Subgroups — #26221087

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20244107	C₂	1	4	yes	maximal, complemented
4	#20243573	C₂	1	4	yes	maximal
5	#20243245	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2283	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689176	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₄×#10625329	—	2	4	no	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688878	C₂²	1	8	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#383	C₄	1	2	yes	—
19	C₄₇×#382	C₄	1	2	yes	—
20	C₄₇×#381	C₄	1	2	yes	—
21	C₄₇×#380	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₄₇×#376	C₂²	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	8	no	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
31	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
32	#13093935	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	4	yes	—
43	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#95×C₄₇	C₂×C₄	1	4	yes	—
48	#95×C₄₇	C₂×C₄	1	4	yes	—
49	#95×C₄₇	C₂×C₄	1	4	yes	—
50	#95×C₄₇	C₂×C₄	1	4	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₂²⋊C₄×C₄₇	—	4	4	no	—
68	C₄²×C₄₇	—	2	8	no	—
69	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
71	C₄²×C₄₇	C₂×D₄	1	2	yes	—
72	C₄²×C₄₇	C₂×D₄	1	2	yes	—
73	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
75	#10625329	—	2	16	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	8	no	—
80	C₂×C₄₇⋊C₈	#41	1	8	yes	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	#131	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
106	C₄₇⋊C₈	—	2	8	no	—
107	C₄₇⋊C₈	—	4	4	no	—
108	#21354	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	2	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	4	4	no	—
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₄×#41	1	2	yes	—
115	C₂²×C₄₇	C₂×#131	1	2	yes	—
116	C₂²×C₄₇	#568	1	2	yes	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	4	8	no	—
122	C₄×C₄₇	#424	1	4	yes	—
123	C₄×C₄₇	#525	1	2	yes	—
124	C₄×C₄₇	#533	1	2	yes	—
125	C₂×#431	—	47	1	no	—
126	C₂×C₄×M₁₆	—	47	1	no	—
127	#2283	D₄₇	1	1	yes	characteristic
128	#1939	—	47	4	no	—
129	#1851	—	47	4	no	—
130	#1768	—	47	4	no	—
131	C₂×C₄₇	—	4	2	no	—
132	C₂×C₄₇	#2296	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
134	C₂×C₄₇	#2905	1	1	yes	characteristic
135	C₂×C₄₇	#2969	1	4	yes	—
136	C₂²×M₁₆	—	47	2	no	—
137	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
138	#431	—	94	4	no	—
139	C₂×C₄⋊C₈	—	47	8	no	—
140	C₂×C₄⋊C₈	—	47	2	no	—
141	C₂×C₂²⋊C₈	—	47	2	no	—
142	C₄×M₁₆	—	94	4	no	—
143	C₂×#97	—	47	2	no	—
144	C₂×C₄×C₈	—	47	2	no	—
145	C₂×C₄×C₈	—	47	2	no	—
146	#383	Dic₄₇	1	2	yes	—
147	#382	Dic₄₇	1	2	yes	—
148	#381	Dic₄₇	1	2	yes	—
149	#380	Dic₄₇	1	2	yes	—
150	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
151	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
152	#376	C₂×D₄₇	1	2	yes	—
153	#336	—	47	8	no	—
154	C₄₇	#21354	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
158	C₂×M₁₆	—	94	8	no	—
159	C₂×M₁₆	—	94	4	no	—
160	C₂²×C₈	—	47	4	no	—
161	C₂²×C₈	—	47	4	no	—
162	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
164	C₂×C₄²	Dic₉₄	1	4	yes	—
165	C₂×C₄²	D₁₈₈	1	4	yes	—
166	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₄⋊C₈	—	94	16	no	—
172	C₄⋊C₈	—	94	4	no	—
173	C₂²⋊C₈	—	94	4	no	—
174	#97	—	94	4	no	—
175	C₄×C₈	—	94	4	no	—
176	C₄×C₈	—	94	4	no	—
177	#95	C₂×Dic₄₇	1	4	yes	—
178	#95	C₂×Dic₄₇	1	4	yes	—
179	#95	C₂×Dic₄₇	1	4	yes	—
180	#95	C₂×Dic₄₇	1	4	yes	—
181	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
186	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
187	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
188	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
190	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	M₁₆	—	94	16	no	—
197	C₂×C₈	—	47	8	no	—
198	C₂×C₈	—	94	4	no	—
199	C₂×C₈	—	94	4	no	—
200	C₂×C₈	—	94	4	no	—
201	C₂×C₈	—	94	8	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊Dic₄₇	1	4	yes	—
206	C₄²	C₄⋊Dic₄₇	1	4	yes	—
207	C₄²	C₂×Dic₉₄	1	2	yes	—
208	C₄²	C₂×Dic₉₄	1	2	yes	—
209	C₄²	C₂×D₁₈₈	1	2	yes	—
210	C₄²	C₂×D₁₈₈	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
215	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
232	C₂×C₄	#13094031	1	4	yes	—
233	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
234	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
235	C₂×C₄	#41×D₄₇	1	4	yes	—
236	C₈	—	94	8	no	—
237	C₈	—	188	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
243	C₂²	#41×Dic₄₇	1	2	yes	—
244	C₂²	C₂×#13094031	1	2	yes	—
245	C₂²	#15689890	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	#15689150	1	4	yes	—
252	C₄	#15689635	1	2	yes	—
253	C₄	#15689676	1	2	yes	—
254	C₂	—	4	2	no	—
255	C₂	#20245872	1	1	yes	characteristic
256	C₂	C₂×#15689150	1	1	yes	characteristic
257	C₂	#20251952	1	1	yes	characteristic
258	C₂	#20252393	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2283	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2283	D₄₇
hypocenter	C₄₇	#21354
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #20244107	C₂	semidirect
5. #20243245	C₂	semidirect
154. C₄₇	#21354	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
C₂×C₄×#10625329	2
#20244107	2
#20243573	2
#20243245	2
C₄₇×#2283	2
#21354	47