Subgroups — #26104668

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	C₂²	1	2	yes	—
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#384	C₄	1	1	yes	characteristic
18	C₄₇×#380	—	2	1	no	—
19	C₄₇×#380	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	4	2	no	—
24	C₂²×C₄₇⋊C₈	—	4	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₇	—	4	2	no	—
32	C₂²⋊Q₈×C₄₇	—	4	4	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	2	no	—
47	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂×Q₈×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₄⋊C₄×C₄₇	—	4	4	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₄⋊C₄×C₄₇	—	4	2	no	—
76	C₂²⋊C₄×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	8	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	#100	1	2	yes	—
93	C₂³×C₄₇	#103	1	2	yes	—
94	C₂³×C₄₇	#103	1	2	yes	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₄₇	—	8	4	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	#99	1	2	yes	—
120	C₂×C₄×C₄₇	#103	1	2	yes	—
121	C₂×C₄×C₄₇	#103	1	2	yes	—
122	C₂×C₄×C₄₇	#104	1	2	yes	—
123	C₂×C₄×C₄₇	#131	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
128	C₄₇⋊C₈	—	4	4	no	—
129	C₄₇⋊C₈	—	8	2	no	—
130	#7667	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	#333	1	4	yes	—
138	C₂²×C₄₇	#408	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
144	C₂²×C₄₇	#421	1	1	yes	characteristic
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	1	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	4	1	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×C₄₇	—	8	2	no	—
154	C₄×C₄₇	#328	1	4	yes	—
155	#2857	D₄₇	1	1	yes	characteristic
156	#1797	—	47	1	no	—
157	C₂×#336	—	47	1	no	—
158	#1272	—	47	4	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	#1471	1	1	yes	characteristic
161	C₂×C₄₇	#1481	1	1	yes	characteristic
162	C₂×C₄₇	#1493	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
164	C₂×C₄₇	#1520	1	1	yes	characteristic
165	C₂×C₄₇	#1530	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
167	C₂³×C₈	—	94	1	no	—
168	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
169	C₂×C₂²⋊Q₈	Dic₄₇	1	2	yes	—
170	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	94	1	no	—
172	C₂×C₂²⋊C₈	—	94	1	no	—
173	#391	C₂×D₄₇	1	2	yes	—
174	#389	—	2	1	no	—
175	#389	—	2	1	no	—
176	#384	Dic₄₇	1	1	yes	characteristic
177	#380	—	2	1	no	—
178	#380	—	2	1	no	—
179	#336	—	47	2	no	—
180	#336	—	47	2	no	—
181	#336	—	94	1	no	—
182	C₄₇	#7667	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
184	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂²×C₈	—	94	4	no	—
188	C₂²×C₈	—	94	2	no	—
189	C₂²×C₈	—	94	2	no	—
190	C₂²×C₈	—	188	2	no	—
191	C₂²×C₈	—	188	2	no	—
192	C₄⋊Q₈	—	4	2	no	—
193	C₂²⋊Q₈	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	4	2	no	—
202	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	C₄⋊C₈	—	188	2	no	—
207	C₂²⋊C₈	—	188	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	#10625336	1	2	yes	—
224	C₂²×C₄	#10625336	1	2	yes	—
225	C₂²×C₄	#10625337	1	2	yes	—
226	C₂²×C₄	#10625337	1	2	yes	—
227	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	188	2	no	—
235	C₂×C₈	—	188	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	—	188	2	no	—
238	C₂×C₈	—	188	8	no	—
239	C₂×C₈	—	188	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#13093941	1	2	yes	—
256	C₂³	#13093948	1	2	yes	—
257	C₂³	#13093948	1	2	yes	—
258	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
259	Q₈	—	8	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	#13093939	1	2	yes	—
283	C₂×C₄	#13093948	1	2	yes	—
284	C₂×C₄	#13093948	1	2	yes	—
285	C₂×C₄	#13093951	1	2	yes	—
286	C₂×C₄	#13094031	1	2	yes	—
287	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
288	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
291	C₈	—	376	2	no	—
292	C₈	—	188	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#15688872	1	4	yes	—
300	C₂²	#15689090	1	1	yes	characteristic
301	C₂²	C₂×#13093939	1	1	yes	characteristic
302	C₂²	C₂×#13093941	1	1	yes	characteristic
303	C₂²	C₂×#13093948	1	1	yes	characteristic
304	C₂²	C₂×#13093948	1	1	yes	characteristic
305	C₂²	C₂×#13093951	1	1	yes	characteristic
306	C₂²	#15689140	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	#15688860	1	4	yes	—
317	C₂	—	2	4	no	—
318	C₂	#20241748	1	1	yes	characteristic
319	C₂	#20241791	1	1	yes	characteristic
320	C₂	#20241843	1	1	yes	characteristic
321	C₂	C₂×#15688860	1	1	yes	characteristic
322	C₂	#20241960	1	1	yes	characteristic
323	C₂	#20241999	1	1	yes	characteristic
324	C₂	C₂×#15688872	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2857	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2857	D₄₇
hypocenter	C₄₇	#7667
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. #20241214	C₂	semidirect
182. C₄₇	#7667	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₂×#15688878	2
#20241214	2
C₄₇×#2857	2
#7667	47