Subgroups — #26135634

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688926	C₂	1	2	yes	maximal
3	C₂×#15688926	C₂	1	2	yes	maximal
4	#20241466	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	—	2	1	no	—
7	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	1	no	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15689503	—	2	4	no	—
11	#15688926	—	2	4	no	—
12	#15688926	—	2	4	no	—
13	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#340	—	2	1	no	—
18	C₂²×Dic₉₄	—	2	2	no	—
19	C₂×Dic₁₈₈	—	2	2	no	—
20	C₂×Dic₁₈₈	—	2	2	no	—
21	C₂×Dic₁₈₈	—	2	2	no	—
22	C₂×Dic₁₈₈	—	2	2	no	—
23	C₂×Dic₁₈₈	—	2	4	no	—
24	C₂×Dic₁₈₈	—	2	2	no	—
25	C₂×Dic₁₈₈	—	4	2	no	—
26	C₂×Dic₁₈₈	—	4	2	no	—
27	C₂×Dic₁₈₈	D₄	1	4	yes	—
28	#13094024	—	2	4	no	—
29	#13094024	—	4	2	no	—
30	C₂×C₄⋊Dic₄₇	—	4	1	no	—
31	C₂×C₄⋊Dic₄₇	—	4	1	no	—
32	C₂×C₄₇⋊C₁₆	—	2	2	no	—
33	C₂×C₄₇⋊C₁₆	—	2	2	no	—
34	C₂×C₄₇⋊C₁₆	—	2	2	no	—
35	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	#107×C₄₇	—	2	1	no	—
42	#107×C₄₇	—	2	2	no	—
43	#107×C₄₇	D₄	1	2	yes	—
44	C₄×C₈×C₄₇	—	2	2	no	—
45	#95×C₄₇	—	4	1	no	—
46	C₂²×C₄×C₄₇	—	4	1	no	—
47	C₂²×C₄×C₄₇	—	4	1	no	—
48	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	4	2	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	—	4	2	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×Dic₉₄	—	4	2	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	2	1	no	—
60	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₄₇	#41	1	2	yes	—
64	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₄⋊C₄×C₄₇	—	4	1	no	—
67	C₄⋊C₄×C₄₇	—	4	1	no	—
68	C₄⋊C₄×C₄₇	—	4	1	no	—
69	C₄⋊C₄×C₄₇	—	8	2	no	—
70	C₄²×C₄₇	—	2	1	no	—
71	C₄²×C₄₇	—	2	1	no	—
72	C₄⋊Dic₄₇	—	4	4	no	—
73	C₄⋊Dic₄₇	—	4	4	no	—
74	Dic₁₈₈	—	2	4	no	—
75	Dic₁₈₈	—	2	4	no	—
76	Dic₁₈₈	—	4	8	no	—
77	Dic₁₈₈	—	4	2	no	—
78	Dic₁₈₈	—	4	2	no	—
79	C₄₇⋊C₁₆	—	4	2	no	—
80	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	8	1	no	—
87	C₂×C₄×C₄₇	—	8	1	no	—
88	C₂×C₄×C₄₇	—	8	1	no	—
89	C₂×C₄×C₄₇	—	8	1	no	—
90	C₂×C₄×C₄₇	—	8	1	no	—
91	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
92	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
93	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	8	2	no	—
101	C₂×Dic₄₇	—	8	2	no	—
102	Dic₉₄	—	4	4	no	—
103	Dic₉₄	—	4	4	no	—
104	Dic₉₄	—	8	2	no	—
105	Dic₉₄	—	8	2	no	—
106	C₈×C₄₇	—	2	2	no	—
107	C₈×C₄₇	#121	1	2	yes	—
108	C₈×C₄₇	#123	1	2	yes	—
109	#13409	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	#447	1	1	yes	characteristic
111	C₂²×C₄₇	#459	1	1	yes	characteristic
112	C₂²×C₄₇	#486	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
115	C₂²×C₄₇	#510	1	1	yes	characteristic
116	C₂²×C₄₇	#510	1	1	yes	characteristic
117	C₄×C₄₇	—	2	2	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	8	2	no	—
120	C₄×C₄₇	—	16	1	no	—
121	C₄×C₄₇	#394	1	1	yes	characteristic
122	C₄×C₄₇	#450	1	1	yes	characteristic
123	C₄×C₄₇	#460	1	1	yes	characteristic
124	C₄×C₄₇	#490	1	1	yes	characteristic
125	Dic₄₇	—	8	4	no	—
126	C₂×#494	—	47	1	no	—
127	C₂×#358	—	47	2	no	—
128	C₂×#358	—	47	2	no	—
129	#1942	D₄₇	1	1	yes	characteristic
130	#1374	—	47	1	no	—
131	C₂×C₄₇	#2041	1	1	yes	characteristic
132	C₂×C₄₇	#2185	1	1	yes	characteristic
133	C₂×C₄₇	#2188	1	1	yes	characteristic
134	C₂×C₄₇	#2208	1	1	yes	characteristic
135	C₂×C₄₇	#2209	1	1	yes	characteristic
136	C₂×C₄₇	#2241	1	1	yes	characteristic
137	C₂×C₄₇	#2245	1	1	yes	characteristic
138	C₂²×Q₁₆	—	47	2	no	—
139	C₂²×Q₁₆	—	94	1	no	—
140	C₂×C₄⋊Q₈	—	94	1	no	—
141	C₂²×C₁₆	—	47	2	no	—
142	#494	—	94	4	no	—
143	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
144	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
146	#389	—	2	1	no	—
147	#358	—	94	4	no	—
148	#358	—	94	4	no	—
149	#340	—	2	1	no	—
150	C₄₇	#13409	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	94	2	no	—
152	C₂×Q₁₆	—	47	4	no	—
153	C₂×Q₁₆	—	94	4	no	—
154	C₂×Q₁₆	—	94	2	no	—
155	C₂×Q₁₆	—	94	2	no	—
156	C₂×Q₁₆	—	94	2	no	—
157	C₂×Q₁₆	—	94	2	no	—
158	C₂×Q₁₆	—	94	2	no	—
159	C₂×Q₁₆	—	188	2	no	—
160	C₂×Q₁₆	—	188	2	no	—
161	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	94	4	no	—
164	C₄⋊Q₈	—	188	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	188	1	no	—
169	C₂×C₄⋊C₄	—	188	1	no	—
170	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂×C₁₆	—	94	2	no	—
172	C₂×C₁₆	—	94	2	no	—
173	C₂×C₁₆	—	94	2	no	—
174	#107	—	2	1	no	—
175	#107	—	2	2	no	—
176	#107	D₁₈₈	1	2	yes	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	4	1	no	—
179	C₂×Q₈	—	94	4	no	—
180	C₂×Q₈	—	188	2	no	—
181	C₂×Q₈	—	188	2	no	—
182	C₂×Q₈	—	188	2	no	—
183	C₂×Q₈	—	188	2	no	—
184	C₂×Q₈	—	188	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	188	2	no	—
188	C₂²×C₄	#10625334	1	1	yes	characteristic
189	C₂²×C₄	#10625334	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	Q₁₆	—	94	4	no	—
192	Q₁₆	—	94	4	no	—
193	Q₁₆	—	188	8	no	—
194	Q₁₆	—	188	2	no	—
195	Q₁₆	—	188	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	#10625336	1	2	yes	—
199	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
200	C₂×C₈	#10625350	1	1	yes	characteristic
201	C₂×C₈	D₄×D₄₇	1	2	yes	—
202	C₂×C₈	#10625354	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	—	188	4	no	—
209	C₄⋊C₄	—	188	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₁₆	—	188	2	no	—
213	C₂³	C₂×#10625334	1	1	yes	characteristic
214	Q₈	—	376	2	no	—
215	Q₈	—	376	2	no	—
216	Q₈	—	188	4	no	—
217	Q₈	—	188	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	376	2	no	—
229	C₂×C₄	—	376	2	no	—
230	C₂×C₄	—	188	4	no	—
231	C₂×C₄	#13093968	1	2	yes	—
232	C₂×C₄	#13093970	1	2	yes	—
233	C₂×C₄	#13093992	1	1	yes	characteristic
234	C₂×C₄	#13093998	1	1	yes	characteristic
235	C₂×C₄	#13094013	1	1	yes	characteristic
236	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
237	C₂×C₄	#13094049	1	1	yes	characteristic
238	C₂×C₄	#13094049	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	#13093999	1	2	yes	—
241	C₈	#13094009	1	2	yes	—
242	C₂²	#15689256	1	1	yes	characteristic
243	C₂²	#15689318	1	1	yes	characteristic
244	C₂²	#15689471	1	1	yes	characteristic
245	C₂²	C₂×#13093968	1	1	yes	characteristic
246	C₂²	C₂×#13093970	1	1	yes	characteristic
247	C₂²	#15689563	1	1	yes	characteristic
248	C₂²	#15689563	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	16	1	no	—
253	C₄	—	376	4	no	—
254	C₄	#15689040	1	1	yes	characteristic
255	C₄	#15689273	1	1	yes	characteristic
256	C₄	#15689323	1	1	yes	characteristic
257	C₄	#15689489	1	1	yes	characteristic
258	C₂	#20244622	1	1	yes	characteristic
259	C₂	#20245304	1	1	yes	characteristic
260	C₂	#20245321	1	1	yes	characteristic
261	C₂	#20245433	1	1	yes	characteristic
262	C₂	#20245438	1	1	yes	characteristic
263	C₂	#20245617	1	1	yes	characteristic
264	C₂	#20245635	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1942	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#1942	D₄₇
hypocenter	C₄₇	#13409
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
150. C₄₇	#13409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689503	2
C₂×#15688926	2
#20241466	2
C₄₇×#1942	2
#13409	47