Subgroups — #26127670

Order 12032.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	1	yes	maximal, characteristic
2	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20241437	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic
5	#15689147	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	—	2	1	no	—
9	#15688876	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₄₇	—	2	1	no	—
12	C₂²×Q₁₆×C₄₇	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
14	C₄₇×#494	—	2	1	no	—
15	C₄₇×#494	—	2	1	no	—
16	C₄₇×#494	C₄	1	2	yes	—
17	C₄₇×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	4	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₄₇	—	2	2	no	—
24	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₂×Q₁₆×C₄₇	—	2	2	no	—
26	C₂×Q₁₆×C₄₇	—	2	2	no	—
27	C₂×Q₁₆×C₄₇	—	4	4	no	—
28	C₂×Q₁₆×C₄₇	—	4	4	no	—
29	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	2	no	—
34	C₄⋊Q₈×C₄₇	—	4	1	no	—
35	C₄⋊Q₈×C₄₇	—	4	1	no	—
36	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₄₇	—	2	4	no	—
44	C₂×Q₈×C₄₇	—	4	4	no	—
45	C₂×Q₈×C₄₇	—	4	4	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₄₇	—	4	4	no	—
52	Q₁₆×C₄₇	—	4	4	no	—
53	Q₁₆×C₄₇	—	4	4	no	—
54	Q₁₆×C₄₇	—	4	4	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₄⋊C₄×C₄₇	—	4	2	no	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₄⋊C₄×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₄₇	D₈	1	1	yes	characteristic
67	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
69	#10625329	—	8	2	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₄₇⋊C₁₆	—	4	2	no	—
72	C₂³×C₄₇	#102	1	1	yes	characteristic
73	Q₈×C₄₇	—	4	4	no	—
74	Q₈×C₄₇	—	4	4	no	—
75	Q₈×C₄₇	—	8	4	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	8	4	no	—
81	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
85	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₄₇	—	2	1	no	—
90	C₈×C₄₇	—	2	1	no	—
91	C₈×C₄₇	#100	1	2	yes	—
92	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
93	C₄₇⋊C₈	—	8	2	no	—
94	#11962	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	#358	1	2	yes	—
98	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	8	4	no	—
102	C₄×C₄₇	#358	1	2	yes	—
103	C₄×C₄₇	#362	1	2	yes	—
104	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
105	C₄×C₄₇	#418	1	1	yes	characteristic
106	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
107	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
108	C₂×#494	D₄₇	1	1	yes	characteristic
109	#2162	—	47	1	no	—
110	#1567	—	47	1	no	—
111	#1362	—	47	4	no	—
112	C₂×C₄₇	—	2	2	no	—
113	C₂×C₄₇	#1545	1	1	yes	characteristic
114	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
115	C₂×C₄₇	C₂×#362	1	1	yes	characteristic
116	C₂²×Q₁₆	—	2	1	no	—
117	C₂²×Q₁₆	Dic₄₇	1	2	yes	—
118	C₂×C₄⋊Q₈	—	2	1	no	—
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	Dic₄₇	1	2	yes	—
122	#494	C₂×D₄₇	1	2	yes	—
123	#423	—	94	1	no	—
124	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
125	C₄⋊C₁₆	—	47	2	no	—
126	C₂²⋊C₁₆	—	94	1	no	—
127	C₄×C₁₆	—	94	1	no	—
128	#335	—	47	2	no	—
129	#335	—	94	1	no	—
130	C₄₇	#11962	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
138	C₂×M₁₆	—	188	1	no	—
139	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂×C₁₆	—	94	2	no	—
149	C₄⋊C₈	—	94	2	no	—
150	C₄⋊C₈	—	188	1	no	—
151	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#10625334	1	1	yes	characteristic
161	C₂²×C₄	#10625335	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	376	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	188	2	no	—
171	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	#10625334	1	1	yes	characteristic
180	C₄²	#10625335	1	1	yes	characteristic
181	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
183	C₁₆	—	188	2	no	—
184	C₂³	#13093945	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#13093945	1	1	yes	characteristic
194	C₂×C₄	#13093945	1	1	yes	characteristic
195	C₂×C₄	#13093945	1	1	yes	characteristic
196	C₂×C₄	#13093967	1	2	yes	—
197	C₂×C₄	#13093970	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
199	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
200	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	376	2	no	—
204	C₈	#13093941	1	2	yes	—
205	C₈	#13094050	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#15688927	1	2	yes	—
209	C₂²	C₂×#13093945	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#15688927	1	2	yes	—
214	C₄	#15688939	1	2	yes	—
215	C₄	C₂×#13093941	1	1	yes	characteristic
216	C₄	#15689128	1	1	yes	characteristic
217	C₄	C₂×#13093967	1	1	yes	characteristic
218	C₄	C₂×#13093970	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#20242056	1	1	yes	characteristic
221	C₂	C₂×#15688927	1	1	yes	characteristic
222	C₂	C₂×#15688939	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#11962
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20241437	C₂	semidirect
130. C₄₇	#11962	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
#20242147	2
#20241437	2
C₂×C₄₇×#494	2
#11962	47