Subgroups — #26218504

Order 12032.

898 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	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
2	#20246004	C₂	1	1	yes	maximal, characteristic
3	#20245744	C₂	1	1	yes	maximal, characteristic
4	#20243974	C₂	1	4	yes	maximal, complemented
5	#20243967	C₂	1	4	yes	maximal, complemented
6	#20243108	C₂	1	1	yes	maximal, characteristic
7	#20243108	C₂	1	1	yes	maximal, characteristic
8	#20243105	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₄	1	8	yes	complemented
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093956	C₂²	1	2	yes	—
17	#15689021	C₂²	1	4	yes	—
18	#15689019	—	2	2	no	—
19	#15688981	—	2	2	no	—
20	C₄⋊C₄×Dic₄₇	—	2	2	no	—
21	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
22	C₂²⋊C₄×Dic₄₇	—	2	2	no	—
23	#15688886	C₂²	1	2	yes	—
24	#15688886	C₂²	1	4	yes	—
25	#15688886	C₂²	1	4	yes	—
26	#15688886	C₂²	1	2	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#336	C₂²	1	2	yes	—
31	C₄₇×#336	C₂²	1	2	yes	—
32	C₂³×Dic₄₇	—	2	1	no	—
33	C₂³×Dic₄₇	—	2	1	no	—
34	#13093983	—	2	2	no	—
35	#13093983	—	2	2	no	—
36	#13093983	—	2	2	no	—
37	#13093983	Q₈	1	4	yes	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	—	2	2	no	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	—	2	4	no	—
49	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
50	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄×Dic₄₇	—	2	1	no	—
52	C₂×C₄×Dic₄₇	—	2	1	no	—
53	C₂×C₄×Dic₄₇	—	2	4	no	—
54	#13093956	C₂×C₄	1	8	yes	—
55	#13093956	C₂×C₄	1	8	yes	—
56	C₄₇⋊#95	—	2	4	no	—
57	C₄₇⋊#95	—	2	4	no	—
58	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₄₇	—	2	2	no	—
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	—	2	2	no	—
62	C₂²×C₈×C₄₇	—	2	2	no	—
63	C₂²×C₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#117×C₄₇	—	2	2	no	—
69	#117×C₄₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₇	#41	1	2	yes	—
79	C₂²×C₄×C₄₇	#41	1	2	yes	—
80	C₂²×C₄×C₄₇	#41	1	2	yes	—
81	C₂²×C₄×C₄₇	#41	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	2	no	—
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	—	2	4	no	—
87	C₂²×Dic₄₇	—	2	4	no	—
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	—	8	2	no	—
90	C₂²×Dic₄₇	—	8	2	no	—
91	M₁₆×C₄₇	—	4	4	no	—
92	C₂×C₈×C₄₇	—	2	8	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	2	1	no	—
96	C₂×C₈×C₄₇	—	2	1	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄⋊C₄×C₄₇	—	2	4	no	—
103	C₂²⋊C₄×C₄₇	—	2	4	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	2	2	no	—
106	C₂²⋊Dic₄₇	—	2	4	no	—
107	C₂²⋊Dic₄₇	—	4	2	no	—
108	C₄⋊Dic₄₇	—	2	4	no	—
109	C₄⋊Dic₄₇	—	2	4	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
116	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
117	C₄⋊Dic₄₇	C₂×Q₈	1	4	yes	—
118	Dic₄₇⋊C₄	—	8	4	no	—
119	C₄×Dic₄₇	—	2	4	no	—
120	C₄×Dic₄₇	—	4	2	no	—
121	C₄×Dic₄₇	—	8	4	no	—
122	C₂³×C₄₇	—	2	2	no	—
123	C₂³×C₄₇	—	2	2	no	—
124	C₂³×C₄₇	C₄×D₄	1	2	yes	—
125	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂³×C₄₇	#123	1	2	yes	—
127	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	#117	1	4	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
134	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₄₇	#123	1	2	yes	—
136	C₂×C₄×C₄₇	#125	1	4	yes	—
137	C₂×C₄×C₄₇	#126	1	4	yes	—
138	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₄₇	—	2	4	no	—
145	C₂×Dic₄₇	—	2	4	no	—
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	4	no	—
149	C₂×Dic₄₇	—	4	4	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	—	4	2	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	C₂×Dic₄₇	—	8	4	no	—
155	C₂×Dic₄₇	—	8	4	no	—
156	C₈×C₄₇	—	2	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	—	4	2	no	—
159	#21038	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	4	no	—
161	C₂²×C₄₇	—	2	4	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	#387	1	4	yes	—
165	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
168	C₂²×C₄₇	#577	1	2	yes	—
169	C₂²×C₄₇	#577	1	2	yes	—
170	C₄×C₄₇	—	4	4	no	—
171	C₄×C₄₇	#387	1	4	yes	—
172	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
173	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
175	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
176	Dic₄₇	—	4	4	no	—
177	Dic₄₇	—	4	4	no	—
178	Dic₄₇	—	4	4	no	—
179	Dic₄₇	—	8	2	no	—
180	C₂²×#106	—	47	1	no	—
181	#2316	—	47	1	no	—
182	#2274	—	47	1	no	—
183	#1920	—	47	4	no	—
184	#1919	—	47	4	no	—
185	#1736	—	47	1	no	—
186	#1736	—	47	1	no	—
187	#1735	—	47	1	no	—
188	#1729	D₄₇	1	1	yes	characteristic
189	C₂×C₄₇	—	2	4	no	—
190	C₂×C₄₇	C₂×#387	1	1	yes	characteristic
191	C₂×C₄₇	#2949	1	1	yes	characteristic
192	C₂×C₄₇	#2949	1	1	yes	characteristic
193	C₂×C₄₇	#3084	1	2	yes	—
194	C₂×C₄₇	#3099	1	2	yes	—
195	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
196	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
197	C₂×#117	—	47	1	no	—
198	C₂×#117	—	47	1	no	—
199	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
200	C₂²×C₄⋊C₄	—	47	1	no	—
201	C₂²×C₄⋊C₄	—	47	1	no	—
202	C₂×#106	—	47	8	no	—
203	C₂×#106	—	47	2	no	—
204	C₂×#106	—	47	2	no	—
205	#389	—	47	4	no	—
206	#389	—	94	2	no	—
207	#380	—	94	2	no	—
208	C₄×C₄⋊C₄	—	47	4	no	—
209	C₄×C₄⋊C₄	—	94	2	no	—
210	C₄×C₂²⋊C₄	—	94	2	no	—
211	#340	—	47	2	no	—
212	#340	—	47	4	no	—
213	#340	—	47	4	no	—
214	#340	—	47	2	no	—
215	#336	C₂×D₄₇	1	2	yes	—
216	#336	C₂×D₄₇	1	2	yes	—
217	C₄₇	#21038	1	1	yes	characteristic, complemented, Sylow
218	C₂³×C₄	—	94	1	no	—
219	C₂³×C₄	—	94	1	no	—
220	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	C₄×D₄₇	1	4	yes	—
227	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	#117	—	2	2	no	—
231	#117	—	2	2	no	—
232	#117	—	47	4	no	—
233	#117	—	94	2	no	—
234	#117	—	94	2	no	—
235	#117	—	94	2	no	—
236	C₂×C₄⋊C₄	—	47	4	no	—
237	C₂×C₄⋊C₄	—	47	4	no	—
238	C₂×C₄⋊C₄	—	47	2	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	—	47	2	no	—
242	C₂×C₄⋊C₄	—	94	1	no	—
243	C₂×C₄⋊C₄	—	94	1	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	—	94	2	no	—
246	C₂×C₄⋊C₄	—	94	4	no	—
247	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
248	C₂×C₂²⋊C₄	—	94	1	no	—
249	C₂×C₂²⋊C₄	—	94	1	no	—
250	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
251	C₂×C₄²	—	94	1	no	—
252	C₂×C₄²	—	94	4	no	—
253	C₂×C₄²	—	94	1	no	—
254	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
255	#106	—	47	8	no	—
256	#106	—	47	8	no	—
257	#95	—	94	4	no	—
258	#95	—	94	4	no	—
259	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
260	C₂²×C₄	—	94	4	no	—
261	C₂²×C₄	—	94	4	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	—	94	2	no	—
265	C₂²×C₄	—	94	2	no	—
266	C₂²×C₄	—	376	2	no	—
267	C₂²×C₄	—	376	2	no	—
268	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
269	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
270	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂²×C₄	#10625350	1	2	yes	—
272	C₂²×C₄	#10625352	1	2	yes	—
273	C₂²×C₄	#10625352	1	2	yes	—
274	C₂²×C₄	#10625352	1	2	yes	—
275	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
276	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
277	M₁₆	—	4	4	no	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	47	4	no	—
290	C₄⋊C₄	—	47	4	no	—
291	C₄⋊C₄	—	47	4	no	—
292	C₄⋊C₄	—	94	4	no	—
293	C₄⋊C₄	—	94	4	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	376	4	no	—
300	C₂²⋊C₄	—	2	4	no	—
301	C₂²⋊C₄	—	94	4	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	94	4	no	—
306	C₄²	—	376	4	no	—
307	C₄²	—	188	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	—	2	2	no	—
310	C₂³	C₄×D₁₈₈	1	2	yes	—
311	C₂³	#13094004	1	2	yes	—
312	C₂³	#13094008	1	2	yes	—
313	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	94	4	no	—
318	C₂×C₄	—	94	4	no	—
319	C₂×C₄	—	94	4	no	—
320	C₂×C₄	—	376	4	no	—
321	C₂×C₄	—	376	4	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	4	no	—
327	C₂×C₄	—	188	4	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	#13093980	1	4	yes	—
330	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
331	C₂×C₄	#13093989	1	4	yes	—
332	C₂×C₄	#13094004	1	2	yes	—
333	C₂×C₄	#13094008	1	2	yes	—
334	C₂×C₄	#13094014	1	4	yes	—
335	C₂×C₄	#13094017	1	4	yes	—
336	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
337	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
339	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
340	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
341	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₂²	—	2	4	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	4	no	—
349	C₂²	#15689008	1	4	yes	—
350	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
351	C₂²	C₂×#13094004	1	1	yes	characteristic
352	C₂²	C₂×#13094008	1	1	yes	characteristic
353	C₂²	#15689935	1	2	yes	—
354	C₂²	#15689935	1	2	yes	—
355	C₄	—	4	4	no	—
356	C₄	—	376	2	no	—
357	C₄	—	188	4	no	—
358	C₄	—	188	4	no	—
359	C₄	—	188	4	no	—
360	C₄	#15689008	1	4	yes	—
361	C₄	C₂×#13093980	1	1	yes	characteristic
362	C₄	C₂×#13093989	1	1	yes	characteristic
363	C₄	C₂×#13094014	1	1	yes	characteristic
364	C₄	C₂×#13094017	1	1	yes	characteristic
365	C₂	—	2	4	no	—
366	C₂	C₂×#15689008	1	1	yes	characteristic
367	C₂	#20252249	1	1	yes	characteristic
368	C₂	#20252249	1	1	yes	characteristic
369	C₂	#20253252	1	2	yes	—
370	C₂	#20253379	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1729	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#21038
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ #1729

Complements

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

N	H ≅ G/N	type
4. #20243974	C₂	semidirect
5. #20243967	C₂	semidirect
14. C₂×#13093956	C₄	semidirect
217. C₄₇	#21038	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093956	2
#20246004	2
#20245744	2
#20243974	2
#20243967	2
#20243108	2
#20243105	2
C₄₇×#1729	2
#21038	47