Subgroups — #26131242

Order 12032.

736 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689107	C₂	1	2	yes	maximal
2	#20241511	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2879	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	#15689107	—	2	2	no	—
6	#15689107	—	2	2	no	—
7	#15689107	C₄	1	4	yes	—
8	#15689107	C₂²	1	4	yes	—
9	C₂×#13093943	—	2	2	no	—
10	C₂×#13093941	C₄	1	4	yes	—
11	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
14	C₄₇×#408	—	2	2	no	—
15	C₄₇×#408	C₄	1	2	yes	—
16	C₄₇×#408	C₂²	1	2	yes	—
17	C₄₇×#407	—	2	2	no	—
18	C₂×#10625329	—	2	4	no	—
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	4	2	no	—
22	C₂×#10625329	—	4	2	no	—
23	C₂×#10625329	—	4	2	no	—
24	C₂²×C₄₇⋊C₈	—	4	2	no	—
25	#13093943	—	2	4	no	—
26	#13093943	—	2	4	no	—
27	#13093941	—	2	4	no	—
28	#13093941	C₂×C₄	1	8	yes	—
29	C₂×#41×C₄₇	—	4	2	no	—
30	C₂×#41×C₄₇	—	4	2	no	—
31	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
34	C₂×#41×C₄₇	D₄	1	2	yes	—
35	C₂×#41×C₄₇	Q₈	1	1	yes	characteristic
36	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	2	yes	—
42	C₂×M₁₆×C₄₇	—	2	1	no	—
43	C₂×M₁₆×C₄₇	—	2	1	no	—
44	C₂×M₁₆×C₄₇	—	2	1	no	—
45	C₂×M₁₆×C₄₇	—	4	2	no	—
46	C₂×M₁₆×C₄₇	—	4	2	no	—
47	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₂²⋊C₈×C₄₇	—	2	4	no	—
51	C₂²⋊C₈×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
53	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
54	#41×C₄₇	—	4	2	no	—
55	#41×C₄₇	—	4	2	no	—
56	#41×C₄₇	—	4	1	no	—
57	#41×C₄₇	—	4	1	no	—
58	#41×C₄₇	—	4	2	no	—
59	#41×C₄₇	—	4	2	no	—
60	#41×C₄₇	—	4	2	no	—
61	#41×C₄₇	—	4	2	no	—
62	#41×C₄₇	—	4	1	no	—
63	#41×C₄₇	—	4	1	no	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂×Q₈×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂×Q₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	2	no	—
70	C₂×D₄×C₄₇	—	4	2	no	—
71	C₂×D₄×C₄₇	C₄²	1	2	yes	—
72	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂×D₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
95	M₁₆×C₄₇	—	4	2	no	—
96	M₁₆×C₄₇	—	4	2	no	—
97	M₁₆×C₄₇	—	4	2	no	—
98	M₁₆×C₄₇	—	4	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	4	2	no	—
102	C₂×C₈×C₄₇	—	4	2	no	—
103	#10625329	—	2	8	no	—
104	#10625329	—	4	4	no	—
105	#10625329	—	4	4	no	—
106	#10625329	—	4	4	no	—
107	C₂×C₄₇⋊C₈	—	4	4	no	—
108	C₂×C₄₇⋊C₈	—	4	4	no	—
109	C₂×C₄₇⋊C₈	—	4	4	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	2	2	no	—
113	C₂³×C₄₇	—	8	2	no	—
114	C₂³×C₄₇	—	8	1	no	—
115	C₂³×C₄₇	#95	1	2	yes	—
116	C₂³×C₄₇	#95	1	1	yes	characteristic
117	C₂³×C₄₇	#95	1	1	yes	characteristic
118	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
119	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	4	1	no	—
122	Q₈×C₄₇	—	4	1	no	—
123	D₄×C₄₇	—	4	2	no	—
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	4	2	no	—
127	D₄×C₄₇	—	4	1	no	—
128	D₄×C₄₇	—	4	2	no	—
129	D₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	1	no	—
142	C₂×C₄×C₄₇	—	2	1	no	—
143	C₂×C₄×C₄₇	—	2	1	no	—
144	C₂×C₄×C₄₇	—	2	1	no	—
145	C₂×C₄×C₄₇	—	2	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
148	C₂×C₄×C₄₇	#95	1	2	yes	—
149	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#99	1	2	yes	—
151	C₂×C₄×C₄₇	#99	1	2	yes	—
152	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₈×C₄₇	—	4	2	no	—
156	C₈×C₄₇	—	4	2	no	—
157	C₄₇⋊C₈	—	4	8	no	—
158	#12579	—	47	1	no	maximal, Sylow
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	4	1	no	—
164	C₂²×C₄₇	—	4	2	no	—
165	C₂²×C₄₇	—	4	1	no	—
166	C₂²×C₄₇	—	4	1	no	—
167	C₂²×C₄₇	—	4	2	no	—
168	C₂²×C₄₇	—	4	4	no	—
169	C₂²×C₄₇	—	8	2	no	—
170	C₂²×C₄₇	—	8	2	no	—
171	C₂²×C₄₇	#342	1	2	yes	—
172	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
173	C₄×C₄₇	—	2	2	no	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	#342	1	2	yes	—
178	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
179	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
180	C₂×#413	—	47	2	no	—
181	#2879	D₄₇	1	1	yes	characteristic
182	#1392	—	47	4	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	2	no	—
185	C₂×C₄₇	—	4	2	no	—
186	C₂×C₄₇	—	4	2	no	—
187	C₂×C₄₇	—	4	1	no	—
188	C₂×C₄₇	—	4	1	no	—
189	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
190	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
191	C₂²×M₁₆	—	94	2	no	—
192	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
193	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
194	#413	—	47	4	no	—
195	#413	—	47	4	no	—
196	#413	—	94	2	no	—
197	#413	—	94	2	no	—
198	C₂×#101	—	94	2	no	—
199	C₂×#100	—	47	4	no	—
200	#408	—	2	2	no	—
201	#408	Dic₄₇	1	2	yes	—
202	#408	C₂×D₄₇	1	2	yes	—
203	#407	—	2	2	no	—
204	C₄₇	#12579	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	4	2	no	—
206	C₂×#41	—	4	2	no	—
207	C₂×#41	Dic₉₄	1	1	yes	characteristic
208	C₂×#41	C₄×D₄₇	1	2	yes	—
209	C₂×#41	D₁₈₈	1	1	yes	characteristic
210	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
211	C₂×#41	C₂²⋊D₄₇	1	2	yes	—
212	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
214	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
215	C₂²×D₄	C₄×D₄₇	1	2	yes	—
216	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	4	2	no	—
222	C₂×M₁₆	—	4	2	no	—
223	C₂×M₁₆	—	94	4	no	—
224	C₂×M₁₆	—	94	4	no	—
225	C₂×M₁₆	—	94	4	no	—
226	C₂×M₁₆	—	188	2	no	—
227	C₂×M₁₆	—	188	2	no	—
228	C₂×M₁₆	—	188	2	no	—
229	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	188	2	no	—
232	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
233	#101	—	94	4	no	—
234	#101	—	94	4	no	—
235	#100	—	47	8	no	—
236	#100	—	94	4	no	—
237	C₂²⋊C₈	—	2	4	no	—
238	C₂²⋊C₈	—	2	4	no	—
239	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
240	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
241	#41	—	4	2	no	—
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	#41	—	4	2	no	—
245	#41	—	4	1	no	—
246	#41	—	4	1	no	—
247	#41	—	4	2	no	—
248	#41	—	4	2	no	—
249	#41	—	4	1	no	—
250	#41	—	4	1	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	C₄⋊Dic₄₇	1	1	yes	characteristic
253	C₂×Q₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
254	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
259	C₂×D₄	Dic₄₇⋊C₄	1	2	yes	—
260	C₂×D₄	Dic₄₇⋊C₄	1	2	yes	—
261	C₂×D₄	C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₂×D₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂×D₄	C₂×Dic₉₄	1	1	yes	characteristic
264	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
272	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
273	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
274	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
275	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
276	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
277	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
278	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
280	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	—	94	8	no	—
287	M₁₆	—	188	4	no	—
288	M₁₆	—	188	4	no	—
289	M₁₆	—	188	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	188	4	no	—
295	C₂×C₈	—	188	4	no	—
296	C₂×C₈	—	188	4	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	8	2	no	—
301	C₂³	—	8	1	no	—
302	C₂³	C₄₇⋊#95	1	1	yes	characteristic
303	C₂³	C₄₇⋊#95	1	2	yes	—
304	C₂³	C₄₇⋊#95	1	1	yes	characteristic
305	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
335	C₂×C₄	C₄₇⋊#95	1	2	yes	—
336	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
337	C₂×C₄	#13093939	1	2	yes	—
338	C₂×C₄	#13093939	1	2	yes	—
339	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
340	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₈	—	188	8	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	2	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	#15688890	1	2	yes	—
358	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	#15688890	1	2	yes	—
364	C₄	C₂×#13093939	1	1	yes	characteristic
365	C₄	C₂×#13093939	1	1	yes	characteristic
366	C₂	—	2	1	no	—
367	C₂	—	2	2	no	—
368	C₂	—	4	2	no	—
369	C₂	—	4	2	no	—
370	C₂	—	4	1	no	—
371	C₂	—	4	1	no	—
372	C₂	C₂×#15688890	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13093939
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂×#41	C₂²×D₄₇
Fitting	C₄₇×#2879	C₂
socle	C₂×C₄₇	C₂×#342
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×#41×C₄₇	C₂³
hypercenter	#2879	D₄₇
hypocenter	C₄₇	#12579
LCS	G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #2879

Complements

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

N	H ≅ G/N	type
2. #20241511	C₂	semidirect
204. C₄₇	#12579	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689107	2
#20241511	2
C₄₇×#2879	2
#12579	47