Subgroups — #26245180

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic, complemented
2	#20251907	C₂	1	2	yes	maximal
3	#20244833	C₂	1	4	yes	maximal, complemented
4	#20244828	C₂	1	4	yes	maximal, complemented
5	#20243126	C₂	1	2	yes	maximal, complemented
6	C₂×#15688888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#408	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689161	—	2	2	no	—
11	#15689161	—	2	2	no	—
12	C₂×#13093958	C₂²	1	2	yes	complemented
13	C₂×#13093956	C₂²	1	2	yes	complemented
14	#15689132	—	2	2	no	—
15	#15689128	—	2	2	no	—
16	#15689124	C₂²	1	8	yes	—
17	#15689119	—	2	4	no	—
18	#15688891	C₂²	1	8	yes	complemented
19	#15688888	C₂²	1	2	yes	complemented
20	#15688888	C₂²	1	2	yes	complemented
21	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#408	—	2	2	no	—
25	C₄₇×#408	C₂²	1	2	yes	—
26	C₄₇×#408	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	#13093983	—	2	2	no	—
34	#13093983	—	2	2	no	—
35	#13093983	—	2	2	no	—
36	#13093983	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	#13093958	—	2	4	no	—
41	#13093956	—	2	4	no	—
42	#13093948	—	2	8	no	—
43	#13093945	—	2	8	no	—
44	C₂×#41×C₄₇	—	2	1	no	—
45	C₂×#41×C₄₇	—	2	1	no	—
46	C₂×#41×C₄₇	—	2	4	no	—
47	C₂×#41×C₄₇	—	2	4	no	—
48	C₂×#41×C₄₇	D₄	1	2	yes	—
49	C₂×#41×C₄₇	C₂³	1	2	yes	—
50	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₄₇	—	2	1	no	—
52	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	—	2	1	no	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₄₇	—	2	1	no	—
56	C₂×M₁₆×C₄₇	—	2	4	no	—
57	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	—	2	4	no	—
59	C₂²×C₈×C₄₇	—	2	1	no	—
60	C₂²×C₈×C₄₇	D₄	1	2	yes	—
61	C₂²×C₈×C₄₇	D₄	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²⋊C₈×C₄₇	—	2	4	no	—
65	C₂²⋊C₈×C₄₇	—	2	4	no	—
66	C₂⁴×C₄₇	—	2	1	no	—
67	C₂⁴×C₄₇	#41	1	1	yes	characteristic
68	#41×C₄₇	—	2	8	no	—
69	#41×C₄₇	—	2	4	no	—
70	#41×C₄₇	—	2	4	no	—
71	#41×C₄₇	—	4	2	no	—
72	#41×C₄₇	—	4	2	no	—
73	#41×C₄₇	—	4	4	no	—
74	C₂×Q₈×C₄₇	—	2	1	no	—
75	C₂×Q₈×C₄₇	—	2	4	no	—
76	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	—	2	1	no	—
80	C₂×D₄×C₄₇	—	2	4	no	—
81	C₂×D₄×C₄₇	—	4	4	no	—
82	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	2	2	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	—	4	2	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	4	yes	—
94	C₂²×C₄×C₄₇	#41	1	4	yes	—
95	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₄₇	—	2	4	no	—
97	M₁₆×C₄₇	—	2	4	no	—
98	M₁₆×C₄₇	—	4	2	no	—
99	C₂×C₈×C₄₇	—	2	4	no	—
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	—	4	2	no	—
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
107	C₂²⋊Dic₄₇	—	4	2	no	—
108	C₂²⋊Dic₄₇	—	4	2	no	—
109	C₄⋊Dic₄₇	—	2	4	no	—
110	C₄⋊Dic₄₇	—	2	4	no	—
111	C₄×Dic₄₇	—	4	2	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	#10625329	—	2	4	no	—
114	#10625329	—	2	4	no	—
115	#10625329	—	4	4	no	—
116	C₂×C₄₇⋊C₈	—	2	4	no	—
117	C₂×C₄₇⋊C₈	—	4	4	no	—
118	C₂³×C₄₇	—	2	2	no	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂³×C₄₇	—	4	4	no	—
121	C₂³×C₄₇	—	4	2	no	—
122	C₂³×C₄₇	#121	1	1	yes	characteristic
123	C₂³×C₄₇	#121	1	1	yes	characteristic
124	C₂³×C₄₇	#126	1	4	yes	—
125	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
126	Q₈×C₄₇	—	2	4	no	—
127	Q₈×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	4	4	no	—
131	D₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	4	no	—
137	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#123	1	2	yes	—
140	C₂×C₄×C₄₇	#123	1	2	yes	—
141	C₂×C₄×C₄₇	#123	1	4	yes	—
142	C₂×C₄×C₄₇	#123	1	4	yes	—
143	C₂×C₄×C₄₇	#124	1	2	yes	—
144	C₂×C₄×C₄₇	#124	1	2	yes	—
145	C₂×C₄×C₄₇	#126	1	4	yes	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
149	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
150	C₂×Dic₄₇	—	4	4	no	—
151	C₂×Dic₄₇	—	8	4	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	—	2	2	no	—
154	C₈×C₄₇	—	4	2	no	—
155	C₄₇⋊C₈	—	4	4	no	—
156	#23646	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	4	4	no	—
161	C₂²×C₄₇	—	4	2	no	—
162	C₂²×C₄₇	—	4	4	no	—
163	C₂²×C₄₇	—	8	4	no	—
164	C₂²×C₄₇	#397	1	2	yes	—
165	C₂²×C₄₇	#397	1	2	yes	—
166	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#126	1	2	yes	—
168	C₄×C₄₇	—	4	2	no	—
169	C₄×C₄₇	#397	1	2	yes	—
170	C₄×C₄₇	#397	1	2	yes	—
171	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
172	C₄×C₄₇	C₂×#123	1	2	yes	—
173	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
174	Dic₄₇	—	8	4	no	—
175	C₂×#427	—	47	1	no	—
176	#2896	—	47	2	no	—
177	C₂×#408	D₄₇	1	1	yes	characteristic
178	#2079	—	47	4	no	—
179	#2078	—	47	4	no	—
180	#1742	—	47	2	no	—
181	C₂×#341	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	—	2	2	no	—
185	C₂×C₄₇	—	8	2	no	—
186	C₂×C₄₇	C₂×#397	1	1	yes	characteristic
187	C₂×C₄₇	#3067	1	2	yes	—
188	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
189	C₂²×M₁₆	—	47	2	no	—
190	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
191	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
192	C₂×#117	—	47	2	no	—
193	#427	—	94	2	no	—
194	#427	—	94	2	no	—
195	C₂×#107	—	47	2	no	—
196	C₂×#106	—	47	2	no	—
197	#419	—	94	2	no	—
198	#418	—	94	2	no	—
199	#417	—	47	8	no	—
200	#416	—	94	4	no	—
201	#408	—	2	2	no	—
202	#408	C₂×D₄₇	1	2	yes	—
203	#408	C₂×D₄₇	1	2	yes	—
204	C₂×C₂²⋊C₈	—	2	2	no	—
205	#343	—	47	8	no	—
206	#341	—	47	2	no	—
207	#341	—	47	2	no	—
208	C₄₇	#23646	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	2	1	no	—
210	C₂×#41	—	2	1	no	—
211	C₂×#41	—	2	4	no	—
212	C₂×#41	—	2	4	no	—
213	C₂×#41	D₁₈₈	1	2	yes	—
214	C₂×#41	C₂²×D₄₇	1	2	yes	—
215	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
218	C₂³×C₄	—	2	1	no	—
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	4	no	—
222	C₂×M₁₆	—	47	4	no	—
223	C₂×M₁₆	—	94	2	no	—
224	C₂×M₁₆	—	94	4	no	—
225	C₂×M₁₆	—	94	4	no	—
226	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
227	C₂²×C₈	—	2	4	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	94	2	no	—
230	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
231	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	#117	—	94	2	no	—
235	#117	—	94	2	no	—
236	#117	—	94	2	no	—
237	#117	—	94	2	no	—
238	C₂×C₄⋊C₄	—	47	4	no	—
239	C₂×C₂²⋊C₄	—	94	2	no	—
240	C₂×C₄²	—	94	2	no	—
241	#107	—	94	4	no	—
242	#106	—	94	4	no	—
243	#103	—	94	8	no	—
244	#102	—	94	8	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	C₂⁴	—	2	1	no	—
248	C₂⁴	#10625350	1	1	yes	characteristic
249	#41	—	2	8	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	#41	—	4	2	no	—
253	#41	—	4	2	no	—
254	#41	—	4	4	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	C₂×D₁₈₈	1	2	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	94	4	no	—
270	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
272	C₂²×C₄	#10625350	1	1	yes	characteristic
273	C₂²×C₄	#10625352	1	4	yes	—
274	C₂²×C₄	#10625352	1	4	yes	—
275	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
277	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
278	M₁₆	—	2	4	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	94	4	no	—
281	M₁₆	—	94	4	no	—
282	M₁₆	—	188	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	4	2	no	—
289	C₂×C₈	—	94	4	no	—
290	C₂×C₈	—	188	4	no	—
291	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
292	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
293	C₄⋊C₄	—	94	4	no	—
294	C₄⋊C₄	—	94	4	no	—
295	C₂²⋊C₄	—	188	2	no	—
296	C₂²⋊C₄	—	188	2	no	—
297	C₄²	—	188	2	no	—
298	C₄²	—	188	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	#13093998	1	1	yes	characteristic
304	C₂³	#13093998	1	1	yes	characteristic
305	C₂³	#13094017	1	4	yes	—
306	C₂³	C₂×#10625350	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	2	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	376	4	no	—
319	C₂×C₄	—	188	4	no	—
320	C₂×C₄	#13093998	1	1	yes	characteristic
321	C₂×C₄	#13093998	1	1	yes	characteristic
322	C₂×C₄	#13094005	1	2	yes	—
323	C₂×C₄	#13094005	1	2	yes	—
324	C₂×C₄	#13094008	1	4	yes	—
325	C₂×C₄	#13094008	1	4	yes	—
326	C₂×C₄	#13094011	1	2	yes	—
327	C₂×C₄	#13094011	1	2	yes	—
328	C₂×C₄	#13094017	1	4	yes	—
329	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625352	1	2	yes	—
332	C₂×C₄	C₂×#10625352	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	188	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	#15689048	1	2	yes	—
345	C₂²	#15689048	1	2	yes	—
346	C₂²	C₂×#13093998	1	1	yes	characteristic
347	C₂²	C₂×#13094017	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	376	4	no	—
350	C₄	#15689048	1	2	yes	—
351	C₄	#15689048	1	2	yes	—
352	C₄	C₂×#13094005	1	1	yes	characteristic
353	C₄	C₂×#13094008	1	2	yes	—
354	C₄	C₂×#13094011	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	C₂×#15689048	1	1	yes	characteristic
360	C₂	#20253108	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093998
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#408	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂²×#41×C₄₇	C₂²
hypercenter	C₂×#408	D₄₇
hypocenter	C₄₇	#23646
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#408

Complements

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

N	H ≅ G/N	type
1. C₂×#15689161	C₂	semidirect
3. #20244833	C₂	semidirect
4. #20244828	C₂	semidirect
5. #20243126	C₂	semidirect
6. C₂×#15688888	C₂	semidirect
12. C₂×#13093958	C₂²	semidirect
13. C₂×#13093956	C₂²	semidirect
18. #15688891	C₂²	semidirect
19. #15688888	C₂²	semidirect
20. #15688888	C₂²	semidirect
208. C₄₇	#23646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20251907	2
#20244833	2
#20244828	2
#20243126	2
C₂×#15688888	2
C₂×C₄₇×#408	2
#23646	47