Subgroups — C₂×C₂₃×#1634 #19166135

Order 5888.

1074 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#476	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1634	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₂₃	—	2	2	no	—
8	C₂₃×#555	—	2	2	no	—
9	C₂₃×#555	—	2	4	no	—
10	C₂₃×#555	C₂²	1	4	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#124	—	2	2	no	—
14	C₂×C₂₃×#121	—	2	2	no	—
15	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂₃×#117	—	2	2	no	—
17	C₂₃×#494	—	2	2	no	—
18	C₂₃×#494	C₂²	1	4	yes	complemented
19	C₂₃×#476	—	2	4	no	—
20	C₂₃×#476	C₂²	1	8	yes	complemented
21	C₂×C₂₃×#106	—	2	2	no	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₂₃×#102	—	2	2	no	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#333	C₂²	1	4	yes	complemented
26	C₂₃×D₄⋊C₈	C₂²	1	8	yes	complemented
27	C₂²×Q₈×C₂₃	—	2	1	no	—
28	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
29	C₂²×D₄×C₂₃	—	2	2	no	—
30	C₂²×D₄×C₂₃	D₄	1	2	yes	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂×Q₁₆×C₂₃	—	2	8	no	—
33	C₂×Q₁₆×C₂₃	—	4	8	no	—
34	C₂²×C₈×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂₃×C₄⋊Q₈	—	2	2	no	—
37	C₂₃×C₄⋊Q₈	—	2	2	no	—
38	C₂₃×C₄⋊Q₈	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
40	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₂₃×#124	—	2	8	no	—
43	C₂₃×#124	—	2	8	no	—
44	C₂₃×#121	—	2	8	no	—
45	C₂₃×#121	—	2	8	no	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	2	2	no	—
48	C₄×D₄×C₂₃	D₄	1	4	yes	—
49	C₄×D₄×C₂₃	C₂³	1	4	yes	—
50	C₂₃×#117	—	2	8	no	—
51	C₂₃×#117	—	2	8	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂×C₄²×C₂₃	—	2	2	no	—
58	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#106	—	2	8	no	—
60	C₂₃×C₄⋊C₈	—	2	2	no	—
61	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
62	C₂₃×#102	—	2	8	no	—
63	C₄×C₈×C₂₃	—	2	1	no	—
64	C₄×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂⁴×C₂₃	—	2	2	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	4	no	—
69	C₂×Q₈×C₂₃	—	4	4	no	—
70	C₂×Q₈×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
72	C₂×D₄×C₂₃	—	2	8	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	2	4	no	—
75	C₂×D₄×C₂₃	—	2	4	no	—
76	C₂×D₄×C₂₃	—	8	4	no	—
77	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	—	2	4	no	—
81	C₂²×C₄×C₂₃	—	2	4	no	—
82	C₂²×C₄×C₂₃	—	4	4	no	—
83	C₂²×C₄×C₂₃	—	4	4	no	—
84	C₂²×C₄×C₂₃	—	4	1	no	—
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
87	Q₁₆×C₂₃	—	4	16	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	2	4	no	—
91	C₂×C₈×C₂₃	—	4	4	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	2	4	no	—
95	C₄⋊C₄×C₂₃	—	2	4	no	—
96	C₄⋊C₄×C₂₃	—	4	2	no	—
97	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
98	C₂²⋊C₄×C₂₃	—	2	16	no	—
99	C₂²⋊C₄×C₂₃	—	2	4	no	—
100	C₂²⋊C₄×C₂₃	—	2	4	no	—
101	C₄²×C₂₃	—	2	8	no	—
102	C₄²×C₂₃	C₂×D₄	1	2	yes	—
103	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂×#1634	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₃	—	2	4	no	—
107	C₂³×C₂₃	—	4	4	no	—
108	C₂³×C₂₃	—	4	4	no	—
109	C₂³×C₂₃	—	4	4	no	—
110	C₂³×C₂₃	#120	1	1	yes	characteristic
111	Q₈×C₂₃	—	4	4	no	—
112	Q₈×C₂₃	—	4	4	no	—
113	Q₈×C₂₃	—	8	2	no	—
114	Q₈×C₂₃	—	8	2	no	—
115	D₄×C₂₃	—	2	4	no	—
116	D₄×C₂₃	—	2	4	no	—
117	D₄×C₂₃	—	4	4	no	—
118	D₄×C₂₃	—	8	8	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	—	4	4	no	—
123	C₂×C₄×C₂₃	—	4	4	no	—
124	C₂×C₄×C₂₃	—	4	4	no	—
125	C₂×C₄×C₂₃	—	4	4	no	—
126	C₂×C₄×C₂₃	—	4	8	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	8	2	no	—
129	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
130	C₂×C₄×C₂₃	#120	1	2	yes	—
131	C₂×C₄×C₂₃	#135	1	4	yes	—
132	C₂×C₄×C₂₃	#137	1	4	yes	—
133	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
135	C₈×C₂₃	—	2	4	no	—
136	C₈×C₂₃	—	4	4	no	—
137	C₂×#555	C₂×C₂₃	1	1	yes	characteristic
138	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#476	C₂×C₂₃	1	2	yes	complemented
140	#1634	C₂×C₂₃	1	8	yes	direct product factor
141	C₂×#333	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₂×D₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
143	C₂²×C₂₃	—	4	4	no	—
144	C₂²×C₂₃	—	4	4	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	—	8	8	no	—
147	C₂²×C₂₃	#452	1	4	yes	—
148	C₂²×C₂₃	#453	1	2	yes	—
149	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	—	8	2	no	—
155	C₄×C₂₃	#452	1	4	yes	—
156	C₄×C₂₃	C₂×#135	1	2	yes	—
157	C₄×C₂₃	C₂×#137	1	2	yes	—
158	C₂²×Q₁₆	—	2	2	no	—
159	#555	—	2	2	no	—
160	#555	—	2	4	no	—
161	#555	C₂²×C₂₃	1	4	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×#124	—	2	2	no	—
165	C₂×#121	—	2	2	no	—
166	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
167	C₂×#117	—	2	2	no	—
168	#494	—	2	2	no	—
169	#494	C₂²×C₂₃	1	4	yes	complemented
170	#476	—	2	4	no	—
171	#476	C₂²×C₂₃	1	8	yes	complemented
172	C₂×#106	—	2	2	no	—
173	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
174	C₂×#102	—	2	2	no	—
175	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
176	#333	C₂²×C₂₃	1	4	yes	complemented
177	D₄⋊C₈	C₂²×C₂₃	1	8	yes	complemented
178	C₂×C₂₃	—	8	4	no	—
179	C₂×C₂₃	#1634	1	4	yes	direct product factor
180	C₂×C₂₃	C₂×#452	1	2	yes	—
181	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
182	C₂²×Q₈	—	2	1	no	—
183	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	D₄×C₂₃	1	2	yes	—
186	C₂³×C₄	—	2	2	no	—
187	C₂×Q₁₆	—	2	8	no	—
188	C₂×Q₁₆	—	4	8	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
195	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
196	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
197	#124	—	2	8	no	—
198	#124	—	2	8	no	—
199	#121	—	2	8	no	—
200	#121	—	2	8	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	D₄×C₂₃	1	4	yes	—
204	C₄×D₄	C₂³×C₂₃	1	4	yes	—
205	#117	—	2	8	no	—
206	#117	—	2	8	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
214	#106	—	2	8	no	—
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
217	#102	—	2	8	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂³×C₂₃	1	2	yes	—
220	C₂₃	C₂×#1634	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
228	C₂×D₄	—	2	8	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	8	4	no	—
233	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
243	Q₁₆	—	4	16	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
254	C₂²⋊C₄	—	2	16	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₄²	—	2	8	no	—
258	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₂₃	1	2	yes	—
260	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
261	C₂³	—	2	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂₃×#120	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	2	no	—
269	Q₈	—	8	2	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	8	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
285	C₂×C₄	C₂₃×#120	1	2	yes	—
286	C₂×C₄	C₂₃×#135	1	4	yes	—
287	C₂×C₄	C₂₃×#137	1	4	yes	—
288	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	8	8	no	—
296	C₂²	C₂₃×#452	1	4	yes	—
297	C₂²	C₂₃×#453	1	2	yes	—
298	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂₃×#452	1	4	yes	—
305	C₄	C₂×C₂₃×#135	1	2	yes	—
306	C₄	C₂×C₂₃×#137	1	2	yes	—
307	C₂	—	8	4	no	—
308	C₂	C₂₃×#1634	1	4	yes	direct product factor
309	C₂	C₂×C₂₃×#452	1	2	yes	—
310	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#494	C₂	semidirect
3. C₂×C₂₃×#476	C₂	semidirect
4. C₂₃×#1634	C₂	direct (N × H)
5. C₂×C₂₃×#333	C₂	semidirect
6. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
18. C₂₃×#494	C₂²	semidirect
20. C₂₃×#476	C₂²	semidirect
25. C₂₃×#333	C₂²	semidirect
26. C₂₃×D₄⋊C₈	C₂²	semidirect
105. C₂×#1634	C₂₃	direct (N × H)
138. C₂×#494	C₂×C₂₃	semidirect
139. C₂×#476	C₂×C₂₃	semidirect
140. #1634	C₂×C₂₃	direct (N × H)
141. C₂×#333	C₂×C₂₃	semidirect
142. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
169. #494	C₂²×C₂₃	semidirect
171. #476	C₂²×C₂₃	semidirect
176. #333	C₂²×C₂₃	semidirect
177. D₄⋊C₈	C₂²×C₂₃	semidirect
179. C₂×C₂₃	#1634	direct (N × H)
220. C₂₃	C₂×#1634	direct (N × H)
308. C₂	C₂₃×#1634	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×#555	2
C₂×C₂₃×#494	2
C₂×C₂₃×#476	2
C₂₃×#1634	2
C₂×C₂₃×#333	2
C₂×C₂₃×D₄⋊C₈	2
C₂×#1634	23

Subgroups — C2×C23×#1634 #19166135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1634 #19166135