Subgroups — #26216757

Order 12032.

890 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×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243541	C₂	1	2	yes	maximal, complemented
4	#20243507	C₂	1	2	yes	maximal, complemented
5	#20243370	C₂	1	4	yes	maximal
6	C₄×Dic₄₇⋊C₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689176	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
17	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
18	#15688969	C₂²	1	1	yes	characteristic
19	#15688969	C₂²	1	1	yes	characteristic
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#431	—	2	4	no	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	#13093983	—	2	8	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	Dic₄₇⋊C₈	—	2	16	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
38	C₄×C₄₇⋊C₈	—	2	4	no	—
39	C₄₇⋊#95	C₂×C₄	1	2	yes	—
40	C₄₇⋊#95	C₂×C₄	1	2	yes	—
41	C₄₇⋊#95	C₂×C₄	1	2	yes	—
42	C₄₇⋊#95	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	4	yes	—
50	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	4	no	—
54	C₂²⋊C₈×C₄₇	—	2	4	no	—
55	C₄×C₈×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂×D₄	1	2	yes	—
78	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
80	C₂²⋊Dic₄₇	—	2	8	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄×Dic₄₇	—	2	8	no	—
83	C₄×Dic₄₇	—	2	8	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	#41	1	8	yes	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	8	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	#131	1	4	yes	—
106	C₂×C₄×C₄₇	#131	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	#117	1	8	yes	—
116	C₈×C₄₇	—	4	4	no	—
117	C₄₇⋊C₈	—	4	4	no	—
118	#20809	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	4	no	—
123	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#117	1	2	yes	—
125	C₂²×C₄₇	C₂×#131	1	2	yes	—
126	C₂²×C₄₇	C₂×#131	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	#424	1	4	yes	—
131	C₄×C₄₇	#424	1	4	yes	—
132	C₄×C₄₇	#532	1	2	yes	—
133	C₄×C₄₇	#533	1	2	yes	—
134	Dic₄₇	—	2	8	no	—
135	Dic₄₇	—	4	4	no	—
136	C₂×#431	—	47	1	no	—
137	C₂×#431	D₄₇	1	1	yes	characteristic
138	C₄×#117	—	47	1	no	—
139	#1843	—	47	2	no	—
140	#1832	—	47	2	no	—
141	#1798	—	47	4	no	—
142	C₄×C₄⋊C₈	—	47	4	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#2285	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
147	C₂×C₄₇	#2922	1	4	yes	—
148	C₂×#117	—	47	2	no	—
149	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	#431	—	2	4	no	—
151	#431	—	94	4	no	—
152	C₂×C₄⋊C₈	—	47	8	no	—
153	C₂×C₄⋊C₈	—	47	2	no	—
154	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
155	C₂×C₂²⋊C₈	—	47	2	no	—
156	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
157	C₂×C₄×C₈	—	47	2	no	—
158	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
159	C₄×C₄⋊C₄	—	47	2	no	—
160	C₄×C₄⋊C₄	—	47	2	no	—
161	C₄×C₂²⋊C₄	—	47	2	no	—
162	C₄×C₂²⋊C₄	—	47	2	no	—
163	#376	—	47	1	no	—
164	#376	—	47	1	no	—
165	C₄³	—	47	2	no	—
166	#336	—	47	8	no	—
167	C₄₇	#20809	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
169	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
173	#117	—	94	8	no	—
174	C₂×C₄⋊C₄	—	47	4	no	—
175	C₂×C₂²⋊C₄	—	47	4	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	4	no	—
178	C₂×C₄²	—	47	4	no	—
179	C₂×C₄²	—	47	2	no	—
180	C₂×C₄²	Dic₉₄	1	4	yes	—
181	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	94	16	no	—
188	C₄⋊C₈	—	94	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	94	4	no	—
191	C₄×C₈	—	2	4	no	—
192	C₄×C₈	—	94	4	no	—
193	#95	—	47	2	no	—
194	#95	—	47	2	no	—
195	#95	—	47	2	no	—
196	#95	—	47	2	no	—
197	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	47	4	no	—
202	C₂²×C₄	—	47	4	no	—
203	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
204	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
205	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	8	no	—
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	—	94	8	no	—
217	C₄⋊C₄	—	94	8	no	—
218	C₂²⋊C₄	—	94	8	no	—
219	C₄²	—	47	8	no	—
220	C₄²	—	94	8	no	—
221	C₄²	—	94	8	no	—
222	C₄²	—	94	4	no	—
223	C₄²	Dic₄₇⋊C₄	1	4	yes	—
224	C₄²	Dic₄₇⋊C₄	1	4	yes	—
225	C₄²	C₂×Dic₉₄	1	2	yes	—
226	C₄²	C₂×Dic₉₄	1	2	yes	—
227	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
228	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
233	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	47	8	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
249	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
250	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
251	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
252	C₂×C₄	#13094030	1	4	yes	—
253	C₂×C₄	#13094032	1	4	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
256	C₂×C₄	#41×D₄₇	1	4	yes	—
257	C₈	—	4	4	no	—
258	C₈	—	188	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
264	C₂²	#117×D₄₇	1	2	yes	—
265	C₂²	C₂×#13094030	1	2	yes	—
266	C₂²	C₂×#13094032	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	94	8	no	—
271	C₄	—	188	4	no	—
272	C₄	#15689149	1	4	yes	—
273	C₄	#15689151	1	4	yes	—
274	C₄	#15689669	1	2	yes	—
275	C₄	#15689676	1	2	yes	—
276	C₂	—	4	2	no	—
277	C₂	#20245791	1	1	yes	characteristic
278	C₂	C₂×#15689149	1	1	yes	characteristic
279	C₂	C₂×#15689151	1	1	yes	characteristic
280	C₂	#20252027	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#431	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#20809
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20243541	C₂	semidirect
4. #20243507	C₂	semidirect
6. C₄×Dic₄₇⋊C₈	C₂	semidirect
167. C₄₇	#20809	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₄×#13093983	2
#20243541	2
#20243507	2
#20243370	2
C₄×Dic₄₇⋊C₈	2
C₂×C₄₇×#431	2
#20809	47