Subgroups — #26106372

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
6	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
7	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
8	#15689180	—	2	2	no	—
9	#15689180	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
12	C₂×#13093935	—	2	1	no	—
13	#15688993	—	2	1	no	—
14	#15688993	—	2	1	no	—
15	#15688993	C₄	1	2	yes	—
16	#15688993	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	—	2	2	no	—
19	C₄₇×#432	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
22	C₂×#97×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	4	4	no	—
31	C₂×C₄⋊Dic₄₇	—	4	4	no	—
32	C₂×C₄⋊Dic₄₇	—	4	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	complemented
36	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
37	#13093935	—	2	4	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	—	2	2	no	—
51	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	complemented
52	C₂²⋊C₈×C₄₇	—	2	4	no	—
53	#97×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊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	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	C₄²	1	2	yes	—
81	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
90	C₄⋊Dic₄₇	—	4	4	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	—	4	4	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	—	4	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	1	no	—
100	C₂³×C₄₇	—	2	1	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	#95	1	2	yes	—
104	C₂³×C₄₇	#100	1	2	yes	—
105	C₂³×C₄₇	#101	1	2	yes	—
106	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	#95	1	2	yes	—
122	C₂×C₄×C₄₇	#95	1	2	yes	—
123	C₂×C₄×C₄₇	#95	1	2	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	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×C₄×C₄₇	#137	1	2	yes	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	8	8	no	—
138	C₈×C₄₇	—	4	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#8040	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	#343	1	4	yes	—
148	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
151	C₂²×C₄₇	#418	1	1	yes	characteristic
152	C₂²×C₄₇	#418	1	1	yes	characteristic
153	C₂²×C₄₇	#419	1	1	yes	characteristic
154	C₂²×C₄₇	#419	1	1	yes	characteristic
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	#417	1	2	yes	—
161	C₄×C₄₇	#417	1	2	yes	—
162	C₄×C₄₇	#428	1	2	yes	—
163	C₄×C₄₇	#428	1	2	yes	—
164	Dic₄₇	—	8	4	no	—
165	C₂×#432	—	47	1	no	—
166	C₂×#432	D₄₇	1	1	yes	characteristic
167	C₂×#384	—	47	1	no	—
168	#1296	—	47	4	no	—
169	C₂×C₄₇	—	2	4	no	—
170	C₂×C₄₇	#1555	1	1	yes	characteristic
171	C₂×C₄₇	#1556	1	1	yes	characteristic
172	C₂×C₄₇	#1556	1	1	yes	characteristic
173	C₂×C₄₇	#1558	1	1	yes	characteristic
174	C₂×C₄₇	#1736	1	1	yes	characteristic
175	C₂×C₄₇	#1736	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	—	47	2	no	—
178	C₂²×C₄⋊C₄	—	94	1	no	—
179	C₂²×C₄⋊C₄	—	94	1	no	—
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	#432	—	2	2	no	—
182	#432	—	47	4	no	—
183	#432	—	94	2	no	—
184	#432	Dic₄₇	1	4	yes	complemented
185	C₂×C₄⋊C₈	—	47	2	no	—
186	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
187	C₂×C₂²⋊C₈	—	2	1	no	—
188	C₂×C₂²⋊C₈	—	94	1	no	—
189	C₂×#97	—	2	1	no	—
190	C₂×#97	—	94	1	no	—
191	#384	—	47	2	no	—
192	#384	—	47	2	no	—
193	#384	—	94	1	no	—
194	#384	—	94	1	no	—
195	C₄₇	#8040	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	94	2	no	—
197	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₂×C₄⋊C₄	—	47	4	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₄⋊C₄	—	94	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	188	4	no	—
209	C₂×C₄⋊C₄	—	188	4	no	—
210	C₂×C₄⋊C₄	—	188	4	no	—
211	C₂×C₄²	Dic₉₄	1	2	yes	—
212	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₄₇	1	2	yes	—
214	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
216	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	47	4	no	—
221	C₄⋊C₈	—	94	2	no	—
222	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	complemented
223	C₂²⋊C₈	—	2	4	no	—
224	C₂²⋊C₈	—	94	4	no	—
225	#97	—	2	4	no	—
226	#97	—	94	4	no	—
227	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	188	4	no	—
236	C₂²×C₄	—	188	4	no	—
237	C₂²×C₄	—	188	4	no	—
238	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
239	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
240	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
241	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
242	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
246	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₄⋊C₄	—	188	4	no	—
257	C₄⋊C₄	—	188	4	no	—
258	C₄⋊C₄	—	188	4	no	—
259	C₄⋊C₄	—	188	4	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₄⋊C₄	—	188	4	no	—
262	C₄²	C₄×Dic₄₇	1	2	yes	—
263	C₄²	Dic₄₇⋊C₄	1	2	yes	—
264	C₄²	Dic₄₇⋊C₄	1	2	yes	—
265	C₄²	C₄⋊Dic₄₇	1	2	yes	—
266	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
267	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
268	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
269	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
271	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	C₄₇⋊#95	1	2	yes	—
277	C₂³	#13093940	1	2	yes	—
278	C₂³	#13093942	1	2	yes	—
279	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	376	8	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	—	188	4	no	—
297	C₂×C₄	—	188	4	no	—
298	C₂×C₄	C₄₇⋊#95	1	2	yes	—
299	C₂×C₄	C₄₇⋊#95	1	2	yes	—
300	C₂×C₄	C₄₇⋊#95	1	2	yes	—
301	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
307	C₂×C₄	#13094049	1	2	yes	—
308	C₂×C₄	#13094052	1	2	yes	—
309	C₂×C₄	#13094054	1	2	yes	—
310	C₂×C₄	#13094057	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₈	—	188	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	#15688891	1	4	yes	—
320	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
321	C₂²	C₂×#13093940	1	1	yes	characteristic
322	C₂²	C₂×#13093942	1	1	yes	characteristic
323	C₂²	#15689127	1	1	yes	characteristic
324	C₂²	#15689129	1	1	yes	characteristic
325	C₂²	#15689131	1	1	yes	characteristic
326	C₂²	#15689133	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	376	4	no	—
333	C₄	#15689124	1	2	yes	—
334	C₄	#15689125	1	2	yes	—
335	C₄	#15689165	1	2	yes	—
336	C₄	#15689166	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	#20242101	1	1	yes	characteristic
339	C₂	#20242104	1	1	yes	characteristic
340	C₂	#20242106	1	1	yes	characteristic
341	C₂	#20242113	1	1	yes	characteristic
342	C₂	#20243108	1	1	yes	characteristic
343	C₂	#20243109	1	1	yes	characteristic
344	C₂	C₂×#15688891	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#8040
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20241272	C₂	semidirect
9. #15689180	C₄	semidirect
19. C₄₇×#432	C₄	semidirect
35. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
51. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
184. #432	Dic₄₇	semidirect
195. C₄₇	#8040	semidirect
222. C₄⋊C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×#15688993	2
#20241272	2
C₂×C₄₇×#432	2
#8040	47