Subgroups — #26213776

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689079	C₂	1	2	yes	maximal
2	#20243140	C₂	1	4	yes	maximal, complemented
3	#20243137	C₂	1	2	yes	maximal, complemented
4	#20243134	C₂	1	2	yes	maximal, complemented
5	#20243085	C₂	1	2	yes	maximal
6	#20243082	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
8	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	2	yes	—
10	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15689079	C₄	1	16	yes	complemented
12	C₂×#13093935	C₂²	1	4	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
14	#15688878	C₂²	1	1	yes	characteristic
15	#15688878	C₂²	1	1	yes	characteristic
16	#15688878	C₂²	1	2	yes	—
17	#15688878	C₂²	1	2	yes	—
18	#15688878	C₂²	1	2	yes	—
19	#15688878	C₂²	1	4	yes	—
20	#15688878	C₂²	1	4	yes	—
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
27	C₄₇×#376	C₂²	1	2	yes	—
28	C₄³×C₄₇	C₂²	1	2	yes	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	—	2	4	no	—
31	C₂×#10625329	C₂×C₄	1	8	yes	—
32	C₂²×C₄₇⋊C₈	—	2	4	no	—
33	C₂²×C₄₇⋊C₈	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	#13093935	C₂×C₄	1	16	yes	complemented
39	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	—	2	4	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	#95×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	#41	1	4	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
77	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	4	no	—
87	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
88	#10625329	—	2	16	no	—
89	C₂×C₄₇⋊C₈	—	2	16	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂³×C₄₇	#117	1	2	yes	—
103	C₂³×C₄₇	#117	1	2	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	8	no	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	#117	1	8	yes	—
120	C₂×C₄×C₄₇	#117	1	8	yes	—
121	C₂×C₄×C₄₇	#117	1	2	yes	—
122	C₂×C₄×C₄₇	#117	1	2	yes	—
123	C₂×C₄×C₄₇	#131	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	—	2	8	no	—
131	#20368	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	#376	1	4	yes	—
137	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#131	1	2	yes	—
141	C₂²×C₄₇	#568	1	2	yes	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	4	2	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	#376	1	4	yes	—
148	C₄×C₄₇	C₂×#117	1	2	yes	—
149	C₄×C₄₇	C₂×#117	1	2	yes	—
150	C₂×#405	—	47	2	no	—
151	C₄×#117	D₄₇	1	1	yes	characteristic
152	#1746	—	47	4	no	—
153	#1745	—	47	2	no	—
154	#1744	—	47	2	no	—
155	#1729	—	47	2	no	—
156	#1728	—	47	1	no	—
157	C₂×#336	—	47	1	no	—
158	C₂×C₄₇	—	2	4	no	—
159	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
160	C₂×C₄₇	C₄×#131	1	1	yes	characteristic
161	C₂×C₄₇	#2875	1	1	yes	characteristic
162	C₂×C₄₇	#2924	1	2	yes	—
163	C₂×C₄₇	#2925	1	2	yes	—
164	C₂²×M₁₆	—	47	2	no	—
165	C₂³×C₈	—	47	2	no	—
166	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
167	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	#405	—	47	16	no	—
170	C₂×#97	—	47	4	no	—
171	C₂×C₄×C₈	—	47	4	no	—
172	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
173	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
174	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
175	#376	C₂×D₄₇	1	2	yes	—
176	C₄³	C₂×D₄₇	1	2	yes	—
177	#336	—	47	1	no	—
178	#336	—	47	1	no	—
179	#336	—	47	2	no	—
180	#336	—	47	2	no	—
181	#336	—	47	2	no	—
182	#336	—	47	4	no	—
183	#336	—	47	4	no	—
184	C₄₇	#20368	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
186	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×M₁₆	—	47	8	no	—
188	C₂×M₁₆	—	94	4	no	—
189	C₂×M₁₆	—	94	4	no	—
190	C₂²×C₈	—	47	8	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	2	no	—
193	C₂²×C₈	—	47	4	no	—
194	C₂²×C₈	—	94	4	no	—
195	C₂²×C₈	—	94	4	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	C₄×D₄₇	1	8	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
212	#97	—	47	16	no	—
213	C₄×C₈	—	47	16	no	—
214	#95	C₂×Dic₄₇	1	4	yes	—
215	#95	C₂×Dic₄₇	1	4	yes	—
216	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
221	C₂²×C₄	#10625350	1	4	yes	—
222	C₂²×C₄	#10625350	1	2	yes	—
223	C₂²×C₄	#10625350	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	M₁₆	—	94	16	no	—
232	C₂×C₈	—	47	4	no	—
233	C₂×C₈	—	47	8	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	94	16	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	94	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
244	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂×C₄×D₄₇	1	8	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
258	C₂³	#13093983	1	2	yes	—
259	C₂³	#13093983	1	2	yes	—
260	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
274	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
275	C₂×C₄	#13093981	1	8	yes	—
276	C₂×C₄	#13093981	1	8	yes	—
277	C₂×C₄	#13093983	1	2	yes	—
278	C₂×C₄	#13093983	1	2	yes	—
279	C₂×C₄	#13094031	1	4	yes	—
280	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
281	C₂×C₄	C₂×#10625350	1	2	yes	—
282	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
283	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
284	C₈	—	94	4	no	—
285	C₈	—	94	4	no	—
286	C₈	—	94	8	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	#15688969	1	4	yes	—
292	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
293	C₂²	C₂×#13093983	1	1	yes	characteristic
294	C₂²	C₂×#13093983	1	1	yes	characteristic
295	C₂²	C₂×#13094031	1	2	yes	—
296	C₂²	#15689890	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	#15688969	1	4	yes	—
303	C₄	C₂×#13093981	1	2	yes	—
304	C₄	C₂×#13093981	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₂×#15688969	1	1	yes	characteristic
307	C₂	C₄×#13094031	1	1	yes	characteristic
308	C₂	#20251816	1	1	yes	characteristic
309	C₂	#20252046	1	2	yes	—
310	C₂	#20252054	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20368
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243140	C₂	semidirect
3. #20243137	C₂	semidirect
4. #20243134	C₂	semidirect
11. #15689079	C₄	semidirect
38. #13093935	C₂×C₄	semidirect
39. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
184. C₄₇	#20368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689079	2
#20243140	2
#20243137	2
#20243134	2
#20243085	2
#20243082	2
C₂×#15688878	2
C₄×#117×C₄₇	2
#20368	47