Subgroups — #26106371

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	2	yes	maximal
2	#20241271	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	4	no	—
5	#15689180	C₄	1	8	yes	complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×#13093935	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#384	—	2	2	no	—
13	C₄₇×#384	C₄	1	2	yes	—
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
18	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
19	#13093935	—	2	8	no	—
20	C₂³×C₄×C₄₇	—	2	2	no	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₄₇	—	4	8	no	—
27	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
28	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
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	—	4	8	no	—
42	C₂²×C₄×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₄⋊C₄×C₄₇	—	4	8	no	—
53	C₄⋊C₄×C₄₇	—	4	4	no	—
54	C₄⋊C₄×C₄₇	—	4	8	no	—
55	C₄²×C₄₇	C₄²	1	2	yes	—
56	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
59	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄₇⋊C₈	—	2	8	no	—
63	C₂×C₄₇⋊C₈	—	4	4	no	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂³×C₄₇	—	2	2	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	#95	1	2	yes	—
68	C₂³×C₄₇	#100	1	2	yes	—
69	C₂³×C₄₇	#101	1	2	yes	—
70	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	4	8	no	—
83	C₂×C₄×C₄₇	—	8	8	no	—
84	C₂×C₄×C₄₇	#95	1	4	yes	—
85	C₂×C₄×C₄₇	#95	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
91	C₂×C₄×C₄₇	#137	1	4	yes	—
92	C₄₇⋊C₈	—	4	8	no	—
93	#8040	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	2	4	no	—
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	#343	1	4	yes	—
99	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
100	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
101	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
102	C₂²×C₄₇	#418	1	2	yes	—
103	C₂²×C₄₇	#419	1	2	yes	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	8	4	no	—
109	C₄×C₄₇	#417	1	4	yes	—
110	C₄×C₄₇	#428	1	4	yes	—
111	C₂×#432	—	47	2	no	—
112	C₂×#384	D₄₇	1	1	yes	characteristic
113	#1296	—	47	4	no	—
114	C₂×C₄₇	—	2	4	no	—
115	C₂×C₄₇	#1555	1	1	yes	characteristic
116	C₂×C₄₇	#1556	1	2	yes	—
117	C₂×C₄₇	#1558	1	1	yes	characteristic
118	C₂×C₄₇	#1736	1	2	yes	—
119	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
120	C₂²×C₄⋊C₄	—	2	2	no	—
121	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
122	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	#432	—	47	8	no	—
124	#432	—	94	4	no	—
125	C₂×C₄⋊C₈	—	47	4	no	—
126	C₂×C₂²⋊C₈	—	94	2	no	—
127	C₂×#97	—	94	2	no	—
128	#384	—	2	2	no	—
129	#384	Dic₄₇	1	2	yes	—
130	#384	C₂×D₄₇	1	2	yes	—
131	C₄₇	#8040	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
135	C₂²×C₈	—	94	4	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	4	4	no	—
140	C₂×C₄⋊C₄	—	4	8	no	—
141	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
142	C₂×C₄²	Dic₉₄	1	2	yes	—
143	C₂×C₄²	C₄×D₄₇	1	4	yes	—
144	C₂×C₄²	D₁₈₈	1	2	yes	—
145	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
147	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
148	C₄⋊C₈	—	47	8	no	—
149	C₄⋊C₈	—	94	4	no	—
150	C₂²⋊C₈	—	94	8	no	—
151	#97	—	94	8	no	—
152	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
161	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
162	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
163	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
164	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
165	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₈	—	94	8	no	—
169	C₂×C₈	—	188	4	no	—
170	C₂×C₈	—	188	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄²	C₄×Dic₄₇	1	2	yes	—
176	C₄²	Dic₄₇⋊C₄	1	4	yes	—
177	C₄²	C₄⋊Dic₄₇	1	2	yes	—
178	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
179	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
180	C₄²	C₂×C₄×D₄₇	1	2	yes	—
181	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₄₇⋊#95	1	2	yes	—
185	C₂³	#13093941	1	2	yes	—
186	C₂³	#13093943	1	2	yes	—
187	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₄₇⋊#95	1	4	yes	—
202	C₂×C₄	C₄₇⋊#95	1	2	yes	—
203	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
207	C₂×C₄	#13094050	1	4	yes	—
208	C₂×C₄	#13094055	1	4	yes	—
209	C₈	—	188	8	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	2	no	—
214	C₂²	#15688892	1	4	yes	—
215	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
216	C₂²	C₂×#13093941	1	1	yes	characteristic
217	C₂²	C₂×#13093943	1	1	yes	characteristic
218	C₂²	#15689128	1	2	yes	—
219	C₂²	#15689132	1	2	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	#15689122	1	4	yes	—
226	C₄	#15689163	1	4	yes	—
227	C₂	—	2	4	no	—
228	C₂	#20242102	1	1	yes	characteristic
229	C₂	#20242105	1	2	yes	—
230	C₂	#20242114	1	1	yes	characteristic
231	C₂	#20243107	1	2	yes	—
232	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	C₂×C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	#8040
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #20241271	C₂	semidirect
5. #15689180	C₄	semidirect
17. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
131. C₄₇	#8040	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
#20241271	2
C₂×C₄₇×#384	2
#8040	47