Subgroups — #26109557

Order 12032.

346 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	#20243526	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20242078	C₂	1	2	yes	maximal
4	#20241262	C₂	1	2	yes	maximal
5	C₄₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15688888	—	2	1	no	—
9	#15688888	—	2	1	no	—
10	#15688876	—	2	1	no	—
11	#15688876	—	2	1	no	—
12	#15688872	—	2	2	no	—
13	#15688869	—	2	2	no	—
14	C₄₇×#533	C₂²	1	1	yes	characteristic
15	C₄₇×#446	—	2	1	no	—
16	C₄₇×#446	C₄	1	2	yes	—
17	C₈×Q₈×C₄₇	—	2	1	no	—
18	C₈×Q₈×C₄₇	C₄	1	2	yes	—
19	C₄₇×#424	—	2	1	no	—
20	C₄₇×#423	C₂²	1	1	yes	characteristic
21	C₄₇×#405	—	2	1	no	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₂×M₁₆×C₄₇	—	2	1	no	—
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
34	#125×C₄₇	—	2	1	no	—
35	C₂²⋊Q₈×C₄₇	—	2	1	no	—
36	C₂²⋊Q₈×C₄₇	Q₈	1	2	yes	—
37	C₄×Q₈×C₄₇	—	2	1	no	—
38	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
39	#117×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	2	no	—
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	—	2	1	no	—
44	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
45	#97×C₄₇	—	2	2	no	—
46	C₄×C₈×C₄₇	—	2	2	no	—
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₇	—	2	1	no	—
52	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
56	M₁₆×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₄⋊C₄×C₄₇	—	2	1	no	—
66	C₄⋊C₄×C₄₇	—	2	1	no	—
67	C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₄⋊C₄×C₄₇	—	2	2	no	—
69	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₄₇	#41	1	1	yes	characteristic
76	#10625329	—	4	2	no	—
77	#10625329	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	8	2	no	—
81	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₄₇	—	4	2	no	—
83	Q₈×C₄₇	—	4	1	no	—
84	Q₈×C₄₇	—	4	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	4	1	no	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#135	1	2	yes	—
97	C₂×C₄×C₄₇	#137	1	2	yes	—
98	C₈×C₄₇	—	2	1	no	—
99	C₈×C₄₇	—	2	1	no	—
100	C₈×C₄₇	—	2	1	no	—
101	C₈×C₄₇	—	2	1	no	—
102	C₈×C₄₇	—	4	2	no	—
103	C₄₇⋊C₈	—	8	2	no	—
104	C₄₇⋊C₈	—	8	2	no	—
105	#8689	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	1	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	#380	1	1	yes	characteristic
109	C₂²×C₄₇	#479	1	2	yes	—
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	—	4	2	no	—
114	C₄×C₄₇	—	4	2	no	—
115	C₄×C₄₇	#416	1	1	yes	characteristic
116	C₄×C₄₇	#419	1	1	yes	characteristic
117	C₄×C₄₇	#464	1	2	yes	—
118	C₄×C₄₇	#472	1	1	yes	characteristic
119	C₄×C₄₇	#473	1	1	yes	characteristic
120	#2960	D₄₇	1	1	yes	characteristic
121	#1837	—	47	1	no	—
122	C₂×#335	—	47	1	no	—
123	#1550	—	47	2	no	—
124	#1292	—	47	2	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	#1805	1	1	yes	characteristic
127	C₂×C₄₇	#1813	1	1	yes	characteristic
128	C₂×C₄₇	#1816	1	1	yes	characteristic
129	#533	C₂×D₄₇	1	1	yes	characteristic
130	#446	—	2	1	no	—
131	#446	Dic₄₇	1	2	yes	—
132	C₈×Q₈	—	2	1	no	—
133	C₈×Q₈	Dic₄₇	1	2	yes	—
134	#424	—	2	1	no	—
135	#423	—	47	2	no	—
136	#423	C₂×D₄₇	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	47	2	no	—
138	#405	—	2	1	no	—
139	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
140	#341	—	94	1	no	—
141	#341	—	94	1	no	—
142	#335	—	94	1	no	—
143	#335	—	94	1	no	—
144	#333	—	94	2	no	—
145	#332	—	94	2	no	—
146	C₄₇	#8689	1	1	yes	characteristic, complemented, Sylow
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	94	2	no	—
149	C₂²×C₈	—	94	2	no	—
150	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
152	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
153	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
154	#125	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₂²⋊Q₈	Dic₉₄	1	2	yes	—
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
159	#117	—	2	1	no	—
160	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	94	2	no	—
165	C₄⋊C₈	—	94	2	no	—
166	C₄⋊C₈	—	94	2	no	—
167	C₄⋊C₈	—	94	2	no	—
168	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
172	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
177	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
178	C₂²×C₄	#10625350	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	188	2	no	—
182	M₁₆	—	188	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	376	2	no	—
187	C₂×C₈	—	188	2	no	—
188	C₂×C₈	—	188	2	no	—
189	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
202	C₄²	#10625350	1	1	yes	characteristic
203	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂³	#13094003	1	1	yes	characteristic
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
216	C₂×C₄	#13093983	1	1	yes	characteristic
217	C₂×C₄	#13093998	1	1	yes	characteristic
218	C₂×C₄	#13093998	1	1	yes	characteristic
219	C₂×C₄	#13094003	1	1	yes	characteristic
220	C₂×C₄	#13094044	1	2	yes	—
221	C₂×C₄	#13094055	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	376	2	no	—
228	C₈	—	376	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#15688983	1	1	yes	characteristic
232	C₂²	#15689430	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#15689119	1	1	yes	characteristic
239	C₄	#15689132	1	1	yes	characteristic
240	C₄	#15689345	1	2	yes	—
241	C₄	#15689393	1	1	yes	characteristic
242	C₄	#15689398	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#20243403	1	1	yes	characteristic
245	C₂	#20243438	1	1	yes	characteristic
246	C₂	#20243452	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688983
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2960	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2960	D₄₇
hypocenter	C₄₇	#8689
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
146. C₄₇	#8689	semidirect

Maximal subgroups

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

subgroup	index
#20243526	2
C₂×#15688876	2
#20242078	2
#20241262	2
C₄₇×#2960	2
#8689	47