Subgroups — C₂×C₂₃×#1315 #19159046

Order 5888.

534 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×#408	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1315	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#41×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×M₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	—	2	2	no	—
8	C₂₃×#408	—	2	2	no	—
9	C₂₃×#408	C₄	1	2	yes	—
10	C₂₃×#408	C₂²	1	2	yes	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
12	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
13	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
14	C₂×#41×C₂₃	—	2	1	no	—
15	C₂×#41×C₂₃	—	2	1	no	—
16	C₂×#41×C₂₃	—	4	2	no	—
17	C₂×#41×C₂₃	—	4	2	no	—
18	C₂×#41×C₂₃	C₈	1	2	yes	—
19	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	—	2	1	no	—
24	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	—	4	2	no	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₂₃	—	2	1	no	—
29	C₂²×C₈×C₂₃	—	4	2	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₂₃	—	2	4	no	—
35	C₂×C₁₆×C₂₃	—	4	4	no	—
36	C₂₃×C₂²⋊C₈	—	2	4	no	—
37	C₂₃×C₂²⋊C₈	—	2	4	no	—
38	C₂⁴×C₂₃	—	2	1	no	—
39	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
40	#41×C₂₃	—	4	4	no	—
41	#41×C₂₃	—	4	2	no	—
42	#41×C₂₃	—	4	2	no	—
43	#41×C₂₃	—	4	4	no	—
44	#41×C₂₃	—	4	2	no	—
45	#41×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	—	2	1	no	—
47	C₂×Q₈×C₂₃	—	4	2	no	—
48	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
49	C₂×D₄×C₂₃	—	2	2	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂×D₄×C₂₃	—	4	4	no	—
54	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₂₃	—	4	2	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
78	C₁₆×C₂₃	—	4	4	no	—
79	C₂×#1315	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	4	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	4	2	no	—
84	C₂³×C₂₃	—	4	4	no	—
85	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
88	Q₈×C₂₃	—	4	2	no	—
89	Q₈×C₂₃	—	4	2	no	—
90	D₄×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	4	no	—
93	D₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#99	1	2	yes	—
109	C₂×C₄×C₂₃	#100	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	—	4	2	no	—
115	C₂×#408	C₂×C₂₃	1	1	yes	characteristic
116	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	2	yes	complemented
117	#1315	C₂×C₂₃	1	4	yes	direct product factor
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	4	4	no	—
123	C₂²×C₂₃	—	4	4	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	8	4	no	—
126	C₂²×C₂₃	#323	1	2	yes	—
127	C₂²×C₂₃	#350	1	4	yes	—
128	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	#323	1	2	yes	—
133	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
134	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
135	C₂²×#41	C₄×C₂₃	1	1	yes	characteristic
136	C₂²×M₁₆	C₄×C₂₃	1	1	yes	characteristic
137	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
138	C₂²×C₁₆	—	2	2	no	—
139	#408	—	2	2	no	—
140	#408	C₄×C₂₃	1	2	yes	—
141	#408	C₂²×C₂₃	1	2	yes	—
142	C₂×C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₁₆	—	2	2	no	—
144	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	complemented
145	C₂×C₂₃	—	4	1	no	—
146	C₂×C₂₃	—	4	1	no	—
147	C₂×C₂₃	—	8	2	no	—
148	C₂×C₂₃	#1315	1	4	yes	direct product factor
149	C₂×C₂₃	C₂×#323	1	1	yes	characteristic
150	C₂×C₂₃	C₂×#350	1	2	yes	—
151	C₂×#41	—	2	1	no	—
152	C₂×#41	—	2	1	no	—
153	C₂×#41	—	4	2	no	—
154	C₂×#41	—	4	2	no	—
155	C₂×#41	C₈×C₂₃	1	2	yes	—
156	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
157	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	4	2	no	—
164	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	4	2	no	—
167	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₂₃	1	2	yes	—
169	C₂²×C₈	D₄×C₂₃	1	2	yes	—
170	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	—	4	4	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	C₂²⋊C₈	—	2	4	no	—
175	C₂₃	C₂×#1315	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	2	1	no	—
177	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
178	#41	—	4	4	no	—
179	#41	—	4	2	no	—
180	#41	—	4	2	no	—
181	#41	—	4	4	no	—
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
216	C₁₆	—	4	4	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	4	no	—
222	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
223	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
224	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
244	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
245	C₂×C₄	C₂₃×#99	1	2	yes	—
246	C₂×C₄	C₂₃×#100	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂₃×#323	1	2	yes	—
261	C₂²	C₂₃×#350	1	4	yes	—
262	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	2	no	—
266	C₄	C₂₃×#323	1	2	yes	—
267	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
268	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	8	2	no	—
272	C₂	C₂₃×#1315	1	4	yes	direct product factor
273	C₂	C₂×C₂₃×#323	1	1	yes	characteristic
274	C₂	C₂×C₂₃×#350	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂³×C₂₃	C₂²⋊C₈
prime radical	C₂²×#41×C₂₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
3. C₂₃×#1315	C₂	direct (N × H)
13. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
79. C₂×#1315	C₂₃	direct (N × H)
116. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
117. #1315	C₂×C₂₃	direct (N × H)
144. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
148. C₂×C₂₃	#1315	direct (N × H)
175. C₂₃	C₂×#1315	direct (N × H)
272. C₂	C₂₃×#1315	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×#408	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#1315	2
C₂×#1315	23

Subgroups — C2×C23×#1315 #19159046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1315 #19159046