Subgroups — #26225755

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252026	C₂	1	2	yes	maximal
2	#20244382	C₂	1	4	yes	maximal, complemented
3	#20244378	C₂	1	4	yes	maximal, complemented
4	#20243471	C₂	1	2	yes	maximal
5	#20243468	C₂	1	2	yes	maximal
6	C₂×C₄₇×#544	C₂	1	1	yes	maximal, characteristic
7	#15689184	—	2	8	no	—
8	#15689147	C₂²	1	4	yes	—
9	#15689087	C₂²	1	4	yes	—
10	C₄×#10625329	C₂²	1	4	yes	—
11	#15688892	C₂²	1	8	yes	—
12	#15688892	C₂²	1	4	yes	—
13	#15688892	C₂²	1	4	yes	—
14	C₄₇×#544	—	2	8	no	—
15	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
17	C₂×#125×C₄₇	C₄	1	2	yes	—
18	C₂×#125×C₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	C₂³	1	8	yes	—
24	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
26	#13093935	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₇	—	2	8	no	—
32	C₄⋊Q₈×C₄₇	—	2	4	no	—
33	C₄⋊Q₈×C₄₇	—	2	4	no	—
34	#125×C₄₇	—	2	8	no	—
35	#125×C₄₇	—	2	4	no	—
36	#125×C₄₇	—	2	4	no	—
37	C₂²⋊Q₈×C₄₇	—	2	16	no	—
38	#117×C₄₇	—	2	4	no	—
39	#117×C₄₇	D₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	—	2	8	no	—
56	C₂×Q₈×C₄₇	—	8	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	8	2	no	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	2	8	no	—
70	C₄⋊C₄×C₄₇	—	2	8	no	—
71	C₄⋊C₄×C₄₇	—	2	8	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
77	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	C₂×D₄	1	4	yes	—
82	C₄²×C₄₇	C₂×D₄	1	4	yes	—
83	#10625329	—	2	16	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	—	4	1	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	Q₈×C₄₇	—	8	8	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	8	4	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
107	C₄₇⋊C₈	—	4	8	no	—
108	#21832	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	1	no	—
110	C₂²×C₄₇	—	2	1	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	4	4	no	—
115	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₄₇	#519	1	2	yes	—
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	4	no	—
119	C₄×C₄₇	—	2	4	no	—
120	C₄×C₄₇	—	4	4	no	—
121	C₄×C₄₇	—	4	2	no	—
122	C₄×C₄₇	—	4	2	no	—
123	C₄×C₄₇	#536	1	4	yes	—
124	C₂×#544	D₄₇	1	1	yes	characteristic
125	#2921	—	47	2	no	—
126	#1998	—	47	4	no	—
127	#1997	—	47	4	no	—
128	#1822	—	47	2	no	—
129	#1821	—	47	2	no	—
130	C₂×C₄₇	—	2	2	no	—
131	C₂×C₄₇	—	4	2	no	—
132	C₂×C₄₇	#2310	1	1	yes	characteristic
133	C₂×C₄₇	#2888	1	1	yes	characteristic
134	C₂×C₄₇	#2889	1	1	yes	characteristic
135	#544	—	2	8	no	—
136	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
137	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
138	C₂×#125	Dic₄₇	1	2	yes	—
139	C₂×#125	C₂×D₄₇	1	2	yes	—
140	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
141	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
142	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
144	#433	—	94	8	no	—
145	#423	—	47	4	no	—
146	#407	—	47	4	no	—
147	C₄×M₁₆	—	47	4	no	—
148	#343	—	47	8	no	—
149	#343	—	47	4	no	—
150	#343	—	47	4	no	—
151	C₄₇	#21832	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	C₂×Dic₄₇	1	4	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	C₄⋊Q₈	—	2	8	no	—
157	C₄⋊Q₈	—	2	4	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	#125	—	2	8	no	—
160	#125	—	2	4	no	—
161	#125	—	2	4	no	—
162	C₂²⋊Q₈	—	2	16	no	—
163	#117	—	2	4	no	—
164	#117	C₂²⋊D₄₇	1	8	yes	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
168	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
169	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
170	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
177	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
179	C₄⋊C₈	—	94	8	no	—
180	C₂²⋊C₈	—	94	8	no	—
181	#97	—	94	4	no	—
182	C₄×C₈	—	94	4	no	—
183	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	8	no	—
185	C₂×Q₈	—	8	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	8	2	no	—
189	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
191	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
197	M₁₆	—	94	16	no	—
198	C₂×C₈	—	94	8	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
208	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
211	C₄²	—	2	4	no	—
212	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
213	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
218	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
219	Q₈	—	8	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₄	#13094072	1	4	yes	—
236	C₈	—	188	8	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
244	C₂²	#15689601	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	2	4	no	—
248	C₄	—	4	4	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	#15689697	1	4	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	4	2	no	—
254	C₂	#20245974	1	1	yes	characteristic
255	C₂	#20251867	1	1	yes	characteristic
256	C₂	#20251871	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#544	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#544	D₄₇
hypocenter	C₄₇	#21832
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#544

Complements

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

N	H ≅ G/N	type
2. #20244382	C₂	semidirect
3. #20244378	C₂	semidirect
151. C₄₇	#21832	semidirect

Maximal subgroups

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

subgroup	index
#20252026	2
#20244382	2
#20244378	2
#20243471	2
#20243468	2
C₂×C₄₇×#544	2
#21832	47