Subgroups — #26106913

Order 12032.

542 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245865	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688853	C₂	1	2	yes	maximal
4	#20241227	C₂	1	2	yes	maximal
5	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15689029	C₄	1	2	yes	—
10	#15689021	C₄	1	2	yes	—
11	#15689007	—	2	2	no	—
12	C₄×Dic₄₇⋊C₄	—	2	1	no	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	#15688969	—	2	1	no	—
15	#15688853	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×#97×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄×Dic₉₄	—	2	2	no	—
23	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₄₇⋊C₄	—	2	1	no	—
27	C₂×Dic₄₇⋊C₄	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	1	no	—
29	C₂×C₄×Dic₄₇	—	2	1	no	—
30	C₄×C₄₇⋊C₈	—	2	8	no	—
31	C₄×C₄₇⋊C₈	—	2	2	no	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	1	no	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	D₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	—	2	2	no	—
44	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
45	#97×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	2	1	no	—
52	C₂²×Dic₄₇	—	2	1	no	—
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂×Dic₉₄	—	2	4	no	—
55	C₂×Dic₉₄	—	2	1	no	—
56	C₂×Dic₉₄	C₄²	1	2	yes	—
57	C₂×C₈×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₄²×C₄₇	—	2	4	no	—
61	C₄²×C₄₇	—	2	4	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	SD₁₆	1	2	yes	—
64	C₄²×C₄₇	Q₁₆	1	2	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₇	#41	1	2	yes	—
67	C₄⋊Dic₄₇	—	2	1	no	—
68	C₄⋊Dic₄₇	—	2	4	no	—
69	C₄⋊Dic₄₇	C₄²	1	2	yes	—
70	Dic₄₇⋊C₄	—	2	4	no	—
71	Dic₄₇⋊C₄	—	4	4	no	—
72	C₄×Dic₄₇	—	2	4	no	—
73	C₄×Dic₄₇	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	#103	1	4	yes	—
83	C₂×C₄×C₄₇	#104	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₇	#131	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#135	1	2	yes	—
92	C₂×Dic₄₇	—	2	2	no	—
93	C₂×Dic₄₇	—	2	2	no	—
94	C₂×Dic₄₇	—	4	2	no	—
95	C₂×Dic₄₇	—	4	2	no	—
96	C₂×Dic₄₇	—	4	2	no	—
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	4	2	no	—
99	Dic₉₄	—	2	4	no	—
100	Dic₉₄	—	4	2	no	—
101	C₈×C₄₇	—	4	4	no	—
102	C₄₇⋊C₈	—	2	8	no	—
103	#8176	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	#408	1	2	yes	—
106	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
107	C₂²×C₄₇	#416	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#104	1	2	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	#405	1	2	yes	—
113	C₄×C₄₇	#433	1	2	yes	—
114	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
115	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
116	Dic₄₇	—	4	2	no	—
117	Dic₄₇	—	4	2	no	—
118	Dic₄₇	—	8	2	no	—
119	#2295	—	47	1	no	—
120	#1767	D₄₇	1	1	yes	characteristic
121	C₄²×C₈	—	47	1	no	—
122	C₂×Q₈⋊C₈	—	47	2	no	—
123	#1279	—	47	2	no	—
124	C₂×C₄₇	#1493	1	2	yes	—
125	C₂×C₄₇	#1494	1	2	yes	—
126	C₂×C₄₇	#1750	1	1	yes	characteristic
127	C₂×C₄₇	C₄×#104	1	1	yes	characteristic
128	C₂×C₄₇	C₄×#103	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	47	1	no	—
130	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
131	C₂×#97	—	2	1	no	—
132	C₂×C₄×C₈	—	47	2	no	—
133	C₂×C₄×C₈	—	94	2	no	—
134	#391	—	47	2	no	—
135	#389	—	47	2	no	—
136	#387	—	94	2	no	—
137	C₄×C₄⋊C₄	—	47	1	no	—
138	C₄×C₄⋊C₄	—	94	1	no	—
139	#376	—	94	1	no	—
140	C₄³	C₂×D₄₇	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	Q₈⋊C₈	—	47	8	no	—
143	C₄₇	#8176	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	47	1	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	94	2	no	—
147	C₄×Q₈	—	47	4	no	—
148	C₄×Q₈	—	94	2	no	—
149	C₂×C₄⋊C₄	—	47	1	no	—
150	C₂×C₄⋊C₄	—	47	2	no	—
151	C₂×C₄⋊C₄	—	94	1	no	—
152	C₂×C₄⋊C₄	—	94	2	no	—
153	C₂×C₄²	—	94	1	no	—
154	C₂×C₄²	—	94	1	no	—
155	C₂×C₄²	C₄×D₄₇	1	2	yes	—
156	C₂×C₄²	D₁₈₈	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
159	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
162	#97	—	4	2	no	—
163	C₄×C₈	—	47	4	no	—
164	C₄×C₈	—	94	8	no	—
165	C₄×C₈	—	94	2	no	—
166	#95	—	94	2	no	—
167	#95	—	94	2	no	—
168	#95	—	94	2	no	—
169	#95	—	94	1	no	—
170	C₂×Q₈	—	47	2	no	—
171	C₂×Q₈	—	94	1	no	—
172	C₂×Q₈	—	94	4	no	—
173	C₂²×C₄	—	94	1	no	—
174	C₂²×C₄	—	94	1	no	—
175	C₂²×C₄	—	94	2	no	—
176	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
177	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	94	8	no	—
185	C₂×C₈	—	94	4	no	—
186	C₄⋊C₄	—	47	2	no	—
187	C₄⋊C₄	—	94	1	no	—
188	C₄⋊C₄	—	94	4	no	—
189	C₄⋊C₄	—	94	4	no	—
190	C₄⋊C₄	—	188	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	94	4	no	—
195	C₄²	—	188	2	no	—
196	C₄²	#10625335	1	2	yes	—
197	C₄²	#10625337	1	2	yes	—
198	C₄²	C₂×C₄×D₄₇	1	2	yes	—
199	C₄²	#10625350	1	2	yes	—
200	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
201	Q₈	—	94	4	no	—
202	Q₈	—	188	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	94	2	no	—
210	C₂×C₄	—	188	2	no	—
211	C₂×C₄	—	188	2	no	—
212	C₂×C₄	—	188	2	no	—
213	C₂×C₄	—	188	2	no	—
214	C₂×C₄	—	188	2	no	—
215	C₂×C₄	#13093947	1	4	yes	—
216	C₂×C₄	#13093950	1	4	yes	—
217	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
218	C₂×C₄	#13093981	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
221	C₂×C₄	#13094032	1	4	yes	—
222	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
223	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
224	C₂×C₄	#13094044	1	2	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	94	8	no	—
227	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂²	#15689091	1	2	yes	—
229	C₂²	C₂×#13093947	1	1	yes	characteristic
230	C₂²	#15689117	1	1	yes	characteristic
231	C₂²	C₂×#13093950	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	376	2	no	—
236	C₄	—	188	2	no	—
237	C₄	—	188	2	no	—
238	C₄	#15689080	1	2	yes	—
239	C₄	#15689185	1	2	yes	—
240	C₄	C₄×#10625335	1	2	yes	—
241	C₄	C₄×#10625337	1	2	yes	—
242	C₂	#20241842	1	2	yes	—
243	C₂	#20241846	1	2	yes	—
244	C₂	#20243157	1	1	yes	characteristic
245	C₂	C₄×#13093950	1	1	yes	characteristic
246	C₂	C₄×#13093947	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#8176
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
15. #15688853	C₄	semidirect
143. C₄₇	#8176	semidirect

Maximal subgroups

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

subgroup	index
#20245865	2
C₄²×C₄₇⋊C₈	2
C₂×#15688853	2
#20241227	2
C₄₇×#1767	2
#8176	47