Subgroups — #26105835

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
12	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#384	C₄	1	2	yes	—
18	C₄₇×#382	—	2	1	no	—
19	C₄₇×#380	—	2	1	no	—
20	C₄₇×#380	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	4	2	no	—
25	C₂×#10625329	—	4	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	8	no	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	2	no	—
35	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	#117×C₄₇	—	4	2	no	—
40	#117×C₄₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	#95×C₄₇	—	2	2	no	—
55	#95×C₄₇	—	2	2	no	—
56	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	D₈	1	2	yes	—
70	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	4	4	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	4	no	—
80	C₄²×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	C₄²	1	2	yes	—
83	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
92	#10625329	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	8	no	—
94	C₂×C₄₇⋊C₈	—	2	16	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	#95	1	2	yes	—
102	C₂³×C₄₇	#102	1	2	yes	—
103	C₂³×C₄₇	#103	1	2	yes	—
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	#95	1	2	yes	—
117	C₂×C₄×C₄₇	#102	1	4	yes	—
118	C₂×C₄×C₄₇	#102	1	2	yes	—
119	C₂×C₄×C₄₇	#103	1	4	yes	—
120	C₂×C₄×C₄₇	#103	1	2	yes	—
121	C₂×C₄×C₄₇	#104	1	4	yes	—
122	C₂×C₄×C₄₇	#106	1	4	yes	—
123	C₂×C₄×C₄₇	#107	1	4	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#131	1	4	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
135	C₄₇⋊C₈	—	2	8	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	#7897	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	#340	1	4	yes	—
143	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
144	C₂²×C₄₇	#408	1	2	yes	—
145	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#104	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	8	4	no	—
152	C₄×C₄₇	#339	1	4	yes	—
153	C₄×C₄₇	#340	1	4	yes	—
154	C₄×C₄₇	#405	1	2	yes	—
155	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
157	C₄×C₄₇	#424	1	2	yes	—
158	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
159	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
160	C₂×#423	—	47	1	no	—
161	C₂²×C₄×C₈	—	47	1	no	—
162	#2285	D₄₇	1	1	yes	characteristic
163	#1279	—	47	4	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	#1492	1	2	yes	—
166	C₂×C₄₇	#1493	1	2	yes	—
167	C₂×C₄₇	#1729	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
170	C₂²×M₁₆	—	94	1	no	—
171	C₂³×C₈	—	94	1	no	—
172	C₂×#117	—	2	1	no	—
173	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
174	#423	—	47	4	no	—
175	#423	—	94	2	no	—
176	C₂×C₄⋊C₈	—	47	2	no	—
177	C₂×C₄⋊C₈	—	94	1	no	—
178	C₂×C₄×C₈	—	47	4	no	—
179	C₂×C₄×C₈	—	47	2	no	—
180	C₂×C₄×C₈	—	94	2	no	—
181	C₂×C₄×C₈	—	94	1	no	—
182	#384	Dic₄₇	1	2	yes	—
183	#382	—	2	1	no	—
184	#380	—	2	1	no	—
185	#380	Dic₄₇	1	2	yes	—
186	C₄×C₄⋊C₄	—	2	1	no	—
187	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
188	C₄₇	#7897	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂×M₁₆	—	94	4	no	—
193	C₂×M₁₆	—	188	2	no	—
194	C₂×M₁₆	—	188	2	no	—
195	C₂²×C₈	—	94	8	no	—
196	C₂²×C₈	—	94	4	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	#117	—	4	2	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	Dic₉₄	1	2	yes	—
207	C₂×C₄²	C₄×D₄₇	1	2	yes	—
208	C₂×C₄²	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	D₁₈₈	1	2	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₄⋊C₈	—	47	4	no	—
215	C₄⋊C₈	—	94	4	no	—
216	C₄⋊C₈	—	94	2	no	—
217	C₄×C₈	—	47	4	no	—
218	C₄×C₈	—	94	4	no	—
219	C₄×C₈	—	94	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
227	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
228	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
232	C₂²×C₄	#10625334	1	2	yes	—
233	C₂²×C₄	#10625335	1	2	yes	—
234	C₂²×C₄	#10625336	1	2	yes	—
235	C₂²×C₄	#10625337	1	2	yes	—
236	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	M₁₆	—	188	8	no	—
243	C₂×C₈	—	94	8	no	—
244	C₂×C₈	—	94	16	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	—	188	2	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₄×Dic₄₇	1	2	yes	—
256	C₄²	Dic₄₇⋊C₄	1	2	yes	—
257	C₄²	Dic₄₇⋊C₄	1	2	yes	—
258	C₄²	C₄⋊Dic₄₇	1	2	yes	—
259	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
260	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
261	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
264	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₄₇⋊#95	1	2	yes	—
268	C₂³	#13093945	1	2	yes	—
269	C₂³	#13093948	1	2	yes	—
270	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₄₇⋊#95	1	2	yes	—
281	C₂×C₄	C₄₇⋊#95	1	2	yes	—
282	C₂×C₄	C₄₇⋊#95	1	2	yes	—
283	C₂×C₄	#13093944	1	4	yes	—
284	C₂×C₄	#13093945	1	2	yes	—
285	C₂×C₄	#13093947	1	4	yes	—
286	C₂×C₄	#13093948	1	2	yes	—
287	C₂×C₄	#13093951	1	4	yes	—
288	C₂×C₄	#13093955	1	4	yes	—
289	C₂×C₄	#13093957	1	4	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	#13094031	1	4	yes	—
297	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
298	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
300	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
301	C₈	—	94	8	no	—
302	C₈	—	188	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	#15688885	1	4	yes	—
308	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
309	C₂²	#15689090	1	2	yes	—
310	C₂²	C₂×#13093945	1	1	yes	characteristic
311	C₂²	C₂×#13093948	1	1	yes	characteristic
312	C₂²	C₂×#13093951	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	#15688882	1	4	yes	—
318	C₄	#15688885	1	4	yes	—
319	C₄	#15689079	1	2	yes	—
320	C₄	C₂×#13093944	1	1	yes	characteristic
321	C₄	C₂×#13093947	1	1	yes	characteristic
322	C₄	#15689150	1	2	yes	—
323	C₄	C₂×#13093955	1	1	yes	characteristic
324	C₄	C₂×#13093957	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	#20241839	1	2	yes	—
327	C₂	#20241843	1	2	yes	—
328	C₂	#20243085	1	1	yes	characteristic
329	C₂	C₂×#15688882	1	1	yes	characteristic
330	C₂	C₂×#15688885	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#7897
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20241229	C₂	semidirect
8. #15689147	C₄	semidirect
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
188. C₄₇	#7897	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂²×C₄×C₄₇⋊C₈	2
#20241229	2
C₄₇×#2285	2
#7897	47