Subgroups — #26122255

Order 12032.

1062 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×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688882	C₂	1	2	yes	maximal
3	C₂×#15688882	C₂	1	2	yes	maximal
4	#20243082	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	—	2	2	no	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	C₄²×Dic₄₇	—	2	4	no	—
10	#15688882	—	2	4	no	—
11	#15688882	C₄	1	8	yes	complemented
12	#15688878	—	2	2	no	—
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#389	C₄	1	4	yes	complemented
17	C₄₇×#385	—	2	2	no	—
18	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	8	no	—
28	C₂×C₄×Dic₄₇	—	2	16	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	—	2	1	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	#95×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×Dic₄₇	—	2	16	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	4	8	no	—
75	C₄⋊C₄×C₄₇	C₄²	1	4	yes	complemented
76	C₂²⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	1	no	—
78	C₂²⋊C₄×C₄₇	—	4	1	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	4	2	no	—
83	C₄×Dic₄₇	—	2	32	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	complemented
88	#10625329	—	2	4	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	4	2	no	—
91	#10625329	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	#120	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	#123	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	#104	1	8	yes	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	#117	1	2	yes	—
113	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#121	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#125	1	2	yes	—
124	C₂×C₄×C₄₇	#126	1	2	yes	—
125	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	16	no	—
127	C₂×Dic₄₇	—	2	16	no	—
128	C₂×Dic₄₇	—	2	16	no	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₄₇⋊C₈	—	4	4	no	—
131	#10878	—	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	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#385	1	2	yes	—
138	C₂²×C₄₇	#386	1	1	yes	characteristic
139	C₂²×C₄₇	#387	1	2	yes	—
140	C₂²×C₄₇	#391	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#104	1	4	yes	—
142	C₄×C₄₇	—	2	8	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	—	8	2	no	—
145	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
146	C₄×C₄₇	#387	1	2	yes	—
147	C₄×C₄₇	#388	1	2	yes	—
148	C₄×C₄₇	#389	1	2	yes	—
149	Dic₄₇	—	2	16	no	—
150	#2296	D₄₇	1	1	yes	characteristic
151	C₂×C₄³	—	47	1	no	—
152	C₂×#339	—	47	2	no	—
153	C₂×#339	—	47	2	no	—
154	#1728	—	47	1	no	—
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	#1436	1	1	yes	characteristic
158	C₂×C₄₇	C₄×#104	1	2	yes	—
159	C₂×C₄₇	#1896	1	2	yes	—
160	C₂×C₄₇	#1994	1	2	yes	—
161	C₂²×M₁₆	—	47	2	no	—
162	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
163	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
164	C₂²×C₄²	—	47	2	no	—
165	C₂²×C₄²	—	94	2	no	—
166	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
167	#389	Dic₄₇	1	4	yes	complemented
168	#385	—	2	2	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄³	—	94	4	no	—
171	#339	—	47	8	no	—
172	#339	—	94	4	no	—
173	#336	—	94	2	no	—
174	C₄₇	#10878	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	94	2	no	—
176	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₄×D₄₇	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	47	4	no	—
199	C₂×C₄²	—	47	4	no	—
200	C₂×C₄²	—	47	4	no	—
201	C₂×C₄²	—	94	8	no	—
202	C₂×C₄²	—	94	16	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
208	#95	—	2	4	no	—
209	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	94	16	no	—
216	C₂²×C₄	—	94	4	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	94	4	no	—
219	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
220	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
221	C₂²×C₄	#10625350	1	2	yes	—
222	C₂²×C₄	D₄×D₄₇	1	2	yes	—
223	C₂²×C₄	#10625352	1	2	yes	—
224	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
225	C₂²×C₄	#10625354	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	M₁₆	—	94	4	no	—
229	M₁₆	—	94	4	no	—
230	M₁₆	—	188	2	no	—
231	M₁₆	—	188	2	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	—	188	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	complemented
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	47	8	no	—
247	C₄²	—	94	32	no	—
248	C₄²	—	94	4	no	—
249	C₄²	—	94	4	no	—
250	C₄²	—	94	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
254	C₂³	#13093992	1	1	yes	characteristic
255	C₂³	#13094000	1	2	yes	—
256	C₂³	#13094005	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	94	16	no	—
266	C₂×C₄	—	94	16	no	—
267	C₂×C₄	—	94	16	no	—
268	C₂×C₄	—	94	8	no	—
269	C₂×C₄	#13093951	1	8	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
272	C₂×C₄	#13093982	1	2	yes	—
273	C₂×C₄	#13093983	1	1	yes	characteristic
274	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
275	C₂×C₄	#13093986	1	2	yes	—
276	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
277	C₂×C₄	#13093989	1	2	yes	—
278	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
279	C₂×C₄	#13093996	1	2	yes	—
280	C₂×C₄	#13094002	1	2	yes	—
281	C₂×C₄	#13094009	1	2	yes	—
282	C₂×C₄	#13094013	1	1	yes	characteristic
283	C₂×C₄	#13094014	1	2	yes	—
284	C₂×C₄	#13094019	1	2	yes	—
285	C₂×C₄	#13094022	1	1	yes	characteristic
286	C₈	—	188	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
292	C₂²	#15688994	1	2	yes	—
293	C₂²	#15689003	1	1	yes	characteristic
294	C₂²	#15689004	1	2	yes	—
295	C₂²	#15689030	1	1	yes	characteristic
296	C₂²	C₂×#13093951	1	4	yes	—
297	C₄	—	2	8	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	94	16	no	—
301	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
302	C₄	#15689009	1	2	yes	—
303	C₄	#15689013	1	2	yes	—
304	C₄	#15689018	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	#20241620	1	1	yes	characteristic
308	C₂	C₄×#13093951	1	2	yes	—
309	C₂	#20243832	1	2	yes	—
310	C₂	#20244366	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2296	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#10878
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2296

Complements

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

N	H ≅ G/N	type
11. #15688882	C₄	semidirect
16. C₄₇×#389	C₄	semidirect
75. C₄⋊C₄×C₄₇	C₄²	semidirect
87. C₄×Dic₄₇	C₄⋊C₄	semidirect
167. #389	Dic₄₇	semidirect
174. C₄₇	#10878	semidirect
238. C₄⋊C₄	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×Dic₄₇	2
C₂×#15688882	2
#20243082	2
C₄₇×#2296	2
#10878	47