Subgroups — #26214922

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689254	C₂	1	2	yes	maximal
2	#20243263	C₂	1	2	yes	maximal
3	#20243263	C₂	1	2	yes	maximal
4	C₄⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal
5	#20243245	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#20243075	C₂	1	2	yes	maximal
8	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2295	C₂	1	1	yes	maximal, characteristic
10	#15689254	C₄	1	16	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	1	yes	characteristic
15	C₂×#13093935	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	2	yes	—
18	C₂×#13093935	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
23	#15688878	C₂²	1	1	yes	characteristic
24	#15688878	C₂²	1	1	yes	characteristic
25	#15688878	C₂²	1	2	yes	—
26	#15688878	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#391	C₄	1	2	yes	—
29	C₄₇×#389	C₄	1	2	yes	—
30	C₄₇×#387	C₄	1	4	yes	—
31	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#376	C₂²	1	2	yes	—
34	C₄³×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
42	#13093935	—	2	2	no	—
43	#13093935	—	2	4	no	—
44	#13093935	C₂×C₄	1	8	yes	—
45	#13093935	C₂×C₄	1	8	yes	—
46	#13093935	Q₈	1	4	yes	complemented
47	C₄×C₄₇⋊C₈	—	2	4	no	—
48	C₄×C₄₇⋊C₈	—	2	4	no	—
49	C₄×C₄₇⋊C₈	—	2	4	no	—
50	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
51	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#95×C₄₇	C₂×C₄	1	4	yes	—
65	#95×C₄₇	C₂×C₄	1	4	yes	—
66	#95×C₄₇	C₂×C₄	1	2	yes	—
67	#95×C₄₇	C₂×C₄	1	4	yes	—
68	C₂×Q₈×C₄₇	—	4	2	no	—
69	C₂×Q₈×C₄₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
81	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
103	C₂×C₄₇⋊C₈	#41	1	4	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	Q₈×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
122	C₂×C₄×C₄₇	#131	1	4	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₄₇⋊C₈	—	2	4	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	C₄₇⋊C₈	—	2	4	no	—
130	C₄₇⋊C₈	C₄×Q₈	1	8	yes	—
131	#20519	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#131	1	2	yes	—
136	C₂²×C₄₇	#568	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	—	2	4	no	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
145	C₄×C₄₇	C₄×#41	1	2	yes	—
146	C₂×#446	—	47	2	no	—
147	#2295	D₄₇	1	1	yes	characteristic
148	#1771	—	47	2	no	—
149	#1771	—	47	2	no	—
150	C₈×C₄⋊C₄	—	47	2	no	—
151	#1768	—	47	2	no	—
152	C₄×C₄⋊C₈	—	47	1	no	—
153	#1726	—	47	2	no	—
154	C₄×#97	—	47	1	no	—
155	C₂×C₄₇	C₄²×Q₈	1	1	yes	characteristic
156	C₂×C₄₇	#2873	1	1	yes	characteristic
157	C₂×C₄₇	C₄×#131	1	1	yes	characteristic
158	C₂×C₄₇	#2962	1	2	yes	—
159	C₂×C₄₇	#2968	1	2	yes	—
160	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
161	#446	—	47	16	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×C₄⋊C₈	—	47	2	no	—
164	C₂×C₄⋊C₈	—	47	2	no	—
165	C₂×#97	—	47	2	no	—
166	C₂×#97	—	47	2	no	—
167	C₂×#97	—	47	1	no	—
168	C₂×#97	—	47	1	no	—
169	C₂×#97	—	47	2	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	C₂×C₄×C₈	—	47	2	no	—
172	C₂×C₄×C₈	—	47	2	no	—
173	C₂×C₄×C₈	—	47	2	no	—
174	#391	Dic₄₇	1	2	yes	—
175	#389	Dic₄₇	1	2	yes	—
176	#387	Dic₄₇	1	4	yes	—
177	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
178	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
179	#376	C₂×D₄₇	1	2	yes	—
180	C₄³	C₂×D₄₇	1	1	yes	characteristic
181	#336	—	47	1	no	—
182	#336	—	47	1	no	—
183	#336	—	47	2	no	—
184	#336	—	47	2	no	—
185	C₄₇	#20519	1	1	yes	characteristic, complemented, Sylow
186	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₄×Q₈	C₄×D₄₇	1	8	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
197	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	47	8	no	—
204	C₄⋊C₈	—	47	8	no	—
205	C₄⋊C₈	—	47	8	no	—
206	#97	—	47	8	no	—
207	#97	—	47	8	no	—
208	#97	—	47	4	no	—
209	#97	—	94	4	no	—
210	#97	—	94	2	no	—
211	C₄×C₈	—	47	8	no	—
212	C₄×C₈	—	94	4	no	—
213	C₄×C₈	—	94	4	no	—
214	C₄×C₈	—	94	4	no	—
215	#95	C₂×Dic₄₇	1	4	yes	—
216	#95	C₂×Dic₄₇	1	4	yes	—
217	#95	C₂×Dic₄₇	1	2	yes	—
218	#95	C₂×Dic₄₇	1	4	yes	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	C₄×Dic₄₇	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	C₂×C₈	—	47	4	no	—
230	C₂×C₈	—	47	4	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	47	4	no	—
233	C₂×C₈	—	47	4	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	94	2	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	2	no	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
244	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×C₄×D₄₇	1	4	yes	—
251	C₄²	C₂×C₄×D₄₇	1	4	yes	—
252	C₄²	C₂×C₄×D₄₇	1	4	yes	—
253	C₄²	Q₈×D₄₇	1	2	yes	—
254	C₄²	#10625354	1	2	yes	—
255	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
269	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
271	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
272	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
273	C₂×C₄	#13094031	1	4	yes	—
274	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
275	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
276	C₂×C₄	#41×D₄₇	1	2	yes	—
277	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
278	C₈	—	47	8	no	—
279	C₈	—	94	4	no	—
280	C₈	—	94	4	no	—
281	C₈	—	94	4	no	—
282	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
283	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
284	C₂²	#41×Dic₄₇	1	1	yes	characteristic
285	C₂²	C₂×#13094031	1	2	yes	—
286	C₂²	#15689890	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	4	4	no	—
294	C₄	C₄×Q₈×D₄₇	1	2	yes	—
295	C₄	C₄×#10625354	1	2	yes	—
296	C₂	C₄×Q₈×Dic₄₇	1	1	yes	characteristic
297	C₂	#20251807	1	1	yes	characteristic
298	C₂	C₄×#13094031	1	1	yes	characteristic
299	C₂	#20252337	1	2	yes	—
300	C₂	#20252384	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2295	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2295	D₄₇
hypocenter	C₄₇	#20519
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
10. #15689254	C₄	semidirect
46. #13093935	Q₈	semidirect
185. C₄₇	#20519	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689254	2
#20243263	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243245	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243075	2
C₄×#13093935	2
C₄₇×#2295	2
#20519	47