	yes	—
209	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
211	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
212	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
213	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
214	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
215	C₄⋊C₄	—	2	32	no	—
216	C₄⋊C₄	—	2	16	no	—
217	C₄⋊C₄	C₂²×C₄×C₂₃	1	64	yes	—
218	C₄²	—	2	32	no	—
219	C₄²	—	2	8	no	—
220	C₄²	C₂²×C₄×C₂₃	1	16	yes	—
221	C₄²	C₂×D₄×C₂₃	1	16	yes	—
222	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
223	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
224	C₂³	C₂×#41×C₂₃	1	2	yes	—
225	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
226	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
227	Q₈	—	2	64	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	32	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	—
242	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
243	C₂×C₄	C₂³×C₄×C₂₃	1	16	yes	—
244	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
245	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
246	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
247	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
248	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
249	C₂²	C₂₃×#519	1	8	yes	—
250	C₂²	C₂₃×#536	1	8	yes	—
251	C₂²	C₂₃×#541	1	8	yes	—
252	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
253	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
254	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
255	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
256	C₄	—	2	16	no	—
257	C₄	—	2	8	no	—
258	C₄	—	2	8	no	—
259	C₄	—	4	8	no	—
260	C₄	—	4	4	no	—
261	C₄	C₂×C₄×D₄×C₂₃	1	8	yes	—
262	C₂	C₂₃×#2327	1	8	yes	direct product factor
263	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
264	C₂	C₂×C₂₃×#519	1	2	yes	—
265	C₂	C₂×C₂₃×#536	1	2	yes	—
266	C₂	C₂×C₂₃×#541	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2327	C₂	direct (N × H)
9. C₂×C₂₃×C₄⋊Q₈	C₄	semidirect
37. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
88. C₂×#2327	C₂₃	direct (N × H)
117. #2327	C₂×C₂₃	direct (N × H)
137. C₂×C₄⋊Q₈	C₄×C₂₃	semidirect
155. C₂×C₂₃	#2327	direct (N × H)
170. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
189. C₂₃	C₂×#2327	direct (N × H)
262. C₂	C₂₃×#2327	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
C₂₃×#2327	2
C₂×C₂₃×#391	2
C₂×C₂₃×#387	2
C₂×C₂₃×#382	2
C₂×C₂₃×#376	2
C₂×#2327	23

Subgroups — C2×C23×#2327 #19181694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2327 #19181694