Subgroups — #26147426

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688989	C₂	1	2	yes	maximal
2	#20241653	C₂	1	4	yes	maximal
3	#20241645	C₂	1	4	yes	maximal
4	#20241641	C₂	1	4	yes	maximal
5	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	#15688989	—	2	8	no	—
10	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
13	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#382	C₄	1	8	yes	complemented
17	C₂×#95×C₄₇	C₂²	1	4	yes	—
18	C₂³×Dic₄₇	C₂³	1	4	yes	—
19	C₂³×Dic₄₇	C₂³	1	4	yes	—
20	C₂×C₄⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄⋊Dic₄₇	—	2	4	no	—
22	C₂×C₄⋊Dic₄₇	—	4	4	no	—
23	C₂×C₄⋊Dic₄₇	—	4	4	no	—
24	C₂×C₄⋊Dic₄₇	—	4	8	no	—
25	C₂×C₄⋊Dic₄₇	—	4	4	no	—
26	C₄₇⋊#95	—	2	8	no	—
27	C₄₇⋊#95	—	2	8	no	—
28	C₄₇⋊#95	—	4	4	no	—
29	C₄₇⋊#95	—	4	8	no	—
30	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	D₄	1	4	yes	—
38	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
39	#95×C₄₇	C₂×C₄	1	16	yes	—
40	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	—	2	1	no	—
42	C₂²×C₄×C₄₇	—	2	1	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	2	8	no	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	4	8	no	—
69	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
70	C₄⋊Dic₄₇	—	4	8	no	—
71	C₄⋊Dic₄₇	—	4	8	no	—
72	C₄⋊Dic₄₇	—	4	16	no	—
73	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
77	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
79	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
80	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
81	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	2	16	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	C₂×Dic₄₇	—	4	4	no	—
102	C₂×Dic₄₇	—	4	4	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	8	no	—
106	C₂×Dic₄₇	—	4	8	no	—
107	C₂×Dic₄₇	—	4	8	no	—
108	C₂×Dic₄₇	—	4	8	no	—
109	#14677	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄₇	#518	1	1	yes	characteristic
112	C₂²×C₄₇	#519	1	1	yes	characteristic
113	C₂²×C₄₇	#520	1	4	yes	—
114	C₂²×C₄₇	#535	1	2	yes	—
115	C₂²×C₄₇	#536	1	2	yes	—
116	C₂²×C₄₇	#537	1	1	yes	characteristic
117	C₂²×C₄₇	#537	1	1	yes	characteristic
118	C₂²×C₄₇	#544	1	4	yes	—
119	C₂²×C₄₇	#544	1	1	yes	characteristic
120	C₂²×C₄₇	#544	1	1	yes	characteristic
121	C₂²×C₄₇	H(3,𝔽₄)	1	2	yes	—
122	C₂²×C₄₇	#562	1	4	yes	—
123	C₂²×C₄₇	#562	1	2	yes	—
124	C₂²×C₄₇	#563	1	2	yes	—
125	C₂²×C₄₇	#563	1	4	yes	—
126	C₂²×C₄₇	#564	1	2	yes	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	—	4	8	no	—
130	Dic₄₇	—	8	4	no	—
131	Dic₄₇	—	8	4	no	—
132	C₂×#382	—	47	2	no	—
133	C₂×#382	D₄₇	1	1	yes	characteristic
134	#1444	—	47	4	no	—
135	#1442	—	47	4	no	—
136	#1441	—	47	4	no	—
137	C₂×C₄₇	#2316	1	1	yes	characteristic
138	C₂×C₄₇	#2317	1	2	yes	—
139	C₂×C₄₇	#2379	1	2	yes	—
140	C₂×C₄₇	#2382	1	2	yes	—
141	C₂×C₄₇	#2819	1	1	yes	characteristic
142	C₂×C₄₇	#2820	1	2	yes	—
143	C₂×C₄₇	#2822	1	1	yes	characteristic
144	C₂×C₄₇	#2844	1	2	yes	—
145	C₂×C₄₇	#2845	1	2	yes	—
146	C₂²×C₄⋊C₄	—	47	2	no	—
147	C₂²×C₄⋊C₄	—	47	2	no	—
148	C₂²×C₄⋊C₄	—	47	4	no	—
149	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	#382	—	94	8	no	—
152	#382	Dic₄₇	1	8	yes	complemented
153	C₂×#95	—	47	4	no	—
154	C₂×#95	—	47	4	no	—
155	C₂×#95	—	47	4	no	—
156	C₂×#95	—	47	8	no	—
157	C₂×#95	C₂×D₄₇	1	4	yes	—
158	C₄₇	#14677	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	47	4	no	—
160	C₂³×C₄	—	47	4	no	—
161	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	4	2	no	—
165	C₂×C₄⋊C₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	94	4	no	—
167	C₂×C₄⋊C₄	—	94	4	no	—
168	C₂×C₄⋊C₄	—	188	4	no	—
169	C₂×C₄⋊C₄	—	188	4	no	—
170	C₂×C₄⋊C₄	—	188	4	no	—
171	C₂×C₄⋊C₄	—	188	8	no	—
172	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
173	C₂×C₄²	Dic₉₄	1	4	yes	—
174	C₂×C₄²	D₁₈₈	1	4	yes	—
175	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
176	#95	—	94	8	no	—
177	#95	—	94	8	no	—
178	#95	—	188	4	no	—
179	#95	—	188	8	no	—
180	#95	C₂×Dic₄₇	1	16	yes	—
181	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	94	4	no	—
193	C₂²×C₄	—	94	4	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	94	8	no	—
197	C₂²×C₄	—	94	8	no	—
198	C₂²×C₄	—	94	8	no	—
199	C₂²×C₄	—	94	8	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
203	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
209	C₄⋊C₄	—	4	8	no	—
210	C₄⋊C₄	—	188	8	no	—
211	C₄⋊C₄	—	188	8	no	—
212	C₄⋊C₄	—	188	16	no	—
213	C₄²	C₄⋊Dic₄₇	1	16	yes	complemented
214	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
215	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
216	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
217	C₂³	#13094069	1	1	yes	characteristic
218	C₂³	#13094069	1	1	yes	characteristic
219	C₂³	#13094070	1	4	yes	—
220	C₂³	#13094071	1	4	yes	—
221	C₂³	#13094072	1	1	yes	characteristic
222	C₂³	#13094072	1	1	yes	characteristic
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	188	8	no	—
239	C₂×C₄	—	188	8	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	—	188	8	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	—	188	4	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
250	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
251	C₂²	#15689598	1	1	yes	characteristic
252	C₂²	#15689601	1	1	yes	characteristic
253	C₂²	#15689606	1	4	yes	—
254	C₂²	#15689692	1	2	yes	—
255	C₂²	#15689699	1	2	yes	—
256	C₂²	#15689704	1	1	yes	characteristic
257	C₂²	#15689704	1	1	yes	characteristic
258	C₂²	#15689751	1	4	yes	—
259	C₂²	#15689754	1	1	yes	characteristic
260	C₂²	#15689754	1	1	yes	characteristic
261	C₂²	#15689865	1	2	yes	—
262	C₂²	#15689867	1	4	yes	—
263	C₂²	#15689871	1	2	yes	—
264	C₂²	#15689873	1	2	yes	—
265	C₂²	#15689877	1	4	yes	—
266	C₂²	C₄₇⋊#564	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	2	8	no	—
269	C₄	—	4	8	no	—
270	C₄	—	376	4	no	—
271	C₄	—	376	4	no	—
272	C₂	#20246006	1	1	yes	characteristic
273	C₂	#20246011	1	2	yes	—
274	C₂	#20246562	1	2	yes	—
275	C₂	#20246574	1	2	yes	—
276	C₂	#20251434	1	1	yes	characteristic
277	C₂	#20251440	1	2	yes	—
278	C₂	#20251456	1	1	yes	characteristic
279	C₂	#20251660	1	2	yes	—
280	C₂	#20251663	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	#14677
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
16. C₄₇×#382	C₄	semidirect
69. C₄²×C₄₇	C₄⋊C₄	semidirect
152. #382	Dic₄₇	semidirect
158. C₄₇	#14677	semidirect
213. C₄²	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×#15688989	2
#20241653	2
#20241645	2
#20241641	2
C₂×C₄₇×#382	2
#14677	47