Subgroups — #26114816

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244107	C₂	1	2	yes	maximal
2	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20241229	C₂	1	2	yes	maximal
5	C₄×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688878	—	2	2	no	—
11	#15688874	—	2	2	no	—
12	C₈×C₄₇⋊C₈	—	2	2	no	—
13	C₂×#106×C₄₇	C₄	1	1	yes	characteristic
14	C₂×#106×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#340	C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
24	C₄×C₄₇⋊C₈	—	2	1	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#106×C₄₇	—	2	2	no	—
35	#106×C₄₇	C₈	1	4	yes	complemented
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₄×C₈×C₄₇	Q₈	1	2	yes	—
38	#95×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄⋊C₄×C₄₇	—	2	2	no	—
52	C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
54	C₄²×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	—	4	2	no	—
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₇	#41	1	1	yes	characteristic
60	C₂×C₄₇⋊C₈	—	2	4	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂×C₄₇⋊C₈	—	4	2	no	—
67	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
68	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	—	4	2	no	—
74	C₂×C₄×C₄₇	—	4	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₄₇	#135	1	2	yes	—
88	C₈×C₄₇	—	2	2	no	—
89	C₈×C₄₇	C₄⋊C₈	1	4	yes	complemented
90	C₄₇⋊C₈	—	2	4	no	—
91	C₄₇⋊C₈	—	2	4	no	—
92	C₄₇⋊C₈	—	8	2	no	—
93	#9593	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	#389	1	1	yes	characteristic
95	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₄₇	#436	1	1	yes	characteristic
97	C₂²×C₄₇	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₄₇	#443	1	2	yes	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	4	4	no	—
102	C₄×C₄₇	—	8	2	no	—
103	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₄₇	#424	1	1	yes	characteristic
105	C₄×C₄₇	C₈×Q₈	1	1	yes	characteristic
106	C₄×C₄₇	#446	1	1	yes	characteristic
107	C₄×C₄₇	#465	1	2	yes	—
108	C₄×C₄₇	#498	1	1	yes	characteristic
109	C₄×C₄₇	#499	1	1	yes	characteristic
110	#1939	—	47	2	no	—
111	C₄×#106	D₄₇	1	1	yes	characteristic
112	C₂×#334	—	47	1	no	—
113	C₂×C₈²	—	47	1	no	—
114	#1279	—	47	2	no	—
115	C₂×C₄₇	C₈×SD₁₆	1	2	yes	—
116	C₂×C₄₇	#1590	1	2	yes	—
117	C₂×C₄₇	#1939	1	1	yes	characteristic
118	C₂×C₄₇	#1941	1	1	yes	characteristic
119	C₂×C₄₇	#1945	1	1	yes	characteristic
120	C₂×#106	Dic₄₇	1	1	yes	characteristic
121	C₂×#106	Dic₄₇	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	47	2	no	—
123	C₂×C₄⋊C₈	—	94	2	no	—
124	C₂×C₄×C₈	—	47	1	no	—
125	C₂×C₄×C₈	—	47	1	no	—
126	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
127	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
128	#340	Dic₄₇	1	2	yes	—
129	#336	—	94	2	no	—
130	#334	—	94	2	no	—
131	C₈²	—	94	2	no	—
132	C₄₇	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	47	2	no	—
134	C₂²×C₈	—	94	1	no	—
135	C₂²×C₈	—	94	2	no	—
136	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
142	#106	—	2	2	no	—
143	#106	C₄₇⋊C₈	1	4	yes	complemented
144	C₄⋊C₈	—	94	2	no	—
145	C₄⋊C₈	—	94	2	no	—
146	C₄⋊C₈	—	188	4	no	—
147	C₄×C₈	—	47	2	no	—
148	C₄×C₈	—	94	1	no	—
149	C₄×C₈	—	94	1	no	—
150	C₄×C₈	—	94	1	no	—
151	C₄×C₈	Dic₉₄	1	2	yes	—
152	C₄×C₈	D₁₈₈	1	2	yes	—
153	#95	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
157	C₂²×C₄	#10625352	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	47	4	no	—
162	C₂×C₈	—	94	4	no	—
163	C₂×C₈	—	94	2	no	—
164	C₂×C₈	—	94	2	no	—
165	C₂×C₈	—	94	2	no	—
166	C₂×C₈	—	188	2	no	—
167	C₂×C₈	—	188	2	no	—
168	C₂×C₈	—	188	2	no	—
169	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
170	C₂×C₈	#10625329	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
172	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
180	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
181	C₄²	Q₈×D₄₇	1	1	yes	characteristic
182	C₄²	#10625354	1	1	yes	characteristic
183	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
193	C₂×C₄	#13093999	1	1	yes	characteristic
194	C₂×C₄	#13094004	1	1	yes	characteristic
195	C₂×C₄	#13094015	1	1	yes	characteristic
196	C₂×C₄	#13094023	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
199	C₂×C₄	#13094031	1	1	yes	characteristic
200	C₂×C₄	#13094031	1	1	yes	characteristic
201	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
202	C₂×C₄	#13094045	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	94	4	no	—
205	C₈	—	94	4	no	—
206	C₈	—	376	2	no	—
207	C₈	C₄⋊(C₄₇⋊C₈)	1	4	yes	complemented
208	C₂²	#15689022	1	1	yes	characteristic
209	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
210	C₂²	#15689201	1	1	yes	characteristic
211	C₂²	SD₁₆×Dic₄₇	1	2	yes	—
212	C₂²	#15689240	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
218	C₄	#15689150	1	1	yes	characteristic
219	C₄	Q₈×C₄₇⋊C₈	1	1	yes	characteristic
220	C₄	#15689254	1	1	yes	characteristic
221	C₄	#15689353	1	2	yes	—
222	C₄	#15689520	1	1	yes	characteristic
223	C₄	#15689523	1	1	yes	characteristic
224	C₂	SD₁₆×C₄₇⋊C₈	1	2	yes	—
225	C₂	#20242279	1	2	yes	—
226	C₂	#20244107	1	1	yes	characteristic
227	C₂	#20244117	1	1	yes	characteristic
228	C₂	#20244137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#106×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#106	D₄₇
hypocenter	C₄₇	#9593
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
35. #106×C₄₇	C₈	semidirect
89. C₈×C₄₇	C₄⋊C₈	semidirect
132. C₄₇	#9593	semidirect
143. #106	C₄₇⋊C₈	semidirect
207. C₈	C₄⋊(C₄₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#20244107	2
C₂×#15688874	2
C₂×C₈×C₄₇⋊C₈	2
#20241229	2
C₄×#106×C₄₇	2
#9593	47