Subgroups — #26106374

Order 12032.

698 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	C₂×#15689147	C₂	1	2	yes	maximal
2	#20241271	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	#15689147	—	2	4	no	—
6	#15689147	C₄	1	8	yes	complemented
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×#117×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#384	C₂²	1	2	yes	—
12	C₄₇×#382	C₄	1	2	yes	—
13	C₄₇×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
15	C₂×#10625329	—	2	8	no	—
16	C₂×#10625329	—	4	8	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	#117×C₄₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#95×C₄₇	—	2	4	no	—
35	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	4	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₄⋊C₄×C₄₇	—	4	4	no	—
50	C₄⋊C₄×C₄₇	—	4	4	no	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₂²⋊C₄×C₄₇	—	4	4	no	—
53	C₄²×C₄₇	—	4	4	no	—
54	C₄²×C₄₇	C₄²	1	2	yes	—
55	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
58	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
59	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	#10625329	—	4	16	no	—
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	4	8	no	—
64	C₂³×C₄₇	—	2	1	no	—
65	C₂³×C₄₇	—	2	1	no	—
66	C₂³×C₄₇	—	2	4	no	—
67	C₂³×C₄₇	#95	1	2	yes	—
68	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	8	no	—
77	C₂×C₄×C₄₇	—	4	4	no	—
78	C₂×C₄×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	#95	1	4	yes	—
80	C₂×C₄×C₄₇	#95	1	2	yes	—
81	C₂×C₄×C₄₇	#100	1	2	yes	—
82	C₂×C₄×C₄₇	#101	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
88	C₂×C₄×C₄₇	#137	1	4	yes	—
89	C₄₇⋊C₈	—	4	8	no	—
90	#8041	—	47	1	no	maximal, Sylow
91	C₂²×C₄₇	—	2	4	no	—
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	—	4	2	no	—
94	C₂²×C₄₇	—	4	2	no	—
95	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
96	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
98	C₂²×C₄₇	#417	1	4	yes	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	—	8	4	no	—
103	C₄×C₄₇	#343	1	4	yes	—
104	C₄×C₄₇	#418	1	2	yes	—
105	C₄×C₄₇	#419	1	2	yes	—
106	C₄×C₄₇	#428	1	4	yes	—
107	C₂×#423	—	47	2	no	—
108	#2285	D₄₇	1	1	yes	characteristic
109	#1296	—	47	4	no	—
110	C₂×C₄₇	—	4	2	no	—
111	C₂×C₄₇	#1560	1	2	yes	—
112	C₂×C₄₇	#1561	1	2	yes	—
113	C₂×C₄₇	#1736	1	2	yes	—
114	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
115	C₂²×M₁₆	—	94	2	no	—
116	C₂×#117	Dic₄₇	1	2	yes	—
117	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	#423	—	47	8	no	—
119	#423	—	94	4	no	—
120	C₂×C₄⋊C₈	—	47	4	no	—
121	C₂×C₄⋊C₈	—	94	2	no	—
122	#384	C₂×D₄₇	1	2	yes	—
123	#382	Dic₄₇	1	2	yes	—
124	#380	—	2	2	no	—
125	C₄×C₄⋊C₄	—	2	2	no	—
126	C₄₇	#8041	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
129	C₂×M₁₆	—	94	8	no	—
130	C₂×M₁₆	—	188	8	no	—
131	C₂²×C₈	—	94	4	no	—
132	#117	—	4	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
135	C₂×C₂²⋊C₄	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	Dic₉₄	1	2	yes	—
138	C₂×C₄²	C₄×D₄₇	1	4	yes	—
139	C₂×C₄²	D₁₈₈	1	2	yes	—
140	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
141	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊C₈	—	47	8	no	—
144	C₄⋊C₈	—	94	8	no	—
145	C₄⋊C₈	—	94	4	no	—
146	#95	—	2	4	no	—
147	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
154	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
156	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
157	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
158	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	M₁₆	—	188	16	no	—
162	C₂×C₈	—	94	8	no	—
163	C₂×C₈	—	188	8	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	4	4	no	—
168	C₄²	—	4	4	no	—
169	C₄²	C₄×Dic₄₇	1	2	yes	—
170	C₄²	Dic₄₇⋊C₄	1	4	yes	—
171	C₄²	C₄⋊Dic₄₇	1	2	yes	—
172	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
173	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
174	C₄²	C₂×C₄×D₄₇	1	2	yes	—
175	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₄₇⋊#95	1	2	yes	—
180	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	C₄₇⋊#95	1	4	yes	—
192	C₂×C₄	C₄₇⋊#95	1	2	yes	—
193	C₂×C₄	#13093941	1	2	yes	—
194	C₂×C₄	#13093943	1	2	yes	—
195	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
196	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
197	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
199	C₂×C₄	#13094049	1	4	yes	—
200	C₂×C₄	#13094054	1	4	yes	—
201	C₈	—	188	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
207	C₂²	C₂×#13093941	1	1	yes	characteristic
208	C₂²	C₂×#13093943	1	1	yes	characteristic
209	C₂²	#15689124	1	4	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	8	4	no	—
214	C₄	#15688892	1	4	yes	—
215	C₄	#15689127	1	2	yes	—
216	C₄	#15689131	1	2	yes	—
217	C₄	#15689166	1	4	yes	—
218	C₂	—	4	2	no	—
219	C₂	#20242123	1	2	yes	—
220	C₂	#20242126	1	2	yes	—
221	C₂	#20243109	1	2	yes	—
222	C₂	C₂×#15688892	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#8041
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #20241271	C₂	semidirect
6. #15689147	C₄	semidirect
126. C₄₇	#8041	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
#20241271	2
C₄₇×#2285	2
#8041	47