Subgroups — #26107347

Order 12032.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252101	C₂	1	1	yes	maximal, characteristic
2	#20242038	C₂	1	1	yes	maximal, characteristic
3	#20241882	C₂	1	1	yes	maximal, characteristic
4	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241291	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241285	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	—	2	1	no	—
9	#15689673	C₂²	1	1	yes	characteristic
10	#15689202	—	2	1	no	—
11	#15689202	C₄	1	2	yes	complemented
12	C₈×D₁₈₈	—	2	1	no	—
13	C₈×D₁₈₈	C₄	1	2	yes	complemented
14	#15689176	C₂²	1	1	yes	characteristic
15	#15689176	C₂²	1	1	yes	characteristic
16	#15689147	C₂²	1	1	yes	characteristic
17	#15689089	—	2	1	no	—
18	#15688870	C₄	1	2	yes	complemented
19	#15688866	C₄	1	2	yes	complemented
20	#15688856	C₄	1	4	yes	—
21	#15688844	—	2	1	no	—
22	C₈×C₄₇⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	—	2	1	no	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#336	—	2	1	no	—
28	C₂×#10625350	—	2	1	no	—
29	#13094030	—	4	2	no	—
30	#13094030	—	4	2	no	—
31	C₂×#10625329	—	2	1	no	—
32	C₂×#10625324	—	2	2	no	—
33	C₂×C₈×D₄₇	—	2	2	no	—
34	#13094024	C₂×C₄	1	1	yes	characteristic
35	#13094021	C₂×C₄	1	1	yes	characteristic
36	#13094012	—	2	1	no	—
37	#13093998	—	2	1	no	—
38	#13093998	C₂×C₄	1	2	yes	—
39	C₄×D₁₈₈	—	2	1	no	—
40	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
44	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
45	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
46	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
47	(C₂×D₄₇)⋊C₈	—	2	2	no	—
48	(C₂×D₄₇)⋊C₈	—	2	2	no	—
49	C₄×C₄₇⋊C₈	—	2	1	no	—
50	C₄×C₄₇⋊C₈	—	2	1	no	—
51	C₂²×C₈×C₄₇	—	2	1	no	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄²×C₄₇	—	2	1	no	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	1	no	—
58	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	—	4	2	no	—
60	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
64	#95×C₄₇	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	2	2	no	—
70	#10625350	—	4	1	no	—
71	#10625350	—	4	1	no	—
72	#10625350	—	4	1	no	—
73	#10625350	—	4	1	no	—
74	C₂×D₁₈₈	—	2	1	no	—
75	C₂×D₁₈₈	—	2	1	no	—
76	C₂×D₁₈₈	C₄²	1	2	yes	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×Dic₉₄	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	1	no	—
83	C₂×C₈×C₄₇	—	4	1	no	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	D₈	1	2	yes	—
89	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	4	1	no	—
92	C₄²×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₇	#41	1	1	yes	characteristic
97	(C₂×D₄₇)⋊C₄	—	2	2	no	—
98	(C₂×D₄₇)⋊C₄	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	1	no	—
100	C₄⋊Dic₄₇	C₄²	1	2	yes	—
101	#10625329	—	4	1	no	—
102	#10625329	—	4	1	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	#10625324	—	4	4	no	—
107	C₈×D₄₇	—	4	4	no	—
108	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂²×D₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
125	C₂²⋊D₄₇	—	4	2	no	—
126	C₂²⋊D₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	2	2	no	—
128	D₁₈₈	—	4	2	no	—
129	D₁₈₈	—	4	1	no	—
130	D₁₈₈	—	4	1	no	—
131	D₁₈₈	—	4	2	no	—
132	C₄×D₄₇	—	4	2	no	—
133	C₄×D₄₇	—	4	2	no	—
134	Dic₉₄	—	4	1	no	—
135	Dic₉₄	—	4	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	—	2	1	no	—
138	C₈×C₄₇	—	4	2	no	—
139	C₈×C₄₇	#102	1	2	yes	—
140	C₈×C₄₇	#102	1	2	yes	—
141	C₄₇⋊C₈	—	4	2	no	—
142	C₄₇⋊C₈	—	4	2	no	—
143	C₄₇⋊C₈	—	4	2	no	—
144	#8251	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₄₇	—	4	2	no	—
151	C₂×D₄₇	—	4	2	no	—
152	C₂×D₄₇	—	4	2	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
158	C₄×C₄₇	#416	1	1	yes	characteristic
159	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
160	C₄×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
161	C₄×C₄₇	#443	1	2	yes	—
162	Dic₄₇	—	4	2	no	—
163	#2931	—	47	1	no	—
164	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
165	#1540	—	47	1	no	—
166	#1502	—	47	1	no	—
167	#1450	—	47	1	no	—
168	#1303	—	47	1	no	—
169	#1301	—	47	1	no	—
170	C₂×C₄₇	—	2	1	no	—
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	#1757	1	1	yes	characteristic
173	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
174	C₂×C₄₇	#1764	1	1	yes	characteristic
175	D₄₇	—	8	2	no	—
176	C₂×#131	—	94	1	no	—
177	#532	—	47	1	no	—
178	#436	—	47	2	no	—
179	#436	—	94	1	no	—
180	C₈×D₄	—	47	2	no	—
181	C₈×D₄	—	94	1	no	—
182	#431	—	47	1	no	—
183	#431	—	47	1	no	—
184	#423	—	47	1	no	—
185	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
186	#408	—	94	1	no	—
187	C₂×C₄×C₈	—	2	1	no	—
188	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
190	#336	—	2	1	no	—
191	#332	—	47	2	no	—
192	#331	—	47	2	no	—
193	#327	—	47	4	no	—
194	#322	—	94	1	no	—
195	C₈²	—	94	1	no	—
196	C₄₇	#8251	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	94	1	no	—
198	#131	—	188	2	no	—
199	#131	—	188	2	no	—
200	C₂×M₁₆	—	94	1	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	94	2	no	—
204	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₄⋊Q₈	—	47	1	no	—
207	#127	—	47	1	no	—
208	#124	—	94	1	no	—
209	#121	—	47	2	no	—
210	#121	—	94	1	no	—
211	C₄×D₄	—	47	2	no	—
212	C₄×D₄	—	94	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	47	2	no	—
218	C₄⋊C₈	—	47	2	no	—
219	C₄⋊C₈	—	94	1	no	—
220	C₄⋊C₈	—	94	1	no	—
221	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
222	C₂²⋊C₈	—	47	2	no	—
223	C₂²⋊C₈	—	94	1	no	—
224	C₂²⋊C₈	—	94	2	no	—
225	C₂²⋊C₈	—	94	2	no	—
226	C₄×C₈	—	4	2	no	—
227	C₄×C₈	—	94	1	no	—
228	C₄×C₈	—	94	1	no	—
229	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
230	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
231	C₄×C₈	D₁₈₈	1	1	yes	characteristic
232	C₄×C₈	D₁₈₈	1	1	yes	characteristic
233	#95	C₄×D₄₇	1	2	yes	—
234	#41	—	188	1	no	—
235	#41	—	188	1	no	—
236	#41	—	188	1	no	—
237	#41	—	188	1	no	—
238	C₂×Q₈	—	94	1	no	—
239	C₂×D₄	—	47	2	no	—
240	C₂×D₄	—	94	2	no	—
241	C₂×D₄	—	94	1	no	—
242	C₂×D₄	—	94	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	94	2	no	—
245	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₂²×C₄	#10625350	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	M₁₆	—	188	1	no	—
249	M₁₆	—	188	1	no	—
250	M₁₆	—	188	4	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	—	94	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	—	188	4	no	—
260	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
261	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	#10625334	1	2	yes	—
265	C₂×C₈	#10625336	1	2	yes	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	47	2	no	—
268	C₄⋊C₄	—	94	1	no	—
269	C₂²⋊C₄	—	94	2	no	—
270	C₂²⋊C₄	—	94	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
276	C₄²	#10625350	1	1	yes	characteristic
277	C₂³	—	94	2	no	—
278	C₂³	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
279	Q₈	—	188	1	no	—
280	Q₈	—	188	1	no	—
281	D₄	—	188	2	no	—
282	D₄	—	188	2	no	—
283	D₄	—	188	2	no	—
284	D₄	—	188	2	no	—
285	D₄	—	188	1	no	—
286	D₄	—	188	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	94	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	#13093981	1	1	yes	characteristic
298	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
300	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
301	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
303	C₂×C₄	#13094044	1	1	yes	characteristic
304	C₂×C₄	#13094044	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	188	2	no	—
309	C₈	—	188	2	no	—
310	C₈	—	188	2	no	—
311	C₈	#13093944	1	2	yes	—
312	C₈	#13093944	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	188	2	no	—
316	C₂²	—	188	2	no	—
317	C₂²	—	188	2	no	—
318	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
319	C₂²	C₄×#10625334	1	1	yes	characteristic
320	C₂²	C₄×#10625336	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	188	2	no	—
326	C₄	C₂×#13093944	1	1	yes	characteristic
327	C₄	#15689117	1	1	yes	characteristic
328	C₄	C₄×#10625334	1	1	yes	characteristic
329	C₄	C₄×#10625336	1	1	yes	characteristic
330	C₄	#15689241	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	376	2	no	—
334	C₂	#20243189	1	1	yes	characteristic
335	C₂	C₄×#13093944	1	1	yes	characteristic
336	C₂	#20243225	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	#15689673	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#8251
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #20241671	C₂	semidirect
5. #20241291	C₂	semidirect
6. #20241285	C₂	semidirect
7. C₈×C₄⋊C₄×C₄₇	C₂	semidirect
11. #15689202	C₄	semidirect
13. C₈×D₁₈₈	C₄	semidirect
18. #15688870	C₄	semidirect
19. #15688866	C₄	semidirect
164. C₈×C₄⋊C₄	D₄₇	semidirect
196. C₄₇	#8251	semidirect

Maximal subgroups

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

subgroup	index
#20252101	2
#20242038	2
#20241882	2
#20241671	2
#20241291	2
#20241285	2
C₈×C₄⋊C₄×C₄₇	2
#8251	47