Subgroups — #26137766

Order 12032.

1546 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₄₇	C₂	1	3	yes	maximal
2	#20241618	C₂	1	3	yes	maximal
3	#20241614	C₂	1	6	yes	maximal
4	#95×Dic₄₇	C₂	1	2	yes	maximal
5	C₄×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
9	C₄×C₄⋊Dic₄₇	C₄	1	24	yes	complemented
10	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
12	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
14	C₂×#95×C₄₇	C₂²	1	3	yes	—
15	C₂×#95×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×Dic₄₇	C₂³	1	6	yes	—
17	C₂³×Dic₄₇	C₂³	1	2	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	6	no	—
19	C₂×C₄⋊Dic₄₇	—	2	6	no	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	12	yes	—
21	C₂×C₄⋊Dic₄₇	C₂×C₄	1	12	yes	—
22	C₂×C₄×Dic₄₇	—	4	12	no	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	24	yes	—
24	C₄₇⋊#95	—	2	12	no	—
25	C₄₇⋊#95	—	4	6	no	—
26	C₄₇⋊#95	C₂×C₄	1	24	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	—	2	12	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
32	#95×C₄₇	—	2	6	no	—
33	#95×C₄₇	Q₈	1	4	yes	—
34	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	6	no	—
36	C₂²×C₄×C₄₇	—	2	6	no	—
37	C₂²×C₄×C₄₇	—	2	6	no	—
38	C₂²×C₄×C₄₇	—	2	6	no	—
39	C₂²×C₄×C₄₇	—	2	3	no	—
40	C₂²×C₄×C₄₇	—	2	3	no	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₄₇	#41	1	6	yes	—
46	C₂²×C₄×C₄₇	#41	1	6	yes	—
47	C₂²×C₄×C₄₇	#41	1	2	yes	—
48	C₂²×Dic₄₇	—	2	6	no	—
49	C₂²×Dic₄₇	—	2	2	no	—
50	C₂²×Dic₄₇	—	2	12	no	—
51	C₂²×Dic₄₇	—	2	6	no	—
52	C₂²×Dic₄₇	—	2	6	no	—
53	C₂²×Dic₄₇	—	2	12	no	—
54	C₂²×Dic₄₇	C₂²×C₄	1	12	yes	—
55	C₂²×Dic₄₇	C₂²×C₄	1	12	yes	—
56	C₄²×C₄₇	—	2	24	no	—
57	C₄⋊Dic₄₇	—	2	24	no	—
58	C₄⋊Dic₄₇	C₄²	1	16	yes	—
59	C₄×Dic₄₇	—	4	24	no	—
60	C₂³×C₄₇	C₂³×C₄	1	3	yes	—
61	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂³×C₄₇	C₂×#41	1	3	yes	—
63	C₂³×C₄₇	C₂×#41	1	3	yes	—
64	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
65	C₂³×C₄₇	H(5,𝔽₂)	1	3	yes	—
66	C₂³×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	6	no	—
68	C₂×C₄×C₄₇	—	2	6	no	—
69	C₂×C₄×C₄₇	—	2	12	no	—
70	C₂×C₄×C₄₇	—	2	12	no	—
71	C₂×C₄×C₄₇	—	2	6	no	—
72	C₂×C₄×C₄₇	—	2	6	no	—
73	C₂×C₄×C₄₇	—	2	6	no	—
74	C₂×C₄×C₄₇	—	4	6	no	—
75	C₂×C₄×C₄₇	—	4	6	no	—
76	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₄₇	#117	1	12	yes	—
78	C₂×C₄×C₄₇	#117	1	12	yes	—
79	C₂×C₄×C₄₇	C₄×Q₈	1	12	yes	—
80	C₂×Dic₄₇	—	2	12	no	—
81	C₂×Dic₄₇	—	2	12	no	—
82	C₂×Dic₄₇	—	2	12	no	—
83	C₂×Dic₄₇	—	2	12	no	—
84	C₂×Dic₄₇	—	2	12	no	—
85	C₂×Dic₄₇	—	4	12	no	—
86	C₂×Dic₄₇	—	4	12	no	—
87	C₂×Dic₄₇	C₂×C₄²	1	8	yes	—
88	#13812	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×#117	1	3	yes	—
91	C₂²×C₄₇	C₂×#117	1	3	yes	—
92	C₂²×C₄₇	C₂×C₄×Q₈	1	3	yes	—
93	C₂²×C₄₇	C₄×#41	1	3	yes	—
94	C₂²×C₄₇	#518	1	3	yes	—
95	C₂²×C₄₇	#520	1	3	yes	—
96	C₂²×C₄₇	#553	1	6	yes	—
97	C₂²×C₄₇	#554	1	3	yes	—
98	C₂²×C₄₇	#556	1	3	yes	—
99	C₂²×C₄₇	#557	1	1	yes	characteristic
100	C₂²×C₄₇	#559	1	3	yes	—
101	C₄×C₄₇	—	2	12	no	—
102	C₄×C₄₇	—	4	6	no	—
103	C₄×C₄₇	#376	1	8	yes	—
104	Dic₄₇	—	2	8	no	—
105	Dic₄₇	—	4	12	no	—
106	C₂×C₄×C₄⋊C₄	—	47	3	no	—
107	C₄⋊#95	—	47	3	no	—
108	#1434	—	47	6	no	—
109	C₄×#95	—	47	2	no	—
110	C₄×#95	D₄₇	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
112	C₂×C₄₇	C₄²×Q₈	1	1	yes	characteristic
113	C₂×C₄₇	#2275	1	1	yes	characteristic
114	C₂×C₄₇	#2345	1	3	yes	—
115	C₂×C₄₇	#2347	1	3	yes	—
116	C₂×C₄₇	#2355	1	3	yes	—
117	C₂×C₄₇	#2357	1	3	yes	—
118	C₂²×C₄⋊C₄	—	47	1	no	—
119	C₂²×C₄⋊C₄	—	47	3	no	—
120	C₂²×C₄²	—	47	6	no	—
121	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
122	C₄×C₄⋊C₄	—	47	24	no	—
123	C₂×#95	—	47	6	no	—
124	C₂×#95	—	47	6	no	—
125	C₂×#95	—	47	6	no	—
126	C₂×#95	C₂×D₄₇	1	3	yes	—
127	C₂×#95	C₂×D₄₇	1	1	yes	characteristic
128	C₄₇	#13812	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	—	47	2	no	—
130	C₂³×C₄	—	47	6	no	—
131	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
132	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
133	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	47	12	no	—
135	C₂×C₄⋊C₄	—	47	12	no	—
136	C₂×C₄⋊C₄	—	94	6	no	—
137	C₂×C₄⋊C₄	—	94	6	no	—
138	C₂×C₄²	—	2	12	no	—
139	C₂×C₄²	—	47	24	no	—
140	C₂×C₄²	—	188	12	no	—
141	C₂×C₄²	C₄×D₄₇	1	12	yes	—
142	#95	—	2	6	no	—
143	#95	—	47	24	no	—
144	#95	—	94	12	no	—
145	#95	—	188	6	no	—
146	#95	Dic₉₄	1	4	yes	—
147	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	6	no	—
149	C₂²×C₄	—	2	6	no	—
150	C₂²×C₄	—	2	6	no	—
151	C₂²×C₄	—	2	6	no	—
152	C₂²×C₄	—	2	3	no	—
153	C₂²×C₄	—	2	3	no	—
154	C₂²×C₄	—	47	12	no	—
155	C₂²×C₄	—	47	12	no	—
156	C₂²×C₄	—	94	6	no	—
157	C₂²×C₄	—	94	2	no	—
158	C₂²×C₄	—	94	6	no	—
159	C₂²×C₄	—	94	12	no	—
160	C₂²×C₄	—	94	6	no	—
161	C₂²×C₄	—	94	12	no	—
162	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
163	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
164	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
165	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
166	C₂²×C₄	#10625350	1	6	yes	—
167	C₂²×C₄	#10625352	1	6	yes	—
168	C₂²×C₄	#10625354	1	2	yes	—
169	C₄⋊C₄	—	47	16	no	—
170	C₄⋊C₄	—	94	24	no	—
171	C₄²	—	2	24	no	—
172	C₄²	—	188	24	no	—
173	C₂³	C₂²×C₄×D₄₇	1	3	yes	—
174	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
175	C₂³	C₂×#10625352	1	3	yes	—
176	C₂³	C₂×#10625350	1	3	yes	—
177	C₂³	C₂×#10625354	1	1	yes	characteristic
178	C₂³	#13094070	1	3	yes	—
179	C₂³	#13094071	1	1	yes	characteristic
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	12	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	2	6	no	—
185	C₂×C₄	—	2	6	no	—
186	C₂×C₄	—	2	6	no	—
187	C₂×C₄	—	4	6	no	—
188	C₂×C₄	—	4	6	no	—
189	C₂×C₄	—	47	8	no	—
190	C₂×C₄	—	94	12	no	—
191	C₂×C₄	—	94	12	no	—
192	C₂×C₄	—	94	12	no	—
193	C₂×C₄	—	94	12	no	—
194	C₂×C₄	—	94	12	no	—
195	C₂×C₄	—	188	12	no	—
196	C₂×C₄	—	188	12	no	—
197	C₂×C₄	C₄²×D₄₇	1	4	yes	—
198	C₂×C₄	#13093980	1	12	yes	—
199	C₂×C₄	#13093982	1	12	yes	—
200	C₂×C₄	C₄×Dic₉₄	1	12	yes	—
201	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
202	C₂²	C₂×#13093980	1	3	yes	—
203	C₂²	C₂×#13093982	1	3	yes	—
204	C₂²	C₂×C₄×Dic₉₄	1	3	yes	—
205	C₂²	C₄×#10625350	1	3	yes	—
206	C₂²	#15689597	1	3	yes	—
207	C₂²	#15689603	1	3	yes	—
208	C₂²	#15689819	1	6	yes	—
209	C₂²	#15689830	1	3	yes	—
210	C₂²	#15689843	1	3	yes	—
211	C₂²	#15689852	1	1	yes	characteristic
212	C₂²	#15689858	1	3	yes	—
213	C₄	—	2	12	no	—
214	C₄	—	4	6	no	—
215	C₄	—	94	8	no	—
216	C₄	—	188	12	no	—
217	C₄	#15688968	1	8	yes	—
218	C₂	C₂×#15688968	1	1	yes	characteristic
219	C₂	C₄²×Dic₉₄	1	1	yes	characteristic
220	C₂	#20245750	1	1	yes	characteristic
221	C₂	#20246263	1	3	yes	—
222	C₂	#20246279	1	3	yes	—
223	C₂	#20246343	1	3	yes	—
224	C₂	#20246359	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#95×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₄×#95	D₄₇
hypocenter	C₄₇	#13812
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
9. C₄×C₄⋊Dic₄₇	C₄	semidirect
128. C₄₇	#13812	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₇	2
#20241618	2
#20241614	2
#95×Dic₄₇	2
C₄×#95×C₄₇	2
#13812	47