Subgroups — #26226640

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	2	yes	maximal
2	#20244112	C₂	1	4	yes	maximal, complemented
3	#20244107	C₂	1	4	yes	maximal, complemented
4	#20243372	C₂	1	2	yes	maximal
5	#20243368	C₂	1	2	yes	maximal
6	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
7	#15689184	—	2	8	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688878	C₂²	1	2	yes	—
16	#15688878	C₂²	1	4	yes	—
17	#15688878	C₂²	1	2	yes	—
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#389	C₄	1	2	yes	—
22	C₄₇×#389	C₄	1	2	yes	—
23	C₄₇×#385	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
29	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
32	#13093935	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	#117×C₄₇	—	4	2	no	—
37	#117×C₄₇	D₄	1	4	yes	—
38	#117×C₄₇	Q₈	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	—	4	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	#95×C₄₇	C₂×C₄	1	4	yes	—
52	#95×C₄₇	C₂×C₄	1	4	yes	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	—	4	4	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
71	C₂²⋊C₄×C₄₇	—	4	2	no	—
72	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	2	no	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₄²×C₄₇	—	4	1	no	—
78	C₄²×C₄₇	C₂×D₄	1	4	yes	—
79	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂³×C₄₇	—	2	1	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	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	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	#131	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
113	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
114	C₄₇⋊C₈	—	4	4	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#21957	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	4	2	no	—
120	C₂²×C₄₇	—	4	1	no	—
121	C₂²×C₄₇	—	4	1	no	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	#518	1	1	yes	characteristic
124	C₂²×C₄₇	#519	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#131	1	2	yes	—
126	C₂²×C₄₇	#568	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	4	4	no	—
131	C₄×C₄₇	—	4	2	no	—
132	C₄×C₄₇	#537	1	2	yes	—
133	C₄×C₄₇	#544	1	2	yes	—
134	C₂×#433	—	47	2	no	—
135	#2296	D₄₇	1	1	yes	characteristic
136	#1940	—	47	4	no	—
137	#1939	—	47	4	no	—
138	#1798	—	47	2	no	—
139	#1797	—	47	2	no	—
140	C₂×C₄₇	—	4	2	no	—
141	C₂×C₄₇	#2316	1	1	yes	characteristic
142	C₂×C₄₇	#2906	1	2	yes	—
143	C₂×C₄₇	#2976	1	2	yes	—
144	C₂×C₄₇	#2986	1	1	yes	characteristic
145	C₂×C₄₇	#2987	1	1	yes	characteristic
146	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
147	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
148	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
149	#433	—	94	8	no	—
150	C₂×C₄⋊C₈	—	47	4	no	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×C₄⋊C₈	—	47	4	no	—
154	C₂×C₂²⋊C₈	—	47	4	no	—
155	C₂×#97	—	47	2	no	—
156	C₂×C₄×C₈	—	47	2	no	—
157	#389	Dic₄₇	1	2	yes	—
158	#389	Dic₄₇	1	2	yes	—
159	#385	Dic₄₇	1	4	yes	—
160	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
161	#336	—	47	4	no	—
162	#336	—	47	2	no	—
163	#336	—	47	2	no	—
164	C₄₇	#21957	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	—	47	4	no	—
168	C₂²×C₈	—	47	4	no	—
169	#117	—	4	2	no	—
170	#117	Dic₉₄	1	4	yes	—
171	#117	D₁₈₈	1	4	yes	—
172	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
178	C₂×C₄²	—	4	2	no	—
179	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
183	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
184	C₄⋊C₈	—	94	8	no	—
185	C₄⋊C₈	—	188	2	no	—
186	C₄⋊C₈	—	188	2	no	—
187	C₄⋊C₈	—	188	4	no	—
188	C₂²⋊C₈	—	94	8	no	—
189	#97	—	94	4	no	—
190	C₄×C₈	—	94	4	no	—
191	#95	C₂×Dic₄₇	1	4	yes	—
192	#95	C₂×Dic₄₇	1	4	yes	—
193	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
200	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
203	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
204	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	—	94	4	no	—
212	C₂×C₈	—	94	4	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	C₄⋊Dic₄₇	1	8	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₄²	C₂×Dic₉₄	1	4	yes	—
225	C₄²	C₂×D₁₈₈	1	4	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
231	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
248	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
249	C₂×C₄	#13094031	1	4	yes	—
250	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
251	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
252	C₂×C₄	#13094069	1	2	yes	—
253	C₂×C₄	#13094072	1	2	yes	—
254	C₈	—	188	4	no	—
255	C₈	—	188	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
262	C₂²	#15689598	1	1	yes	characteristic
263	C₂²	#15689601	1	1	yes	characteristic
264	C₂²	C₂×#13094031	1	2	yes	—
265	C₂²	#15689890	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	2	8	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	4	no	—
271	C₄	#15689704	1	2	yes	—
272	C₄	#15689754	1	2	yes	—
273	C₂	—	4	2	no	—
274	C₂	#20246006	1	1	yes	characteristic
275	C₂	#20251956	1	2	yes	—
276	C₂	#20252436	1	2	yes	—
277	C₂	#20252490	1	1	yes	characteristic
278	C₂	#20252494	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2296	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#21957
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2296

Complements

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

N	H ≅ G/N	type
2. #20244112	C₂	semidirect
3. #20244107	C₂	semidirect
164. C₄₇	#21957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
#20244112	2
#20244107	2
#20243372	2
#20243368	2
C₄₇×#2296	2
#21957	47