Subgroups — #26215714

Order 12032.

1314 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	2	yes	maximal
2	C₂²×#13093943	C₂	1	2	yes	maximal
3	#20243384	C₂	1	8	yes	maximal, complemented
4	#20243127	C₂	1	2	yes	maximal
5	C₂×C₄₇×#519	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	4	yes	—
7	#15689150	—	2	8	no	—
8	#15689147	C₂²	1	8	yes	—
9	C₂×#13093943	—	2	8	no	—
10	C₂×#13093943	C₂²	1	8	yes	—
11	#15688892	C₂²	1	8	yes	—
12	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#519	—	2	8	no	—
14	C₂×C₄×Q₈×C₄₇	C₄	1	4	yes	—
15	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
16	C₂×#117×C₄₇	C₄	1	4	yes	—
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	2	4	no	—
20	C₂×#10625329	—	2	8	no	—
21	C₂×#10625329	—	2	8	no	—
22	C₂×#10625329	C₂³	1	8	yes	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
25	#13093943	—	2	16	no	—
26	C₂²×Q₈×C₄₇	—	2	4	no	—
27	C₂²×Q₈×C₄₇	—	2	4	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×Q₈×C₄₇	D₄	1	4	yes	—
30	C₂²×Q₈×C₄₇	Q₈	1	4	yes	—
31	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	—	2	16	no	—
35	C₄×Q₈×C₄₇	—	2	16	no	—
36	#117×C₄₇	—	2	4	no	—
37	#117×C₄₇	—	2	16	no	—
38	#117×C₄₇	D₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	2	4	no	—
50	C₂×Q₈×C₄₇	—	2	16	no	—
51	C₂×Q₈×C₄₇	—	2	16	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
56	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	2	16	no	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
72	C₂²⋊C₄×C₄₇	—	2	4	no	—
73	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₄₇	—	2	16	no	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	C₂×D₄	1	8	yes	—
77	#10625329	—	2	8	no	—
78	#10625329	—	2	8	no	—
79	#10625329	—	4	8	no	—
80	C₂×C₄₇⋊C₈	—	2	8	no	—
81	C₂×C₄₇⋊C₈	—	4	8	no	—
82	C₂³×C₄₇	—	4	2	no	—
83	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	#117	1	2	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₄₇	—	2	8	no	—
88	Q₈×C₄₇	—	2	8	no	—
89	Q₈×C₄₇	—	2	16	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	16	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	#117	1	2	yes	—
101	C₂×C₄×C₄₇	#121	1	8	yes	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₄₇⋊C₈	—	4	8	no	—
108	#20639	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	4	2	no	—
112	C₂²×C₄₇	—	4	2	no	—
113	C₂²×C₄₇	#380	1	4	yes	—
114	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
117	C₄×C₄₇	—	2	8	no	—
118	C₄×C₄₇	—	2	8	no	—
119	C₄×C₄₇	—	4	4	no	—
120	C₄×C₄₇	#380	1	4	yes	—
121	C₄×C₄₇	C₂×#121	1	2	yes	—
122	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
123	C₂×#519	D₄₇	1	1	yes	characteristic
124	#2905	—	47	2	no	—
125	C₂²×#101	—	47	2	no	—
126	#1801	—	47	8	no	—
127	#1742	—	47	2	no	—
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	—	2	1	no	—
130	C₂×C₄₇	—	2	4	no	—
131	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
132	C₂×C₄₇	#2889	1	2	yes	—
133	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
134	C₂²×M₁₆	—	47	4	no	—
135	#519	—	2	8	no	—
136	C₂×C₄×Q₈	Dic₄₇	1	4	yes	—
137	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
138	C₂×#117	Dic₄₇	1	4	yes	—
139	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
140	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
141	#424	—	94	8	no	—
142	#423	—	47	8	no	—
143	C₂×#101	—	47	8	no	—
144	C₂×#101	—	94	8	no	—
145	#343	—	47	8	no	—
146	C₄₇	#20639	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	4	no	—
148	C₂²×Q₈	—	2	4	no	—
149	C₂²×Q₈	Dic₉₄	1	4	yes	—
150	C₂²×Q₈	D₁₈₈	1	4	yes	—
151	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
152	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
153	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
154	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
155	C₂×M₁₆	—	47	8	no	—
156	C₂×M₁₆	—	94	4	no	—
157	C₂×M₁₆	—	94	8	no	—
158	C₂×M₁₆	—	94	8	no	—
159	C₂²×C₈	—	94	4	no	—
160	C₄×Q₈	—	2	16	no	—
161	C₄×Q₈	—	2	16	no	—
162	#117	—	2	4	no	—
163	#117	—	2	16	no	—
164	#117	C₂²⋊D₄₇	1	8	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
170	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
171	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
173	C₄⋊C₈	—	94	16	no	—
174	#101	—	94	16	no	—
175	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	2	16	no	—
179	C₂×Q₈	—	2	16	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	C₄⋊Dic₄₇	1	8	yes	—
183	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
184	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
188	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
189	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
190	C₂²×C₄	#10625350	1	4	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
192	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
197	M₁₆	—	94	8	no	—
198	M₁₆	—	94	8	no	—
199	M₁₆	—	188	8	no	—
200	C₂×C₈	—	94	8	no	—
201	C₂×C₈	—	188	8	no	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
207	C₄²	—	2	16	no	—
208	C₄²	—	2	4	no	—
209	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
212	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
213	C₂³	#13093983	1	2	yes	—
214	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
215	Q₈	—	2	16	no	—
216	Q₈	—	2	8	no	—
217	Q₈	—	2	8	no	—
218	C₂×C₄	—	2	16	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
225	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
228	C₂×C₄	#13093983	1	2	yes	—
229	C₂×C₄	#13093998	1	8	yes	—
230	C₂×C₄	#13094003	1	8	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
232	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
234	C₂×C₄	C₂×#10625350	1	2	yes	—
235	C₈	—	188	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#15688983	1	4	yes	—
241	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
243	C₂²	C₂×#13093983	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	4	4	no	—
247	C₄	#15688983	1	4	yes	—
248	C₄	C₂×#13093998	1	2	yes	—
249	C₄	C₂×#13094003	1	2	yes	—
250	C₂	—	2	4	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₂×#15688983	1	1	yes	characteristic
254	C₂	#20251871	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#519	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#519	D₄₇
hypocenter	C₄₇	#20639
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
3. #20243384	C₂	semidirect
146. C₄₇	#20639	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂²×#13093943	2
#20243384	2
#20243127	2
C₂×C₄₇×#519	2
#20639	47