Subgroups — #26186029

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252354	C₂	1	2	yes	maximal, complemented
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20242286	C₂	1	2	yes	maximal, complemented
4	#20242280	C₂	1	4	yes	maximal, complemented
5	#20242274	C₂	1	2	yes	maximal, complemented
6	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241771	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₄	1	1	yes	characteristic
10	C₄×#10625350	C₂²	1	2	yes	—
11	#15689253	C₂²	1	4	yes	—
12	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
14	C₄×#10625325	C₄	1	2	yes	—
15	C₄×#10625325	C₂²	1	2	yes	—
16	C₄×D₃₇₆	C₄	1	1	yes	characteristic
17	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
18	#15689202	C₂²	1	4	yes	—
19	#15689194	—	2	4	no	—
20	#15689176	C₂²	1	2	yes	—
21	#15689174	—	2	2	no	—
22	#15689161	C₄	1	1	yes	characteristic
23	#15689120	C₄	1	2	yes	—
24	C₄×#10625324	—	2	2	no	—
25	#15688876	C₂²	1	2	yes	—
26	#15688874	C₂²	1	2	yes	—
27	#15688855	C₂²	1	4	yes	—
28	#15688852	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₈²×C₄₇	C₂²	1	2	yes	—
33	C₈²×C₄₇	C₂²	1	2	yes	—
34	C₂×#10625350	C₂×C₄	1	2	yes	—
35	#13094046	—	2	4	no	—
36	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625325	C₂×C₄	1	2	yes	—
38	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#10625324	—	2	4	no	—
40	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
41	C₄×Dic₉₄	C₂³	1	2	yes	—
42	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
43	C₄×D₁₈₈	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₄₇	—	2	2	no	—
45	C₄×C₂²⋊D₄₇	—	2	2	no	—
46	#13093983	C₂×C₄	1	2	yes	—
47	#13093981	—	2	2	no	—
48	C₄²×D₄₇	—	2	2	no	—
49	#13093958	C₂×C₄	1	1	yes	characteristic
50	#13093958	C₂×C₄	1	1	yes	characteristic
51	#13093956	C₂×C₄	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
53	Dic₄₇⋊C₈	—	2	4	no	—
54	#13093949	C₂×C₄	1	2	yes	—
55	#13093949	C₂×C₄	1	2	yes	—
56	#13093946	C₂×C₄	1	2	yes	—
57	#13093946	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
59	(C₂×D₄₇)⋊C₈	—	2	4	no	—
60	#13093934	—	2	4	no	—
61	C₄×C₄₇⋊C₈	—	2	2	no	—
62	C₂²×C₈×C₄₇	—	2	1	no	—
63	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	D₄	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	—	2	1	no	—
68	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
72	C₄×C₈×C₄₇	C₂³	1	2	yes	—
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₇	C₂³	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	—	2	2	no	—
80	#10625350	—	2	4	no	—
81	#10625350	—	2	4	no	—
82	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	—	2	2	no	—
85	C₂×C₄×D₄₇	—	2	2	no	—
86	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	4	no	—
91	C₂×C₈×C₄₇	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₇	#41	1	1	yes	characteristic
101	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₂²⋊Dic₄₇	—	2	2	no	—
103	(C₂×D₄₇)⋊C₄	—	2	2	no	—
104	(C₂×D₄₇)⋊C₄	—	2	2	no	—
105	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
106	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
107	Dic₄₇⋊C₄	—	2	2	no	—
108	Dic₄₇⋊C₄	—	2	2	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₂×C₄₇⋊C₈	—	2	4	no	—
113	Dic₁₈₈	—	2	2	no	—
114	D₃₇₆	—	2	2	no	—
115	#10625325	—	2	4	no	—
116	#10625324	—	4	8	no	—
117	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₄₇	#131	1	2	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	D₁₈₈	—	2	2	no	—
134	D₁₈₈	—	2	2	no	—
135	C₄×D₄₇	—	4	4	no	—
136	C₄×D₄₇	—	4	4	no	—
137	C₄×D₄₇	—	4	4	no	—
138	Dic₉₄	—	2	2	no	—
139	Dic₉₄	—	2	2	no	—
140	C₈×C₄₇	—	2	4	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	2	1	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
145	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
146	C₈×C₄₇	#135	1	4	yes	complemented
147	C₄₇⋊C₈	—	4	4	no	—
148	#17886	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	#436	1	2	yes	—
152	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₄₇	—	4	2	no	—
154	C₂×D₄₇	—	4	2	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	#436	1	2	yes	—
159	C₄×C₄₇	#443	1	2	yes	—
160	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
162	C₄×C₄₇	C₂×#135	1	2	yes	—
163	Dic₄₇	—	4	2	no	—
164	Dic₄₇	—	4	2	no	—
165	Dic₄₇	—	4	2	no	—
166	#2965	—	47	2	no	—
167	C₄×#135	—	47	1	no	—
168	#1591	—	47	2	no	—
169	#1590	—	47	4	no	—
170	#1589	—	47	2	no	—
171	#1564	—	47	1	no	—
172	#1476	—	47	2	no	—
173	C₂×C₈²	D₄₇	1	1	yes	characteristic, complemented
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	C₂×#436	1	1	yes	characteristic
176	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
178	D₄₇	—	8	2	no	—
179	C₂×#135	—	47	1	no	—
180	C₄×#41	—	47	2	no	—
181	#446	—	47	4	no	—
182	C₄×Q₁₆	—	47	1	no	—
183	C₄×Q₁₆	—	47	1	no	—
184	C₄×SD₁₆	—	47	2	no	—
185	C₄×SD₁₆	—	47	2	no	—
186	C₄×D₈	—	47	1	no	—
187	C₄×D₈	—	47	1	no	—
188	#436	—	47	4	no	—
189	#435	—	94	4	no	—
190	#431	—	47	2	no	—
191	#431	—	94	2	no	—
192	#427	—	47	1	no	—
193	#416	—	47	2	no	—
194	C₄×M₁₆	—	94	2	no	—
195	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
198	#335	—	47	2	no	—
199	#334	—	47	2	no	—
200	Q₈⋊C₈	—	47	4	no	—
201	D₄⋊C₈	—	47	4	no	—
202	C₈²	C₂×D₄₇	1	2	yes	—
203	C₈²	C₂×D₄₇	1	2	yes	—
204	C₄₇	#17886	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	47	2	no	—
206	#135	—	94	4	no	—
207	C₂×Q₁₆	—	47	1	no	—
208	C₂×SD₁₆	—	47	2	no	—
209	C₂×D₈	—	47	1	no	—
210	C₂×M₁₆	—	94	4	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	D₁₈₈	1	2	yes	—
215	C₄×Q₈	—	47	2	no	—
216	C₄×Q₈	—	47	2	no	—
217	C₄×D₄	—	47	2	no	—
218	C₄×D₄	—	47	2	no	—
219	C₄×D₄	—	94	2	no	—
220	C₄×D₄	—	94	2	no	—
221	#117	—	47	2	no	—
222	#117	—	94	2	no	—
223	C₂×C₄²	—	94	2	no	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	#107	—	47	1	no	—
226	#107	—	47	1	no	—
227	#106	—	47	2	no	—
228	C₄⋊C₈	—	47	4	no	—
229	C₄⋊C₈	—	94	4	no	—
230	#103	—	47	2	no	—
231	#103	—	47	2	no	—
232	#102	—	47	2	no	—
233	#102	—	47	2	no	—
234	C₂²⋊C₈	—	94	2	no	—
235	C₂²⋊C₈	—	94	4	no	—
236	#97	—	94	4	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	94	2	no	—
239	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
240	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
241	C₄×C₈	D₁₈₈	1	1	yes	characteristic
242	C₄×C₈	D₁₈₈	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₇	1	2	yes	—
244	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₄₇	1	2	yes	—
247	#41	—	94	4	no	—
248	#41	—	94	4	no	—
249	C₂×Q₈	—	47	2	no	—
250	C₂×D₄	—	47	2	no	—
251	C₂×D₄	—	94	2	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	94	2	no	—
254	C₂²×C₄	—	94	2	no	—
255	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
257	C₂²×C₄	#10625350	1	1	yes	characteristic
258	Q₁₆	—	94	2	no	—
259	SD₁₆	—	94	4	no	—
260	D₈	—	94	2	no	—
261	M₁₆	—	188	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	94	4	no	—
267	C₂×C₈	#10625324	1	4	yes	complemented
268	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
273	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
274	C₄⋊C₄	—	47	2	no	—
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	94	2	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₂²⋊C₄	—	94	2	no	—
279	C₂²⋊C₄	—	94	2	no	—
280	C₂²⋊C₄	—	94	2	no	—
281	C₄²	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₄²	—	94	2	no	—
284	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
286	C₄²	#10625350	1	1	yes	characteristic
287	C₄²	C₂³×D₄₇	1	1	yes	characteristic
288	C₂³	—	94	2	no	—
289	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
290	Q₈	—	94	2	no	—
291	Q₈	—	94	2	no	—
292	D₄	—	94	2	no	—
293	D₄	—	94	2	no	—
294	D₄	—	188	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	188	4	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	—	188	4	no	—
303	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
304	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
305	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
306	C₂×C₄	C₂×#10625324	1	2	yes	—
307	C₂×C₄	#13094030	1	2	yes	—
308	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	188	4	no	—
316	C₈	C₂×#10625324	1	2	yes	—
317	C₈	C₂×#10625324	1	2	yes	—
318	C₈	#13094046	1	4	yes	complemented
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	188	2	no	—
322	C₂²	—	188	2	no	—
323	C₂²	—	188	2	no	—
324	C₂²	#15689202	1	2	yes	—
325	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	188	2	no	—
329	C₄	—	188	2	no	—
330	C₄	—	188	2	no	—
331	C₄	#15689202	1	2	yes	—
332	C₄	#15689241	1	2	yes	—
333	C₄	C₂²×#10625324	1	1	yes	characteristic
334	C₄	C₂×#13094030	1	1	yes	characteristic
335	C₄	C₂×#13094046	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	376	2	no	—
338	C₂	C₂×#15689202	1	1	yes	characteristic
339	C₂	C₄×#13094046	1	1	yes	characteristic
340	C₂	C₂×#15689241	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#17886
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #20252354	C₂	semidirect
3. #20242286	C₂	semidirect
4. #20242280	C₂	semidirect
5. #20242274	C₂	semidirect
6. #20242136	C₂	semidirect
7. #20241771	C₂	semidirect
8. C₂×C₈²×C₄₇	C₂	semidirect
91. C₂×C₈×C₄₇	M₁₆	semidirect
146. C₈×C₄₇	#135	semidirect
173. C₂×C₈²	D₄₇	semidirect
204. C₄₇	#17886	semidirect
267. C₂×C₈	#10625324	semidirect
318. C₈	#13094046	semidirect

Maximal subgroups

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

subgroup	index
#20252354	2
C₄×#13094046	2
#20242286	2
#20242280	2
#20242274	2
#20242136	2
#20241771	2
C₂×C₈²×C₄₇	2
#17886	47