Subgroups — #26109147

Order 12032.

368 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20242136	C₂	1	2	yes	maximal, complemented
2	#20241702	C₂	1	2	yes	maximal
3	#20241219	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₇×#433	C₂	1	1	yes	maximal, characteristic
5	#15689176	C₂²	1	4	yes	—
6	#15688876	—	2	2	no	—
7	#15688874	—	2	2	no	—
8	#15688849	—	2	2	no	—
9	#15688847	—	2	2	no	—
10	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#433	—	2	2	no	—
12	C₄₇×#433	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₄×C₄₇⋊C₈	—	2	4	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	#117×C₄₇	—	2	2	no	—
28	#117×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	1	no	—
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	1	no	—
38	C₂²⋊C₈×C₄₇	—	2	1	no	—
39	C₂²⋊C₈×C₄₇	—	2	2	no	—
40	C₂²⋊C₈×C₄₇	—	2	1	no	—
41	#97×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	D₄	1	2	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₂²⋊C₄×C₄₇	—	2	2	no	—
67	C₂²⋊C₄×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	1	no	—
69	C₄²×C₄₇	—	2	1	no	—
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	C₄²×C₄₇	#41	1	2	yes	—
72	C₂×C₄₇⋊C₈	—	4	4	no	—
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	—	4	1	no	—
76	C₂³×C₄₇	—	4	1	no	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	4	1	no	—
83	C₂×C₄×C₄₇	—	4	1	no	—
84	C₂×C₄×C₄₇	—	4	1	no	—
85	C₂×C₄×C₄₇	—	4	1	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	1	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#121	1	2	yes	—
92	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	2	yes	—
94	C₂×C₄×C₄₇	#131	1	2	yes	—
95	C₈×C₄₇	—	2	1	no	—
96	C₈×C₄₇	—	2	2	no	—
97	C₈×C₄₇	—	2	1	no	—
98	C₈×C₄₇	—	4	1	no	—
99	C₈×C₄₇	—	4	1	no	—
100	C₄₇⋊C₈	—	8	4	no	—
101	#8603	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	—	4	1	no	—
105	C₂²×C₄₇	—	4	1	no	—
106	C₂²×C₄₇	—	4	1	no	—
107	C₂²×C₄₇	—	8	2	no	—
108	C₂²×C₄₇	#380	1	1	yes	characteristic
109	C₂²×C₄₇	#424	1	2	yes	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	—	4	1	no	—
113	C₄×C₄₇	—	4	1	no	—
114	C₄×C₄₇	—	4	1	no	—
115	C₄×C₄₇	#408	1	1	yes	characteristic
116	C₄×C₄₇	#408	1	1	yes	characteristic
117	C₄×C₄₇	#433	1	2	yes	—
118	C₂×#433	D₄₇	1	1	yes	characteristic
119	#1564	—	47	2	no	—
120	#1458	—	47	2	no	—
121	#1275	—	47	2	no	—
122	C₂×C₄₇	—	2	2	no	—
123	C₂×C₄₇	—	8	1	no	—
124	C₂×C₄₇	#1798	1	1	yes	characteristic
125	C₂×C₄₇	#1812	1	1	yes	characteristic
126	C₂×C₄₇	#1813	1	1	yes	characteristic
127	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
128	#433	—	2	2	no	—
129	#433	—	2	2	no	—
130	#431	—	47	4	no	—
131	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
132	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
133	C₂×C₂²⋊C₈	Dic₄₇	1	1	yes	characteristic
134	C₂×C₂²⋊C₈	Dic₄₇	1	1	yes	characteristic
135	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
137	#335	—	94	2	no	—
138	#334	—	94	2	no	—
139	#324	—	94	2	no	—
140	#323	—	94	2	no	—
141	C₄₇	#8603	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
147	#117	—	2	2	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
150	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	Dic₉₄	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	D₁₈₈	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	94	4	no	—
159	C₂²⋊C₈	—	2	1	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₂²⋊C₈	—	94	4	no	—
163	#97	—	2	2	no	—
164	C₄×C₈	—	94	4	no	—
165	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
166	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
167	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	188	4	no	—
185	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	#10625350	1	2	yes	—
195	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂×C₄	#13093983	1	1	yes	characteristic
214	C₂×C₄	#13093998	1	2	yes	—
215	C₂×C₄	#13094003	1	2	yes	—
216	C₂×C₄	#13094031	1	2	yes	—
217	C₂×C₄	#13094031	1	2	yes	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	1	no	—
222	C₈	—	4	1	no	—
223	C₈	—	376	4	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	#15688983	1	1	yes	characteristic
231	C₂²	#15689150	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	#15689090	1	1	yes	characteristic
238	C₄	#15689090	1	1	yes	characteristic
239	C₄	#15689184	1	2	yes	—
240	C₂	—	2	2	no	—
241	C₂	—	8	1	no	—
242	C₂	#20243372	1	1	yes	characteristic
243	C₂	#20243434	1	1	yes	characteristic
244	C₂	#20243438	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#433	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#433	D₄₇
hypocenter	C₄₇	#8603
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#433

Complements

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

N	H ≅ G/N	type
1. #20242136	C₂	semidirect
3. #20241219	C₂	semidirect
141. C₄₇	#8603	semidirect

Maximal subgroups

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

subgroup	index
#20242136	2
#20241702	2
#20241219	2
C₂×C₄₇×#433	2
#8603	47