Subgroups — #15686724

Order 2944.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₂₃	C₂	1	2	yes	maximal, complemented
2	#13092994	C₂	1	4	yes	maximal, complemented
3	#13092992	C₂	1	2	yes	maximal, complemented
4	#13092987	C₂	1	2	yes	maximal, complemented
5	#13092985	C₂	1	2	yes	maximal, complemented
6	#13092980	C₂	1	2	yes	maximal, complemented
7	#13092975	C₂	1	4	yes	maximal, complemented
8	#13092967	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092963	C₂	1	2	yes	maximal, complemented
10	#13092926	C₂	1	8	yes	maximal, complemented
11	C₂×#10625043	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
14	#41×D₂₃	—	2	16	no	—
15	C₂×#67292	C₂²	1	2	yes	—
16	C₂×Q₈×D₂₃	C₂²	1	2	yes	complemented
17	C₂×#67290	C₂²	1	2	yes	—
18	C₂×#67290	C₂²	1	4	yes	complemented
19	C₂×D₄×D₂₃	C₂²	1	4	yes	—
20	C₂×D₄×D₂₃	C₂²	1	2	yes	complemented
21	C₂×#67288	C₂²	1	4	yes	—
22	C₂×#67288	C₂²	1	2	yes	complemented
23	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₂₃	C₂²	1	4	yes	complemented
25	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#124	C₂²	1	2	yes	complemented
28	C₂₃×#121	C₂²	1	4	yes	complemented
29	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	#10625118	C₂²	1	2	yes	complemented
32	#10625115	C₂²	1	2	yes	complemented
33	#10625111	C₂²	1	2	yes	complemented
34	#10625110	C₂²	1	8	yes	complemented
35	#10625110	C₂²	1	8	yes	complemented
36	#10625108	C₂²	1	8	yes	complemented
37	#10625107	C₂²	1	2	yes	complemented
38	#10625101	C₂²	1	4	yes	complemented
39	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
40	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
41	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
42	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
43	#10625067	C₂²	1	8	yes	complemented
44	#10625065	C₂²	1	8	yes	complemented
45	#10625055	C₂²	1	8	yes	complemented
46	#10625054	C₂²	1	8	yes	complemented
47	#10625053	C₂²	1	8	yes	complemented
48	#10625049	C₂²	1	8	yes	complemented
49	#10625046	C₂²	1	2	yes	complemented
50	#10625045	C₂²	1	1	yes	characteristic, complemented
51	C₄×D₉₂	C₂²	1	4	yes	complemented
52	#10625043	C₂²	1	4	yes	complemented
53	#10625043	C₂²	1	2	yes	complemented
54	#10625043	C₂²	1	2	yes	complemented
55	#10625040	C₂²	1	1	yes	characteristic, complemented
56	C₂³×D₂₃	C₂³	1	4	yes	—
57	C₂³×D₂₃	C₂³	1	1	yes	characteristic
58	#41×C₂₃	—	2	8	no	—
59	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂×D₄×C₂₃	C₂³	1	4	yes	—
62	C₂×D₄×C₂₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
67	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
68	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
69	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
70	C₂²×Dic₂₃	C₂³	1	4	yes	—
71	C₂²×Dic₂₃	C₂³	1	2	yes	—
72	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
73	#67292	—	2	8	no	—
74	Q₈×D₂₃	—	2	8	no	—
75	#67290	—	2	16	no	—
76	#67290	—	2	8	no	—
77	D₄×D₂₃	—	2	16	no	—
78	D₄×D₂₃	—	2	8	no	—
79	#67288	—	2	8	no	—
80	#67288	—	2	16	no	—
81	C₂×D₉₂	C₂³	1	2	yes	—
82	C₂×D₉₂	C₂³	1	4	yes	—
83	C₂×D₉₂	C₂³	1	4	yes	—
84	C₂×C₄×D₂₃	—	2	16	no	—
85	C₂×C₄×D₂₃	D₄	1	8	yes	complemented
86	C₂×C₄×D₂₃	C₂³	1	4	yes	—
87	C₂×C₄×D₂₃	C₂³	1	4	yes	—
88	C₂×C₄×D₂₃	C₂³	1	4	yes	—
89	C₂×C₄×D₂₃	C₂³	1	2	yes	—
90	C₂×C₄×D₂₃	C₂³	1	2	yes	—
91	C₂×Dic₄₆	C₂³	1	2	yes	—
92	C₂×Dic₄₆	C₂³	1	4	yes	—
93	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
95	C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₄²×C₂₃	C₂³	1	2	yes	—
97	C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
99	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
100	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
101	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
102	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
103	C₄⋊Dic₂₃	C₂³	1	4	yes	—
104	Dic₂₃⋊C₄	C₂³	1	4	yes	—
105	Dic₂₃⋊C₄	C₂³	1	4	yes	—
106	Dic₂₃⋊C₄	C₂³	1	8	yes	—
107	C₄×Dic₂₃	C₂³	1	2	yes	—
108	C₄×Dic₂₃	C₂³	1	2	yes	—
109	C₂³×C₂₃	C₂⁴	1	4	yes	—
110	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
111	C₂²×D₂₃	—	2	8	no	—
112	C₂²×D₂₃	—	4	2	no	—
113	C₂²×D₂₃	—	4	8	no	—
114	C₂²×D₂₃	C₂×D₄	1	4	yes	—
115	C₂²×D₂₃	C₂⁴	1	4	yes	—
116	C₂²×D₂₃	C₂⁴	1	2	yes	—
117	Q₈×C₂₃	—	2	4	no	—
118	D₄×C₂₃	—	2	8	no	—
119	D₄×C₂₃	—	2	4	no	—
120	D₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	2	8	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
124	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
125	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
126	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
127	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₂₃	—	2	8	no	—
130	C₂²⋊D₂₃	—	2	8	no	—
131	C₂²⋊D₂₃	—	2	8	no	—
132	C₂²⋊D₂₃	—	4	8	no	—
133	C₂×Dic₂₃	—	2	8	no	—
134	C₂×Dic₂₃	—	4	2	no	—
135	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
136	C₂×Dic₂₃	C₂⁴	1	4	yes	—
137	C₂×Dic₂₃	C₂⁴	1	4	yes	—
138	C₂×Dic₂₃	C₂⁴	1	2	yes	—
139	D₉₂	—	2	8	no	—
140	D₉₂	—	2	4	no	—
141	D₉₂	—	4	4	no	—
142	C₄×D₂₃	—	2	8	no	—
143	C₄×D₂₃	—	2	8	no	—
144	C₄×D₂₃	—	2	8	no	—
145	C₄×D₂₃	C₂×D₄	1	8	yes	—
146	C₄×D₂₃	C₂×D₄	1	8	yes	—
147	Dic₄₆	—	2	4	no	—
148	Dic₄₆	—	2	8	no	—
149	#3449	—	23	1	no	maximal, Sylow
150	C₂²×C₂₃	—	2	2	no	—
151	C₂²×C₂₃	—	2	4	no	—
152	C₂²×C₂₃	—	2	8	no	—
153	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
154	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
155	C₂×D₂₃	—	2	4	no	—
156	C₂×D₂₃	—	2	4	no	—
157	C₂×D₂₃	—	2	8	no	—
158	C₂×D₂₃	—	4	4	no	—
159	C₂×D₂₃	—	4	16	no	—
160	C₂×D₂₃	C₂²×D₄	1	4	yes	—
161	C₄×C₂₃	—	2	4	no	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	2	2	no	—
164	C₄×C₂₃	C₂²×D₄	1	2	yes	—
165	C₄×C₂₃	C₂²×D₄	1	2	yes	—
166	Dic₂₃	—	2	4	no	—
167	Dic₂₃	—	2	4	no	—
168	Dic₂₃	C₂²×D₄	1	4	yes	—
169	C₂²×#41	—	23	2	no	—
170	#537	—	23	4	no	—
171	#536	—	23	2	no	—
172	#536	—	23	2	no	—
173	#535	—	23	2	no	—
174	#535	—	23	2	no	—
175	#534	—	23	4	no	—
176	#532	—	23	2	no	—
177	#532	—	23	1	no	—
178	#532	D₂₃	1	1	yes	characteristic, complemented
179	#525	—	23	8	no	—
180	C₂×#117	—	23	1	no	—
181	C₂×C₂₃	—	2	2	no	—
182	C₂×C₂₃	—	4	4	no	—
183	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
184	C₂×C₂₃	#585	1	2	yes	—
185	D₂₃	—	2	4	no	—
186	D₂₃	—	4	4	no	—
187	C₂×#41	—	23	2	no	—
188	C₂×#41	—	23	2	no	—
189	C₂×#41	—	23	4	no	—
190	C₂×#41	—	23	4	no	—
191	C₂×#41	—	23	2	no	—
192	C₂×#41	—	46	16	no	—
193	C₂×#41	C₂×D₂₃	1	2	yes	complemented
194	C₂²×Q₈	—	23	2	no	—
195	C₂²×D₄	—	23	4	no	—
196	C₂²×D₄	—	23	2	no	—
197	C₂³×C₄	—	23	1	no	—
198	C₂³×C₄	—	23	4	no	—
199	C₄⋊Q₈	—	23	2	no	—
200	C₄⋊Q₈	—	23	1	no	—
201	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
202	#127	—	23	2	no	—
203	#127	—	23	1	no	—
204	#127	C₂×D₂₃	1	1	yes	characteristic, complemented
205	#124	—	23	2	no	—
206	#124	—	23	2	no	—
207	#124	—	23	2	no	—
208	#124	C₂×D₂₃	1	2	yes	complemented
209	#123	—	23	8	no	—
210	#123	—	23	8	no	—
211	C₂²⋊Q₈	—	23	8	no	—
212	C₂²⋊Q₈	—	23	8	no	—
213	#121	—	23	8	no	—
214	#121	—	23	8	no	—
215	#121	—	23	4	no	—
216	#121	—	23	8	no	—
217	#121	C₂×D₂₃	1	4	yes	complemented
218	#120	—	23	8	no	—
219	#120	—	23	8	no	—
220	C₄×D₄	—	23	8	no	—
221	C₄×D₄	—	23	4	no	—
222	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
223	#117	—	23	4	no	—
224	#117	—	23	2	no	—
225	#117	—	23	2	no	—
226	C₂×C₄⋊C₄	—	23	2	no	—
227	C₂×C₂²⋊C₄	—	23	2	no	—
228	C₂×C₄²	—	23	1	no	—
229	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
230	C₂₃	#3449	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	—	23	1	no	—
232	C₂⁴	—	23	4	no	—
233	#41	—	2	8	no	—
234	#41	—	46	16	no	—
235	#41	—	46	16	no	—
236	#41	—	46	8	no	—
237	#41	—	46	8	no	—
238	#41	—	46	8	no	—
239	C₂×Q₈	—	23	4	no	—
240	C₂×Q₈	—	23	2	no	—
241	C₂×Q₈	—	46	8	no	—
242	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
243	C₂×D₄	—	23	4	no	—
244	C₂×D₄	—	23	4	no	—
245	C₂×D₄	—	23	4	no	—
246	C₂×D₄	—	23	8	no	—
247	C₂×D₄	—	23	4	no	—
248	C₂×D₄	—	23	2	no	—
249	C₂×D₄	—	23	4	no	—
250	C₂×D₄	—	46	8	no	—
251	C₂×D₄	—	46	16	no	—
252	C₂×D₄	C₂²×D₂₃	1	4	yes	—
253	C₂×D₄	C₂²×D₂₃	1	2	yes	—
254	C₂×D₄	C₂²×D₂₃	1	4	yes	—
255	C₂²×C₄	—	23	1	no	—
256	C₂²×C₄	—	23	8	no	—
257	C₂²×C₄	—	23	2	no	—
258	C₂²×C₄	—	23	2	no	—
259	C₂²×C₄	—	23	4	no	—
260	C₂²×C₄	—	23	2	no	—
261	C₂²×C₄	—	23	4	no	—
262	C₂²×C₄	—	23	4	no	—
263	C₂²×C₄	—	23	4	no	—
264	C₂²×C₄	—	46	16	no	—
265	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
266	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
267	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
268	C₄⋊C₄	—	23	4	no	—
269	C₄⋊C₄	—	23	4	no	—
270	C₄⋊C₄	—	23	8	no	—
271	C₄⋊C₄	—	23	4	no	—
272	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
273	C₂²⋊C₄	—	23	8	no	—
274	C₂²⋊C₄	—	23	4	no	—
275	C₂²⋊C₄	—	23	4	no	—
276	C₂²⋊C₄	—	23	8	no	—
277	C₂²⋊C₄	—	23	8	no	—
278	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
279	C₄²	—	23	2	no	—
280	C₄²	—	23	2	no	—
281	C₄²	C₂²×D₂₃	1	1	yes	characteristic
282	C₄²	C₂²×D₂₃	1	1	yes	characteristic
283	C₄²	C₂²×D₂₃	1	2	yes	—
284	C₂³	—	23	2	no	—
285	C₂³	—	23	4	no	—
286	C₂³	—	23	4	no	—
287	C₂³	—	46	8	no	—
288	C₂³	—	92	2	no	—
289	C₂³	—	92	8	no	—
290	C₂³	C₂³×D₂₃	1	4	yes	—
291	C₂³	C₂³×D₂₃	1	1	yes	characteristic
292	Q₈	—	2	4	no	—
293	Q₈	—	46	8	no	—
294	Q₈	—	46	4	no	—
295	D₄	—	2	8	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	46	8	no	—
299	D₄	—	46	8	no	—
300	D₄	—	46	8	no	—
301	D₄	—	46	8	no	—
302	D₄	—	46	4	no	—
303	D₄	—	92	4	no	—
304	D₄	—	92	8	no	—
305	C₂×C₄	—	2	8	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	23	4	no	—
308	C₂×C₄	—	23	2	no	—
309	C₂×C₄	—	23	8	no	—
310	C₂×C₄	—	23	8	no	—
311	C₂×C₄	—	23	4	no	—
312	C₂×C₄	—	23	4	no	—
313	C₂×C₄	—	46	8	no	—
314	C₂×C₄	—	46	8	no	—
315	C₂×C₄	—	46	8	no	—
316	C₂×C₄	—	46	8	no	—
317	C₂×C₄	—	92	2	no	—
318	C₂×C₄	D₄×D₂₃	1	4	yes	—
319	C₂×C₄	C₂³×D₂₃	1	2	yes	—
320	C₂×C₄	C₂³×D₂₃	1	2	yes	—
321	C₂×C₄	C₂³×D₂₃	1	4	yes	—
322	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
324	C₂²	—	2	2	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	8	no	—
327	C₂²	—	23	4	no	—
328	C₂²	—	46	4	no	—
329	C₂²	—	46	4	no	—
330	C₂²	—	46	8	no	—
331	C₂²	—	92	4	no	—
332	C₂²	—	92	16	no	—
333	C₂²	C₂×D₄×D₂₃	1	2	yes	—
334	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	23	4	no	—
339	C₄	—	46	4	no	—
340	C₄	—	46	4	no	—
341	C₄	C₂×D₄×D₂₃	1	2	yes	—
342	C₄	C₂×D₄×D₂₃	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	4	4	no	—
345	C₂	—	46	4	no	—
346	C₂	—	92	4	no	—
347	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
348	C₂	#13093253	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂₃×#532	C₂
hypercenter	#532	D₂₃
hypocenter	C₂₃	#3449
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₂₃	C₂	semidirect
2. #13092994	C₂	semidirect
3. #13092992	C₂	semidirect
4. #13092987	C₂	semidirect
5. #13092985	C₂	semidirect
6. #13092980	C₂	semidirect
7. #13092975	C₂	semidirect
8. #13092967	C₂	semidirect
9. #13092963	C₂	semidirect
10. #13092926	C₂	semidirect
11. C₂×#10625043	C₂	semidirect
12. C₂₃×#532	C₂	semidirect
13. C₂×#41×C₂₃	C₂²	semidirect
16. C₂×Q₈×D₂₃	C₂²	semidirect
18. C₂×#67290	C₂²	semidirect
20. C₂×D₄×D₂₃	C₂²	semidirect
22. C₂×#67288	C₂²	semidirect
24. C₂²×C₄×D₂₃	C₂²	semidirect
25. C₂₃×C₄⋊Q₈	C₂²	semidirect
26. C₂₃×#127	C₂²	semidirect
27. C₂₃×#124	C₂²	semidirect
28. C₂₃×#121	C₂²	semidirect
29. C₄×D₄×C₂₃	C₂²	semidirect
30. C₂×C₄²×C₂₃	C₂²	semidirect
31. #10625118	C₂²	semidirect
32. #10625115	C₂²	semidirect
33. #10625111	C₂²	semidirect
34. #10625110	C₂²	semidirect
35. #10625110	C₂²	semidirect
36. #10625108	C₂²	semidirect
37. #10625107	C₂²	semidirect
38. #10625101	C₂²	semidirect
39. C₄×C₂²⋊D₂₃	C₂²	semidirect
41. C₂×Dic₂₃⋊C₄	C₂²	semidirect
42. C₂×C₄×Dic₂₃	C₂²	semidirect
43. #10625067	C₂²	semidirect
44. #10625065	C₂²	semidirect
45. #10625055	C₂²	semidirect
46. #10625054	C₂²	semidirect
47. #10625053	C₂²	semidirect
48. #10625049	C₂²	semidirect
49. #10625046	C₂²	semidirect
50. #10625045	C₂²	semidirect
51. C₄×D₉₂	C₂²	semidirect
52. #10625043	C₂²	semidirect
53. #10625043	C₂²	semidirect
54. #10625043	C₂²	semidirect
55. #10625040	C₂²	semidirect
85. C₂×C₄×D₂₃	D₄	semidirect
178. #532	D₂₃	semidirect
193. C₂×#41	C₂×D₂₃	semidirect
201. C₄⋊Q₈	C₂×D₂₃	semidirect
204. #127	C₂×D₂₃	semidirect
208. #124	C₂×D₂₃	semidirect
217. #121	C₂×D₂₃	semidirect
222. C₄×D₄	C₂×D₂₃	semidirect
229. C₂×C₄²	C₂×D₂₃	semidirect
230. C₂₃	#3449	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₂₃	2
#13092994	2
#13092992	2
#13092987	2
#13092985	2
#13092980	2
#13092975	2
#13092967	2
#13092963	2
#13092926	2
C₂×#10625043	2
C₂₃×#532	2
#3449	23