Subgroups — #15681941

Order 2944.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093253	C₂	1	1	yes	maximal, characteristic
2	#13093183	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#67288	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092434	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092433	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092432	C₂	1	4	yes	maximal, complemented
7	C₂×#10624987	C₂	1	1	yes	maximal, characteristic, complemented
8	#104×D₂₃	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#104	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	#10625161	—	2	1	no	—
12	#10625161	C₄	1	2	yes	complemented
13	#10625160	—	2	1	no	—
14	#10625160	C₄	1	2	yes	complemented
15	#41×D₂₃	—	2	2	no	—
16	#41×D₂₃	C₂²	1	4	yes	—
17	#10625158	C₄	1	2	yes	complemented
18	#10625155	—	2	2	no	—
19	#10625155	C₄	1	2	yes	complemented
20	C₂×#67288	—	2	2	no	—
21	C₂×#67288	C₂²	1	1	yes	characteristic
22	C₂×#67288	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₄×C₂²⋊D₂₃	—	2	2	no	—
26	#10625097	C₂²	1	1	yes	characteristic, complemented
27	C₂×#67267	C₂²	1	1	yes	characteristic, complemented
28	#10625078	—	2	2	no	—
29	M₁₆×D₂₃	C₂²	1	4	yes	complemented
30	#10625072	—	2	2	no	—
31	C₄×D₉₂	—	2	1	no	—
32	#10625043	C₂²	1	2	yes	complemented
33	C₄²×D₂₃	C₂²	1	2	yes	complemented
34	C₄×Dic₄₆	—	2	1	no	—
35	C₂₃×#104	C₂²	1	2	yes	complemented
36	C₂₃×#104	C₂²	1	2	yes	complemented
37	#10625019	C₂²	1	4	yes	complemented
38	#10625007	C₂²	1	4	yes	complemented
39	#10624987	C₂²	1	2	yes	complemented
40	#10624987	C₂²	1	2	yes	complemented
41	#41×C₂₃	—	2	1	no	—
42	#41×C₂₃	—	2	1	no	—
43	#41×C₂₃	C₂×C₄	1	2	yes	complemented
44	#41×C₂₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₂₃	—	2	1	no	—
52	C₂×C₂²⋊D₂₃	—	2	4	no	—
53	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
54	#67292	—	2	2	no	—
55	#67292	C₂×C₄	1	4	yes	complemented
56	Q₈×D₂₃	—	2	2	no	—
57	Q₈×D₂₃	C₂×C₄	1	4	yes	complemented
58	#67290	—	2	4	no	—
59	#67290	—	2	4	no	—
60	#67290	—	2	2	no	—
61	#67290	C₂×C₄	1	4	yes	complemented
62	D₄×D₂₃	—	2	4	no	—
63	D₄×D₂₃	—	2	4	no	—
64	D₄×D₂₃	—	2	2	no	—
65	D₄×D₂₃	C₂×C₄	1	4	yes	complemented
66	#67288	—	2	1	no	—
67	#67288	—	2	1	no	—
68	#67288	—	2	4	no	—
69	#67288	—	2	2	no	—
70	#67288	—	2	4	no	—
71	#67288	—	2	2	no	—
72	#67288	C₂×C₄	1	2	yes	—
73	#67288	C₂×C₄	1	2	yes	—
74	#67288	C₂×C₄	1	2	yes	complemented
75	#67288	D₄	1	2	yes	—
76	#67288	D₄	1	2	yes	—
77	#67288	C₂³	1	2	yes	—
78	C₂×D₉₂	—	2	2	no	—
79	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₉₂	D₄	1	1	yes	characteristic
81	C₂×C₄×D₂₃	—	2	1	no	—
82	C₂×C₄×D₂₃	—	2	4	no	—
83	C₂×C₄×D₂₃	—	2	2	no	—
84	C₂×C₄×D₂₃	C₂³	1	2	yes	—
85	C₂×Dic₄₆	—	2	2	no	—
86	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic, complemented
87	C₂×Dic₄₆	D₄	1	1	yes	characteristic
88	M₁₆×C₂₃	—	2	1	no	—
89	M₁₆×C₂₃	C₂³	1	2	yes	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₄²×C₂₃	—	2	1	no	—
92	C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₂₃	—	2	1	no	—
95	(C₂×D₂₃)⋊C₄	—	2	2	no	—
96	C₄⋊Dic₂₃	—	2	1	no	—
97	Dic₂₃⋊C₄	—	2	2	no	—
98	C₄×Dic₂₃	—	2	2	no	—
99	C₄×Dic₂₃	C₂³	1	2	yes	—
100	#67267	—	2	1	no	—
101	#67267	C₂³	1	2	yes	—
102	C₂×C₂₃⋊C₈	—	2	1	no	—
103	#67262	—	2	4	no	—
104	C₈×D₂₃	—	2	4	no	—
105	C₂³×C₂₃	—	2	1	no	—
106	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₂₃	—	2	1	no	—
108	C₂²×D₂₃	—	2	4	no	—
109	C₂²×D₂₃	C₂×D₄	1	2	yes	—
110	Q₈×C₂₃	—	2	1	no	—
111	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
112	D₄×C₂₃	—	2	2	no	—
113	D₄×C₂₃	—	2	2	no	—
114	D₄×C₂₃	—	2	1	no	—
115	D₄×C₂₃	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₂₃	—	2	2	no	—
129	C₂²⋊D₂₃	—	2	2	no	—
130	C₂²⋊D₂₃	—	2	4	no	—
131	C₂²⋊D₂₃	—	2	4	no	—
132	C₂²⋊D₂₃	C₂²⋊C₄	1	4	yes	—
133	C₂²⋊D₂₃	C₂²⋊C₄	1	4	yes	—
134	C₂×Dic₂₃	—	2	1	no	—
135	C₂×Dic₂₃	—	2	4	no	—
136	C₂×Dic₂₃	—	2	2	no	—
137	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
138	D₉₂	—	2	4	no	—
139	D₉₂	—	2	1	no	—
140	D₉₂	C₂²⋊C₄	1	2	yes	—
141	D₉₂	C₂²⋊C₄	1	2	yes	—
142	D₉₂	C₂²×C₄	1	2	yes	—
143	C₄×D₂₃	—	2	4	no	—
144	C₄×D₂₃	—	2	2	no	—
145	C₄×D₂₃	—	2	2	no	—
146	C₄×D₂₃	—	2	4	no	—
147	C₄×D₂₃	—	4	4	no	—
148	C₄×D₂₃	C₂²×C₄	1	4	yes	—
149	C₄×D₂₃	C₂×D₄	1	4	yes	—
150	Dic₄₆	—	2	1	no	—
151	Dic₄₆	—	2	4	no	—
152	Dic₄₆	C₂²⋊C₄	1	2	yes	—
153	Dic₄₆	C₂²⋊C₄	1	2	yes	—
154	Dic₄₆	C₂²×C₄	1	2	yes	—
155	C₈×C₂₃	—	2	2	no	—
156	C₂₃⋊C₈	—	2	2	no	—
157	#2901	—	23	1	no	maximal, Sylow
158	C₂²×C₂₃	—	2	2	no	—
159	C₂²×C₂₃	—	4	1	no	—
160	C₂²×C₂₃	—	8	1	no	—
161	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₂₃	—	2	2	no	—
165	C₂×D₂₃	—	4	2	no	—
166	C₂×D₂₃	—	8	4	no	—
167	C₂×D₂₃	—	8	1	no	—
168	C₂×D₂₃	C₂×C₂²⋊C₄	1	4	yes	—
169	C₄×C₂₃	—	2	2	no	—
170	C₄×C₂₃	—	2	2	no	—
171	C₄×C₂₃	—	4	1	no	—
172	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
175	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	Dic₂₃	—	2	2	no	—
177	Dic₂₃	—	4	2	no	—
178	Dic₂₃	C₂×C₂²⋊C₄	1	4	yes	—
179	#585	—	23	1	no	—
180	#568	—	23	1	no	—
181	C₄×#41	—	23	1	no	—
182	#421	—	23	1	no	—
183	#421	—	23	1	no	—
184	#421	—	23	4	no	—
185	C₂×#104	—	23	1	no	—
186	C₂×#104	—	23	4	no	—
187	C₂×#104	D₂₃	1	1	yes	characteristic, complemented
188	C₂×C₂₃	—	2	1	no	—
189	C₂×C₂₃	—	2	1	no	—
190	C₂×C₂₃	—	2	1	no	—
191	C₂×C₂₃	—	4	2	no	—
192	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	D₂₃	—	2	4	no	—
194	D₂₃	—	4	2	no	—
195	2₋¹⁺⁴	—	23	2	no	—
196	2₋¹⁺⁴	—	23	2	no	—
197	2₋¹⁺⁴	—	46	1	no	—
198	H(5,𝔽₂)	—	23	2	no	—
199	H(5,𝔽₂)	—	23	2	no	—
200	H(5,𝔽₂)	—	46	2	no	—
201	H(5,𝔽₂)	—	46	1	no	—
202	C₂×#41	—	23	1	no	—
203	C₂×#41	—	23	4	no	—
204	C₂×#41	—	23	1	no	—
205	C₂×#41	—	46	2	no	—
206	C₂×#41	—	46	2	no	—
207	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
208	#131	—	46	2	no	—
209	#131	—	46	2	no	—
210	C₂×M₁₆	—	23	1	no	—
211	C₂×M₁₆	—	23	4	no	—
212	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
213	C₄×Q₈	—	46	1	no	—
214	C₄×D₄	—	46	1	no	—
215	C₄×D₄	—	46	2	no	—
216	#117	—	23	1	no	—
217	#117	—	23	2	no	—
218	C₂×C₄²	—	23	2	no	—
219	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
220	#104	—	23	4	no	—
221	#104	—	23	4	no	—
222	#104	—	23	2	no	—
223	#104	—	23	2	no	—
224	#104	C₂×D₂₃	1	2	yes	complemented
225	#104	C₂×D₂₃	1	2	yes	complemented
226	C₂₃	#2901	1	1	yes	characteristic, complemented, Sylow
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	23	2	no	—
230	#41	—	23	2	no	—
231	#41	—	23	2	no	—
232	#41	—	23	2	no	—
233	#41	—	23	4	no	—
234	#41	—	23	4	no	—
235	#41	—	23	2	no	—
236	#41	—	23	2	no	—
237	#41	—	46	4	no	—
238	#41	—	46	4	no	—
239	#41	—	46	2	no	—
240	#41	—	46	2	no	—
241	#41	—	46	4	no	—
242	#41	—	46	4	no	—
243	#41	—	46	2	no	—
244	#41	—	46	2	no	—
245	#41	—	46	1	no	—
246	#41	—	46	1	no	—
247	#41	C₄×D₂₃	1	2	yes	complemented
248	#41	C₂²×D₂₃	1	2	yes	—
249	C₂×Q₈	—	23	1	no	—
250	C₂×Q₈	—	23	4	no	—
251	C₂×Q₈	—	23	1	no	—
252	C₂×Q₈	—	46	2	no	—
253	C₂×Q₈	—	46	2	no	—
254	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic, complemented
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	23	2	no	—
257	C₂×D₄	—	23	4	no	—
258	C₂×D₄	—	23	1	no	—
259	C₂×D₄	—	23	1	no	—
260	C₂×D₄	—	46	2	no	—
261	C₂×D₄	—	46	4	no	—
262	C₂×D₄	—	46	2	no	—
263	C₂×D₄	—	46	4	no	—
264	C₂×D₄	—	46	4	no	—
265	C₂×D₄	—	46	1	no	—
266	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic, complemented
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	23	2	no	—
270	C₂²×C₄	—	46	4	no	—
271	C₂²×C₄	—	46	2	no	—
272	C₂²×C₄	—	46	1	no	—
273	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
274	M₁₆	—	2	1	no	—
275	M₁₆	—	23	2	no	—
276	M₁₆	—	46	4	no	—
277	M₁₆	—	46	1	no	—
278	M₁₆	C₂²×D₂₃	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	46	4	no	—
281	C₂×C₈	—	46	1	no	—
282	C₄⋊C₄	—	46	2	no	—
283	C₄⋊C₄	—	46	1	no	—
284	C₂²⋊C₄	—	46	2	no	—
285	C₂²⋊C₄	—	46	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	23	2	no	—
288	C₄²	—	46	2	no	—
289	C₄²	C₂²×D₂₃	1	1	yes	characteristic
290	C₄²	C₂²×D₂₃	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	23	2	no	—
293	C₂³	—	46	4	no	—
294	C₂³	—	46	1	no	—
295	C₂³	D₄×D₂₃	1	1	yes	characteristic
296	Q₈	—	2	1	no	—
297	Q₈	—	23	2	no	—
298	Q₈	—	23	2	no	—
299	Q₈	—	23	2	no	—
300	Q₈	—	46	4	no	—
301	Q₈	—	46	1	no	—
302	Q₈	C₂×C₄×D₂₃	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	23	4	no	—
307	D₄	—	23	4	no	—
308	D₄	—	23	2	no	—
309	D₄	—	23	2	no	—
310	D₄	—	23	2	no	—
311	D₄	—	46	4	no	—
312	D₄	—	46	4	no	—
313	D₄	—	46	4	no	—
314	D₄	—	46	2	no	—
315	D₄	—	46	2	no	—
316	D₄	—	46	1	no	—
317	D₄	C₂×C₄×D₂₃	1	2	yes	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	23	2	no	—
325	C₂×C₄	—	23	4	no	—
326	C₂×C₄	—	23	4	no	—
327	C₂×C₄	—	46	4	no	—
328	C₂×C₄	—	46	4	no	—
329	C₂×C₄	—	46	4	no	—
330	C₂×C₄	—	46	1	no	—
331	C₂×C₄	—	46	2	no	—
332	C₂×C₄	—	46	2	no	—
333	C₂×C₄	—	46	2	no	—
334	C₂×C₄	—	92	4	no	—
335	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
337	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
338	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
339	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	46	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	23	4	no	—
347	C₂²	—	46	2	no	—
348	C₂²	—	92	2	no	—
349	C₂²	—	184	1	no	—
350	C₂²	—	184	4	no	—
351	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
352	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
353	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	23	4	no	—
358	C₄	—	46	2	no	—
359	C₄	—	92	2	no	—
360	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
361	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
362	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
363	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	4	2	no	—
368	C₂	—	46	4	no	—
369	C₂	—	92	2	no	—
370	C₂	C₂×C₂²⋊C₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#104	C₂
socle	C₂×C₂₃	C₂²×C₂²⋊C₄
prime radical	#13093253	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#104	D₂₃
hypocenter	C₂₃	#2901
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #13093183	C₂	semidirect
3. C₄×#67288	C₂	semidirect
4. #13092434	C₂	semidirect
5. #13092433	C₂	semidirect
6. #13092432	C₂	semidirect
7. C₂×#10624987	C₂	semidirect
8. #104×D₂₃	C₂	semidirect
9. C₂×C₂₃×#104	C₂	semidirect
12. #10625161	C₄	semidirect
14. #10625160	C₄	semidirect
17. #10625158	C₄	semidirect
19. #10625155	C₄	semidirect
23. C₂×M₁₆×C₂₃	C₂²	semidirect
24. C₂×C₄²×C₂₃	C₂²	semidirect
26. #10625097	C₂²	semidirect
27. C₂×#67267	C₂²	semidirect
29. M₁₆×D₂₃	C₂²	semidirect
32. #10625043	C₂²	semidirect
33. C₄²×D₂₃	C₂²	semidirect
35. C₂₃×#104	C₂²	semidirect
36. C₂₃×#104	C₂²	semidirect
37. #10625019	C₂²	semidirect
38. #10625007	C₂²	semidirect
39. #10624987	C₂²	semidirect
40. #10624987	C₂²	semidirect
43. #41×C₂₃	C₂×C₄	semidirect
45. C₂×Q₈×C₂₃	C₂×C₄	semidirect
47. C₂×D₄×C₂₃	C₂×C₄	semidirect
55. #67292	C₂×C₄	semidirect
57. Q₈×D₂₃	C₂×C₄	semidirect
61. #67290	C₂×C₄	semidirect
65. D₄×D₂₃	C₂×C₄	semidirect
74. #67288	C₂×C₄	semidirect
79. C₂×D₉₂	C₂×C₄	semidirect
86. C₂×Dic₄₆	C₂×C₄	semidirect
187. C₂×#104	D₂₃	semidirect
212. C₂×M₁₆	C₂×D₂₃	semidirect
219. C₂×C₄²	C₂×D₂₃	semidirect
224. #104	C₂×D₂₃	semidirect
225. #104	C₂×D₂₃	semidirect
226. C₂₃	#2901	semidirect
247. #41	C₄×D₂₃	semidirect
254. C₂×Q₈	C₄×D₂₃	semidirect
266. C₂×D₄	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#13093253	2
#13093183	2
C₄×#67288	2
#13092434	2
#13092433	2
#13092432	2
C₂×#10624987	2
#104×D₂₃	2
C₂×C₂₃×#104	2
#2901	23