Subgroups — #26218529

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251978	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689151	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
4	#20243986	C₂	1	8	yes	maximal, complemented
5	#20243099	C₂	1	1	yes	maximal, characteristic
6	#20243099	C₂	1	1	yes	maximal, characteristic
7	#20243086	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1738	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#13093983	C₂²	1	1	yes	characteristic
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	#15689165	C₄	1	4	yes	complemented
14	#15689165	C₄	1	4	yes	complemented
15	#15689161	C₂²	1	4	yes	—
16	#15689151	—	2	1	no	—
17	#15689151	—	2	1	no	—
18	#15689151	C₂²	1	4	yes	—
19	C₂×Dic₄₇⋊C₈	—	2	2	no	—
20	#15689081	—	2	2	no	—
21	#15689081	C₂²	1	4	yes	—
22	C₂×#13093934	—	2	1	no	—
23	C₂×C₈×Dic₄₇	—	2	1	no	—
24	#15688884	C₂²	1	4	yes	—
25	#15688884	C₂²	1	4	yes	—
26	#15688877	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#341	C₂²	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13093983	—	2	1	no	—
38	#13093983	—	2	1	no	—
39	#13093983	—	2	1	no	—
40	#13093983	—	2	1	no	—
41	#13093983	—	2	2	no	—
42	#13093983	—	2	2	no	—
43	#13093983	C₂×C₄	1	2	yes	—
44	#13093983	C₂×C₄	1	2	yes	—
45	#13093983	C₂³	1	2	yes	—
46	#13093983	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	—	2	1	no	—
48	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
51	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
52	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
53	C₂×C₄×Dic₄₇	—	2	1	no	—
54	C₂×C₄×Dic₄₇	—	2	1	no	—
55	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
56	#13093958	C₂×C₄	1	8	yes	—
57	#13093956	C₂×C₄	1	8	yes	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	Dic₄₇⋊C₈	—	2	4	no	—
60	#13093934	—	2	4	no	—
61	C₈×Dic₄₇	—	2	4	no	—
62	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₄₇	—	2	1	no	—
64	C₂×M₁₆×C₄₇	—	2	2	no	—
65	C₂×M₁₆×C₄₇	—	2	2	no	—
66	C₂×M₁₆×C₄₇	—	2	2	no	—
67	C₂×M₁₆×C₄₇	—	2	1	no	—
68	C₂×M₁₆×C₄₇	—	2	2	no	—
69	C₂×M₁₆×C₄₇	—	2	2	no	—
70	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
71	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
72	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
73	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
74	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
75	C₂²×C₈×C₄₇	—	2	1	no	—
76	C₂²×C₈×C₄₇	—	2	1	no	—
77	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
78	C₂⁴×C₄₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	#41	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₄₇	—	2	2	no	—
90	C₂²×Dic₄₇	—	2	2	no	—
91	C₂²×Dic₄₇	—	2	2	no	—
92	M₁₆×C₄₇	—	2	4	no	—
93	M₁₆×C₄₇	—	2	4	no	—
94	M₁₆×C₄₇	—	2	4	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	2	2	no	—
98	M₁₆×C₄₇	—	4	2	no	—
99	M₁₆×C₄₇	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₄₇	—	2	4	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	4	2	no	—
106	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
107	C₂²⋊Dic₄₇	—	2	2	no	—
108	C₂²⋊Dic₄₇	—	2	2	no	—
109	C₂²⋊Dic₄₇	—	2	2	no	—
110	C₂²⋊Dic₄₇	—	2	2	no	—
111	C₂²⋊Dic₄₇	—	4	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	4	2	no	—
115	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
116	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₄×Dic₄₇	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	4	no	—
124	C₂×C₄₇⋊C₈	—	2	2	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	C₂³×C₄₇	—	4	2	no	—
131	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
132	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
133	C₂³×C₄₇	#125	1	2	yes	—
134	C₂³×C₄₇	#126	1	2	yes	—
135	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#117	1	2	yes	—
137	C₂×C₄×C₄₇	#117	1	2	yes	—
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₄₇	#123	1	2	yes	—
145	C₂×C₄×C₄₇	#123	1	2	yes	—
146	C₂×C₄×C₄₇	#125	1	2	yes	—
147	C₂×C₄×C₄₇	#126	1	2	yes	—
148	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₄₇	—	2	4	no	—
155	C₂×Dic₄₇	—	2	4	no	—
156	C₂×Dic₄₇	—	4	4	no	—
157	C₂×Dic₄₇	—	4	4	no	—
158	C₂×Dic₄₇	—	4	4	no	—
159	C₂×Dic₄₇	—	4	2	no	—
160	C₈×C₄₇	—	2	4	no	—
161	C₈×C₄₇	—	2	4	no	—
162	C₈×C₄₇	—	4	2	no	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	2	2	no	—
165	#21041	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	—	2	2	no	—
167	C₂²×C₄₇	—	2	2	no	—
168	C₂²×C₄₇	—	2	2	no	—
169	C₂²×C₄₇	—	4	2	no	—
170	C₂²×C₄₇	—	4	2	no	—
171	C₂²×C₄₇	#387	1	2	yes	—
172	C₂²×C₄₇	#387	1	2	yes	—
173	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
175	C₂²×C₄₇	C₂×#126	1	1	yes	characteristic
176	C₄×C₄₇	#387	1	2	yes	—
177	C₄×C₄₇	#387	1	2	yes	—
178	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
179	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
182	Dic₄₇	—	4	4	no	—
183	Dic₄₇	—	4	4	no	—
184	Dic₄₇	—	8	2	no	—
185	#2911	—	47	1	no	—
186	C₂×#424	—	47	1	no	—
187	C₂×#405	—	47	1	no	—
188	#1922	—	47	8	no	—
189	#1738	D₄₇	1	1	yes	characteristic
190	#1733	—	47	1	no	—
191	#1733	—	47	1	no	—
192	#1729	—	47	1	no	—
193	C₂×C₄₇	—	2	1	no	—
194	C₂×C₄₇	—	2	1	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	—	2	2	no	—
197	C₂×C₄₇	C₂×#387	1	1	yes	characteristic
198	C₂×C₄₇	#2955	1	2	yes	—
199	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
200	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
201	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
202	C₂³×C₈	—	47	1	no	—
203	C₂×#117	—	47	1	no	—
204	C₂×#117	—	47	1	no	—
205	C₂×#117	—	47	1	no	—
206	#428	—	47	4	no	—
207	#428	—	47	4	no	—
208	#427	—	47	4	no	—
209	#424	—	47	4	no	—
210	#424	—	94	1	no	—
211	#424	—	94	1	no	—
212	C₂×C₄⋊C₈	—	94	2	no	—
213	#405	—	47	4	no	—
214	#405	—	94	2	no	—
215	C₂×#97	—	94	1	no	—
216	C₂×C₄×C₈	—	94	1	no	—
217	#341	C₂×D₄₇	1	4	yes	—
218	#339	—	47	4	no	—
219	#339	—	47	4	no	—
220	#336	—	47	4	no	—
221	C₄₇	#21041	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	Dic₉₄	1	4	yes	—
231	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
232	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
233	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
234	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	—	47	1	no	—
238	C₂²×C₈	—	47	1	no	—
239	C₂²×C₈	—	47	2	no	—
240	C₂²×C₈	—	47	2	no	—
241	C₂²×C₈	—	94	2	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
244	#117	—	47	2	no	—
245	#117	—	47	2	no	—
246	#117	—	47	2	no	—
247	#117	—	47	2	no	—
248	#117	—	94	1	no	—
249	#117	—	94	1	no	—
250	#117	—	94	1	no	—
251	#117	—	94	1	no	—
252	#117	—	94	2	no	—
253	#117	—	94	2	no	—
254	C₂×C₄⋊C₄	—	47	2	no	—
255	C₂×C₄⋊C₄	—	47	2	no	—
256	C₂×C₄⋊C₄	—	94	1	no	—
257	C₂×C₂²⋊C₄	—	94	1	no	—
258	C₂×C₂²⋊C₄	—	94	1	no	—
259	C₂×C₂²⋊C₄	—	94	1	no	—
260	C₂×C₄²	—	47	2	no	—
261	C₂×C₄²	—	94	1	no	—
262	C₂×C₄²	—	94	1	no	—
263	#107	—	47	8	no	—
264	#106	—	47	8	no	—
265	C₄⋊C₈	—	94	4	no	—
266	C₄⋊C₈	—	94	4	no	—
267	#97	—	94	4	no	—
268	C₄×C₈	—	94	4	no	—
269	C₂⁴	#10625354	1	1	yes	characteristic
270	C₂²×C₄	—	94	2	no	—
271	C₂²×C₄	—	94	2	no	—
272	C₂²×C₄	—	94	2	no	—
273	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
274	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
276	C₂²×C₄	#10625350	1	2	yes	—
277	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
278	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
279	C₂²×C₄	#10625352	1	2	yes	—
280	C₂²×C₄	#10625352	1	2	yes	—
281	C₂²×C₄	#10625354	1	1	yes	characteristic
282	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	4	2	no	—
290	M₁₆	C₂×Dic₉₄	1	4	yes	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	47	2	no	—
298	C₂×C₈	—	47	2	no	—
299	C₂×C₈	—	94	4	no	—
300	C₂×C₈	—	94	2	no	—
301	C₂×C₈	—	94	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	—	94	2	no	—
304	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
305	C₄⋊C₄	—	47	4	no	—
306	C₄⋊C₄	—	47	4	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	94	2	no	—
311	C₂²⋊C₄	—	94	2	no	—
312	C₂²⋊C₄	—	94	2	no	—
313	C₂²⋊C₄	—	94	2	no	—
314	C₂²⋊C₄	—	188	2	no	—
315	C₄²	—	94	2	no	—
316	C₄²	—	94	2	no	—
317	C₄²	—	94	2	no	—
318	C₄²	—	94	2	no	—
319	C₄²	—	94	2	no	—
320	C₄²	—	94	2	no	—
321	C₂³	—	4	2	no	—
322	C₂³	#13093986	1	1	yes	characteristic
323	C₂³	#13093986	1	1	yes	characteristic
324	C₂³	#13094015	1	2	yes	—
325	C₂³	#13094019	1	2	yes	—
326	C₂³	C₂×#10625354	1	1	yes	characteristic
327	C₂×C₄	—	94	4	no	—
328	C₂×C₄	—	94	4	no	—
329	C₂×C₄	—	188	4	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	4	no	—
332	C₂×C₄	—	188	4	no	—
333	C₂×C₄	#13093980	1	2	yes	—
334	C₂×C₄	#13093980	1	2	yes	—
335	C₂×C₄	#13093986	1	1	yes	characteristic
336	C₂×C₄	#13093986	1	1	yes	characteristic
337	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
338	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
339	C₂×C₄	#13094001	1	2	yes	—
340	C₂×C₄	#13094001	1	2	yes	—
341	C₂×C₄	#13094006	1	2	yes	—
342	C₂×C₄	#13094006	1	2	yes	—
343	C₂×C₄	#13094015	1	2	yes	—
344	C₂×C₄	#13094019	1	2	yes	—
345	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
347	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
350	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	94	2	no	—
355	C₈	—	94	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	#15689006	1	2	yes	—
362	C₂²	#15689006	1	2	yes	—
363	C₂²	C₂×#13093986	1	1	yes	characteristic
364	C₂²	C₂×#13094015	1	1	yes	characteristic
365	C₂²	C₂×#13094019	1	1	yes	characteristic
366	C₄	—	376	2	no	—
367	C₄	—	188	4	no	—
368	C₄	—	188	4	no	—
369	C₄	#15689006	1	2	yes	—
370	C₄	#15689006	1	2	yes	—
371	C₄	C₂×#13093980	1	1	yes	characteristic
372	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
373	C₄	C₂×#13094001	1	1	yes	characteristic
374	C₄	C₂×#13094006	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	2	no	—
377	C₂	—	2	2	no	—
378	C₂	—	2	1	no	—
379	C₂	C₂×#15689006	1	1	yes	characteristic
380	C₂	#20252290	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1738	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1738	D₄₇
hypocenter	C₄₇	#21041
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1738

Complements

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

N	H ≅ G/N	type
4. #20243986	C₂	semidirect
13. #15689165	C₄	semidirect
14. #15689165	C₄	semidirect
221. C₄₇	#21041	semidirect

Maximal subgroups

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

subgroup	index
#20251978	2
C₂×#15689151	2
C₂×#15689081	2
#20243986	2
#20243099	2
#20243086	2
C₄₇×#1738	2
#21041	47