Subgroups — #26105743

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#20241222	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂³×C₄₇⋊C₈	C₄	1	2	yes	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₄₇	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	2	no	—
18	C₄₇×#423	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	4	2	no	—
23	C₂×#10625329	—	4	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
31	C₄×C₄₇⋊C₈	—	2	4	no	—
32	C₄×C₄₇⋊C₈	—	2	2	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	—	4	2	no	—
39	C₂×M₁₆×C₄₇	—	4	2	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₄⋊C₈×C₄₇	—	2	2	no	—
51	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	D₈	1	2	yes	—
65	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	M₁₆×C₄₇	—	4	8	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₄²×C₄₇	C₄²	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	#10625329	—	4	8	no	—
86	C₂×C₄₇⋊C₈	—	2	16	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	complemented
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	#95	1	2	yes	—
96	C₂³×C₄₇	#106	1	2	yes	—
97	C₂³×C₄₇	#107	1	2	yes	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	#95	1	2	yes	—
106	C₂×C₄×C₄₇	#95	1	2	yes	—
107	C₂×C₄×C₄₇	#95	1	2	yes	—
108	C₂×C₄×C₄₇	#102	1	4	yes	—
109	C₂×C₄×C₄₇	#102	1	4	yes	—
110	C₂×C₄×C₄₇	#103	1	4	yes	—
111	C₂×C₄×C₄₇	#103	1	4	yes	—
112	C₂×C₄×C₄₇	#106	1	2	yes	—
113	C₂×C₄×C₄₇	#107	1	2	yes	—
114	C₂×C₄×C₄₇	#108	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
126	C₈×C₄₇	—	4	4	no	—
127	C₄₇⋊C₈	—	2	8	no	—
128	C₄₇⋊C₈	—	4	4	no	—
129	#7874	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	#340	1	4	yes	—
135	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₇	#423	1	2	yes	—
137	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#108	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	#340	1	4	yes	—
144	C₄×C₄₇	#341	1	4	yes	—
145	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
146	C₄×C₄₇	#407	1	2	yes	—
147	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
151	C₂×#423	—	47	1	no	—
152	C₂×#423	D₄₇	1	1	yes	characteristic
153	C₂²×C₄×C₈	—	47	1	no	—
154	#1277	—	47	4	no	—
155	C₂×C₄₇	—	2	4	no	—
156	C₂×C₄₇	#1566	1	2	yes	—
157	C₂×C₄₇	#1567	1	2	yes	—
158	C₂×C₄₇	#1728	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
161	C₂²×M₁₆	—	2	1	no	—
162	C₂²×M₁₆	—	94	1	no	—
163	C₂³×C₈	—	47	2	no	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	#423	—	2	2	no	—
166	#423	—	47	4	no	—
167	#423	—	94	2	no	—
168	#423	Dic₄₇	1	4	yes	—
169	C₂×C₄⋊C₈	—	2	1	no	—
170	C₂×C₄⋊C₈	—	47	2	no	—
171	C₂×C₄⋊C₈	—	94	1	no	—
172	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
173	C₂×C₄×C₈	—	47	2	no	—
174	C₂×C₄×C₈	—	47	2	no	—
175	C₂×C₄×C₈	—	94	2	no	—
176	C₂×C₄×C₈	—	94	2	no	—
177	C₄₇	#7874	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
179	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
180	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	4	2	no	—
183	C₂×M₁₆	—	4	2	no	—
184	C₂×M₁₆	—	94	4	no	—
185	C₂×M₁₆	—	188	2	no	—
186	C₂×M₁₆	—	188	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	94	8	no	—
190	C₂²×C₈	—	94	4	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₄₇	1	2	yes	—
195	C₂×C₄²	D₁₈₈	1	2	yes	—
196	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	47	4	no	—
203	C₄⋊C₈	—	94	4	no	—
204	C₄⋊C₈	—	94	2	no	—
205	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
206	C₄×C₈	—	94	4	no	—
207	C₄×C₈	—	94	2	no	—
208	C₄×C₈	—	94	2	no	—
209	C₂⁴	Dic₄₇⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
213	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
215	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
220	C₂²×C₄	#10625334	1	2	yes	—
221	C₂²×C₄	#10625335	1	2	yes	—
222	C₂²×C₄	#10625336	1	2	yes	—
223	C₂²×C₄	#10625337	1	2	yes	—
224	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	M₁₆	—	4	8	no	—
229	M₁₆	—	188	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	47	8	no	—
234	C₂×C₈	—	94	16	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₄²	C₄×Dic₄₇	1	2	yes	—
241	C₄²	Dic₄₇⋊C₄	1	2	yes	—
242	C₄²	C₄⋊Dic₄₇	1	2	yes	—
243	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
244	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
245	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
246	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
248	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₄₇⋊#95	1	2	yes	—
253	C₂³	#13093955	1	2	yes	—
254	C₂³	#13093957	1	2	yes	—
255	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	C₄₇⋊#95	1	2	yes	—
263	C₂×C₄	C₄₇⋊#95	1	2	yes	—
264	C₂×C₄	C₄₇⋊#95	1	2	yes	—
265	C₂×C₄	#13093944	1	4	yes	—
266	C₂×C₄	#13093945	1	4	yes	—
267	C₂×C₄	#13093947	1	4	yes	—
268	C₂×C₄	#13093948	1	4	yes	—
269	C₂×C₄	#13093955	1	2	yes	—
270	C₂×C₄	#13093957	1	2	yes	—
271	C₂×C₄	#13093959	1	4	yes	—
272	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
279	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
281	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	94	8	no	—
285	C₈	—	188	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	#15688885	1	4	yes	—
291	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
292	C₂²	#15689146	1	2	yes	—
293	C₂²	C₂×#13093955	1	1	yes	characteristic
294	C₂²	C₂×#13093957	1	1	yes	characteristic
295	C₂²	C₂×#13093959	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	#15688885	1	4	yes	—
300	C₄	#15688887	1	4	yes	—
301	C₄	M₁₆×Dic₄₇	1	2	yes	—
302	C₄	#15689086	1	2	yes	—
303	C₄	C₂×#13093944	1	1	yes	characteristic
304	C₄	C₂×#13093945	1	1	yes	characteristic
305	C₄	C₂×#13093947	1	1	yes	characteristic
306	C₄	C₂×#13093948	1	1	yes	characteristic
307	C₂	—	2	4	no	—
308	C₂	#20242143	1	2	yes	—
309	C₂	#20242146	1	2	yes	—
310	C₂	#20243081	1	1	yes	characteristic
311	C₂	C₂×#15688885	1	1	yes	characteristic
312	C₂	C₂×#15688887	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	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#7874
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20241222	C₂	semidirect
92. C₂×C₄₇⋊C₈	M₁₆	semidirect
177. C₄₇	#7874	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
#20241222	2
C₂×C₄₇×#423	2
#7874	47