Subgroups — #26220982

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245862	C₂	1	1	yes	maximal, characteristic
2	#20244328	C₂	1	1	yes	maximal, characteristic
3	#20244327	C₂	1	1	yes	maximal, characteristic
4	#20244112	C₂	1	1	yes	maximal, characteristic
5	#20243955	C₂	1	2	yes	maximal
6	#20243949	C₂	1	2	yes	maximal
7	#20243247	C₂	1	2	yes	maximal
8	#20243243	C₂	1	1	yes	maximal, characteristic
9	#20243138	C₂	1	2	yes	maximal
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1940	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	1	yes	characteristic
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×#13093934	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15689028	C₄	1	2	yes	—
22	#15689019	C₄	1	2	yes	—
23	#15689005	C₄	1	4	yes	—
24	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	#15688969	C₂²	1	2	yes	—
26	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
27	#15688878	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	#15688877	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
35	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#336	C₂²	1	2	yes	—
38	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₄×Dic₉₄	—	2	2	no	—
43	C₄×Dic₉₄	—	2	2	no	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
54	#13093935	—	2	2	no	—
55	#13093934	—	2	2	no	—
56	#13093934	—	2	4	no	—
57	#13093934	—	2	2	no	—
58	C₈×Dic₄₇	—	2	2	no	—
59	C₈×Dic₄₇	Q₈	1	4	yes	complemented
60	C₄₇⋊#95	C₂×C₄	1	4	yes	—
61	C₄₇⋊#95	C₂×C₄	1	4	yes	—
62	C₄₇⋊#95	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	4	no	—
70	C₄⋊C₈×C₄₇	—	2	2	no	—
71	#97×C₄₇	—	2	2	no	—
72	#95×C₄₇	C₂×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₉₄	—	2	2	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	1	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₄⋊C₄×C₄₇	—	4	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	#41	1	2	yes	—
96	C₄²×C₄₇	#41	1	2	yes	—
97	C₄⋊Dic₄₇	—	2	2	no	—
98	Dic₄₇⋊C₄	—	2	4	no	—
99	Dic₄₇⋊C₄	—	4	4	no	—
100	C₄×Dic₄₇	—	2	1	no	—
101	C₄×Dic₄₇	—	2	1	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	4	2	no	—
105	C₄×Dic₄₇	—	4	2	no	—
106	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
107	C₄×Dic₄₇	#41	1	2	yes	—
108	C₂×C₄₇⋊C₈	—	2	1	no	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	—	2	1	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
116	C₂×C₄₇⋊C₈	#41	1	2	yes	—
117	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	#117	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	2	no	—
137	C₂×Dic₄₇	—	2	2	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	C₄×Q₈	1	4	yes	—
141	Dic₉₄	—	4	4	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	—	4	2	no	—
145	C₄₇⋊C₈	—	2	2	no	—
146	C₄₇⋊C₈	—	2	2	no	—
147	C₄₇⋊C₈	—	4	2	no	—
148	#21342	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
152	C₂²×C₄₇	#568	1	2	yes	—
153	C₂²×C₄₇	#568	1	2	yes	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	#529	1	1	yes	characteristic
158	C₄×C₄₇	#529	1	1	yes	characteristic
159	C₄×C₄₇	#533	1	1	yes	characteristic
160	C₄×C₄₇	#533	1	1	yes	characteristic
161	Dic₄₇	—	2	4	no	—
162	Dic₄₇	—	4	2	no	—
163	Dic₄₇	—	4	2	no	—
164	Dic₄₇	—	4	2	no	—
165	#2295	—	47	1	no	—
166	#1985	—	47	1	no	—
167	#1985	—	47	1	no	—
168	#1940	—	47	1	no	—
169	#1940	D₄₇	1	1	yes	characteristic
170	#1917	—	47	2	no	—
171	#1917	—	47	2	no	—
172	#1768	—	47	2	no	—
173	#1767	—	47	1	no	—
174	#1746	—	47	2	no	—
175	C₄×#97	—	47	1	no	—
176	C₂×C₄₇	#2295	1	1	yes	characteristic
177	C₂×C₄₇	#2921	1	1	yes	characteristic
178	C₂×C₄₇	#2923	1	1	yes	characteristic
179	C₂×C₄₇	#2961	1	2	yes	—
180	C₂×C₄₇	#2967	1	2	yes	—
181	C₂×C₄×Q₈	—	47	1	no	—
182	C₂×C₄⋊C₈	—	47	2	no	—
183	C₂×C₄⋊C₈	—	47	1	no	—
184	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
185	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
186	C₂×#97	—	47	1	no	—
187	C₂×#97	—	47	1	no	—
188	C₂×#97	—	47	2	no	—
189	C₂×#97	—	47	1	no	—
190	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
191	C₂×C₄×C₈	—	47	1	no	—
192	C₂×C₄×C₈	—	47	1	no	—
193	#391	—	47	2	no	—
194	#389	—	47	2	no	—
195	#387	—	47	4	no	—
196	C₄×C₄⋊C₄	—	47	2	no	—
197	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
198	#376	—	47	2	no	—
199	C₄³	—	47	1	no	—
200	#336	—	47	2	no	—
201	#336	—	47	2	no	—
202	#336	—	47	2	no	—
203	#336	—	47	2	no	—
204	#336	—	47	2	no	—
205	#336	—	47	2	no	—
206	#336	C₂×D₄₇	1	2	yes	—
207	C₄₇	#21342	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	47	1	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
213	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	C₄×Q₈	—	94	2	no	—
216	C₄×Q₈	—	94	2	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	47	1	no	—
219	C₂×C₄⋊C₄	—	47	4	no	—
220	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
221	C₂×C₄²	—	47	1	no	—
222	C₂×C₄²	—	47	2	no	—
223	C₂×C₄²	—	47	1	no	—
224	C₂×C₄²	—	47	2	no	—
225	C₂×C₄²	—	47	2	no	—
226	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
227	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
228	C₄⋊C₈	—	2	4	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	94	4	no	—
231	C₄⋊C₈	—	94	2	no	—
232	#97	—	2	2	no	—
233	#97	—	94	4	no	—
234	#97	—	94	2	no	—
235	#97	—	94	2	no	—
236	#97	—	94	2	no	—
237	C₄×C₈	—	47	4	no	—
238	C₄×C₈	—	94	2	no	—
239	#95	—	47	4	no	—
240	#95	—	47	4	no	—
241	#95	—	47	4	no	—
242	#95	C₄×D₄₇	1	2	yes	—
243	C₂×Q₈	—	94	2	no	—
244	C₂×Q₈	—	188	2	no	—
245	C₂²×C₄	—	47	2	no	—
246	C₂²×C₄	—	47	2	no	—
247	C₂²×C₄	—	47	2	no	—
248	C₂²×C₄	—	47	2	no	—
249	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
250	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	47	2	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	94	1	no	—
263	C₂×C₈	—	94	1	no	—
264	C₂×C₈	—	94	1	no	—
265	C₂×C₈	—	94	1	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	#10625352	1	2	yes	—
270	C₂×C₈	Q₈×D₄₇	1	2	yes	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	—	188	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	47	2	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	94	1	no	—
280	C₄²	—	94	1	no	—
281	C₄²	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₄²	—	188	2	no	—
284	C₄²	—	188	2	no	—
285	C₄²	#10625352	1	2	yes	—
286	C₄²	#10625354	1	2	yes	—
287	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
288	Q₈	—	188	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	47	4	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	2	no	—
301	C₂×C₄	—	94	4	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	#13093982	1	2	yes	—
305	C₂×C₄	#13093982	1	2	yes	—
306	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
307	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
308	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
311	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	94	2	no	—
316	C₈	—	94	2	no	—
317	C₈	—	188	2	no	—
318	C₂²	C₂×#13093982	1	1	yes	characteristic
319	C₂²	C₄×Q₈×D₄₇	1	1	yes	characteristic
320	C₂²	C₄×#10625352	1	1	yes	characteristic
321	C₂²	#15689888	1	2	yes	—
322	C₂²	#15689889	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	94	4	no	—
327	C₄	—	188	2	no	—
328	C₄	—	188	2	no	—
329	C₄	—	188	2	no	—
330	C₄	#15689654	1	1	yes	characteristic
331	C₄	#15689658	1	1	yes	characteristic
332	C₄	#15689675	1	1	yes	characteristic
333	C₄	#15689678	1	1	yes	characteristic
334	C₂	#20245859	1	1	yes	characteristic
335	C₂	#20252025	1	1	yes	characteristic
336	C₂	#20252038	1	1	yes	characteristic
337	C₂	#20252324	1	2	yes	—
338	C₂	#20252370	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1940	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1940	D₄₇
hypocenter	C₄₇	#21342
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1940

Complements

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

N	H ≅ G/N	type
59. C₈×Dic₄₇	Q₈	semidirect
207. C₄₇	#21342	semidirect

Maximal subgroups

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

subgroup	index
#20245862	2
#20244328	2
#20244327	2
#20244112	2
#20243955	2
#20243949	2
#20243247	2
#20243243	2
#20243138	2
#97×Dic₄₇	2
C₄₇×#1940	2
#21342	47