Subgroups — #26115335

Order 12032.

498 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251766	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	2	yes	maximal
4	C₂×#15688870	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13094024	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#15689029	—	2	1	no	—
11	#15689029	—	2	1	no	—
12	#15689029	C₄	1	4	yes	—
13	#15688989	—	2	2	no	—
14	#15688871	—	2	4	no	—
15	#15688870	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
44	C₄×C₈×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	—	4	2	no	—
55	C₂×Dic₉₄	—	4	1	no	—
56	C₂×Dic₉₄	—	4	1	no	—
57	C₂×Dic₉₄	—	4	4	no	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	1	no	—
63	C₄²×C₄₇	—	2	1	no	—
64	C₄²×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	SD₁₆	1	4	yes	—
66	C₄²×C₄₇	Q₁₆	1	4	yes	—
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₄⋊Dic₄₇	—	2	4	no	—
70	C₄⋊Dic₄₇	—	4	2	no	—
71	C₄⋊Dic₄₇	—	4	1	no	—
72	C₄⋊Dic₄₇	—	4	1	no	—
73	C₄⋊Dic₄₇	—	4	1	no	—
74	C₄⋊Dic₄₇	—	4	1	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	4	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
83	C₂×C₄×C₄₇	#103	1	4	yes	—
84	C₂×C₄×C₄₇	#103	1	4	yes	—
85	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₄₇	#121	1	2	yes	—
87	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	#135	1	2	yes	—
92	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	2	no	—
95	C₂×Dic₄₇	—	4	2	no	—
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	4	2	no	—
99	Dic₉₄	—	8	2	no	—
100	Dic₉₄	—	8	2	no	—
101	Dic₉₄	—	8	4	no	—
102	C₈×C₄₇	—	4	4	no	—
103	C₄₇⋊C₈	—	8	2	no	—
104	#9697	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	#390	1	1	yes	characteristic
106	C₂²×C₄₇	#413	1	2	yes	—
107	C₂²×C₄₇	C₂×#103	1	2	yes	—
108	C₂²×C₄₇	#416	1	1	yes	characteristic
109	C₂²×C₄₇	#417	1	1	yes	characteristic
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	2	2	no	—
113	C₄×C₄₇	#463	1	2	yes	—
114	C₄×C₄₇	#464	1	2	yes	—
115	C₄×C₄₇	#487	1	2	yes	—
116	C₄×C₄₇	#492	1	1	yes	characteristic
117	C₄×C₄₇	#494	1	1	yes	characteristic
118	Dic₄₇	—	8	2	no	—
119	Dic₄₇	—	8	2	no	—
120	#2862	—	47	1	no	—
121	#1842	—	47	1	no	—
122	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
123	C₂×#333	—	47	2	no	—
124	C₂×#332	—	47	2	no	—
125	C₂×C₄₇	#1542	1	2	yes	—
126	C₂×C₄₇	#1550	1	2	yes	—
127	C₂×C₄₇	#1956	1	1	yes	characteristic
128	C₂×C₄₇	#1967	1	1	yes	characteristic
129	C₂×C₄₇	#1983	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	—	47	1	no	—
131	C₂×C₄⋊Q₈	—	47	1	no	—
132	C₂×C₄⋊C₈	—	47	2	no	—
133	C₂×C₄⋊C₈	—	94	2	no	—
134	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
135	C₂×C₄×C₈	—	2	1	no	—
136	#391	—	47	4	no	—
137	#391	—	94	1	no	—
138	#391	—	94	1	no	—
139	#382	—	94	2	no	—
140	C₄³	C₂×D₄₇	1	1	yes	characteristic
141	#336	—	2	1	no	—
142	#333	—	94	4	no	—
143	#332	—	94	4	no	—
144	C₄₇	#9697	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	47	2	no	—
146	C₂²×Q₈	—	94	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	94	2	no	—
149	C₄⋊Q₈	—	94	2	no	—
150	C₄⋊Q₈	—	94	2	no	—
151	C₄⋊Q₈	—	94	2	no	—
152	C₄⋊Q₈	—	94	2	no	—
153	C₂×C₄⋊C₄	—	47	2	no	—
154	C₂×C₄⋊C₄	—	94	1	no	—
155	C₂×C₄⋊C₄	—	94	1	no	—
156	C₂×C₄⋊C₄	—	94	1	no	—
157	C₂×C₄²	C₄×D₄₇	1	2	yes	—
158	C₂×C₄²	D₁₈₈	1	2	yes	—
159	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	94	2	no	—
164	C₄⋊C₈	—	94	2	no	—
165	C₄⋊C₈	—	188	4	no	—
166	C₄⋊C₈	D₁₈₈	1	4	yes	—
167	C₄×C₈	—	4	2	no	—
168	#95	—	94	4	no	—
169	#95	—	94	4	no	—
170	C₂×Q₈	—	94	4	no	—
171	C₂×Q₈	—	188	4	no	—
172	C₂×Q₈	—	188	2	no	—
173	C₂×Q₈	—	188	2	no	—
174	C₂×Q₈	—	188	1	no	—
175	C₂×Q₈	—	188	1	no	—
176	C₂²×C₄	—	94	2	no	—
177	C₂²×C₄	—	94	2	no	—
178	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
179	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	188	2	no	—
187	C₂×C₈	—	188	4	no	—
188	C₄⋊C₄	—	94	4	no	—
189	C₄⋊C₄	—	188	2	no	—
190	C₄⋊C₄	—	188	1	no	—
191	C₄⋊C₄	—	188	1	no	—
192	C₄⋊C₄	—	188	1	no	—
193	C₄⋊C₄	—	188	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#10625325	1	4	yes	—
199	C₄²	Dic₁₈₈	1	4	yes	—
200	C₄²	C₂×D₁₈₈	1	2	yes	—
201	C₄²	#10625350	1	2	yes	—
202	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
203	Q₈	—	376	4	no	—
204	Q₈	—	376	2	no	—
205	Q₈	—	376	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	188	2	no	—
212	C₂×C₄	—	188	2	no	—
213	C₂×C₄	—	188	4	no	—
214	C₂×C₄	—	188	2	no	—
215	C₂×C₄	—	188	2	no	—
216	C₂×C₄	#13093949	1	4	yes	—
217	C₂×C₄	#13093949	1	4	yes	—
218	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
219	C₂×C₄	#13093998	1	2	yes	—
220	C₂×C₄	#13094012	1	1	yes	characteristic
221	C₂×C₄	#13094021	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625325	1	2	yes	—
223	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
224	C₂×C₄	#13094044	1	2	yes	—
225	C₂×C₄	#13094050	1	1	yes	characteristic
226	C₂×C₄	#13094055	1	1	yes	characteristic
227	C₈	—	4	4	no	—
228	C₈	—	376	2	no	—
229	C₂²	#15689025	1	1	yes	characteristic
230	C₂²	#15689105	1	2	yes	—
231	C₂²	C₂×#13093949	1	2	yes	—
232	C₂²	#15689117	1	1	yes	characteristic
233	C₂²	#15689122	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	376	2	no	—
238	C₄	—	376	2	no	—
239	C₄	#15689338	1	2	yes	—
240	C₄	#15689345	1	2	yes	—
241	C₄	#15689474	1	2	yes	—
242	C₄	#15689496	1	1	yes	characteristic
243	C₄	#15689503	1	1	yes	characteristic
244	C₂	#20242043	1	2	yes	—
245	C₂	#20242076	1	2	yes	—
246	C₂	#20244190	1	1	yes	characteristic
247	C₂	#20244237	1	1	yes	characteristic
248	C₂	#20244320	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9697
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
144. C₄₇	#9697	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20243540	2
C₂×#15688871	2
C₂×#15688870	2
C₄×C₄⋊C₈×C₄₇	2
#9697	47