Subgroups — #26216662

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 246 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	1	yes	maximal, characteristic
2	C₂×#15689146	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243539	C₂	1	8	yes	maximal, complemented
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₄	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	#15689147	—	2	4	no	—
10	#15689146	—	2	16	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	16	yes	—
13	C₄²×Dic₄₇	C₄	1	2	yes	—
14	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
15	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
16	C₄²×Dic₄₇	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#423	—	2	4	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
25	C₂×C₄×Dic₄₇	—	2	16	no	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
32	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
35	Dic₄₇⋊C₈	—	2	32	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	8	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	4	yes	—
44	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	16	no	—
59	C₂²×Dic₄₇	C₄⋊C₄	1	16	yes	—
60	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
61	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
62	M₁₆×C₄₇	—	4	8	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	8	no	—
65	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₄₇	C₂×D₄	1	4	yes	—
68	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
69	C₄×Dic₄₇	—	2	8	no	—
70	C₄×Dic₄₇	—	2	16	no	—
71	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
72	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₄₇	M₁₆	1	16	yes	complemented
74	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
75	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
76	#10625329	—	4	8	no	—
77	C₂×C₄₇⋊C₈	—	2	8	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂³×C₄₇	—	2	4	no	—
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	16	no	—
85	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₄₇	#127	1	4	yes	—
90	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
91	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	32	no	—
99	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
100	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
101	C₂×Dic₄₇	C₂×M₁₆	1	16	yes	—
102	C₈×C₄₇	—	4	4	no	—
103	C₄₇⋊C₈	—	4	4	no	—
104	#20791	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	—	2	8	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	#384	1	4	yes	—
108	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
109	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	#384	1	4	yes	—
113	C₄×C₄₇	#423	1	8	yes	complemented
114	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
117	Dic₄₇	—	2	8	no	—
118	Dic₄₇	#423	1	16	yes	complemented
119	C₂×#423	—	47	1	no	—
120	C₂×#423	—	47	4	no	—
121	C₂×#423	D₄₇	1	1	yes	characteristic
122	C₂×C₄³	—	47	1	no	—
123	#1842	—	47	8	no	—
124	C₂×C₄₇	—	2	4	no	—
125	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#423	1	4	yes	—
127	C₂×C₄₇	C₂×#423	1	2	yes	—
128	C₂²×M₁₆	—	47	2	no	—
129	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
130	C₂²×C₄²	—	47	2	no	—
131	C₂²×C₄²	—	47	4	no	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	#423	—	2	4	no	—
134	#423	—	94	16	no	—
135	#423	—	94	4	no	—
136	C₂×C₄⋊C₈	—	47	16	no	—
137	C₂×C₄⋊C₈	—	47	4	no	—
138	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
139	C₄³	—	47	1	no	—
140	C₄³	—	47	2	no	—
141	C₄³	—	47	1	no	—
142	C₄³	—	47	4	no	—
143	C₄₇	#20791	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	—	47	4	no	—
145	C₂³×C₄	C₄×D₄₇	1	2	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×M₁₆	—	2	8	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	94	8	no	—
150	C₂×M₁₆	—	94	4	no	—
151	C₂²×C₈	—	47	4	no	—
152	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
153	C₂×C₄²	—	47	2	no	—
154	C₂×C₄²	—	47	2	no	—
155	C₂×C₄²	—	47	4	no	—
156	C₂×C₄²	—	47	8	no	—
157	C₂×C₄²	—	47	8	no	—
158	C₂×C₄²	—	47	4	no	—
159	C₂×C₄²	—	47	8	no	—
160	C₂×C₄²	—	47	4	no	—
161	C₂×C₄²	—	94	16	no	—
162	C₂×C₄²	Dic₉₄	1	4	yes	—
163	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
166	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
167	C₄⋊C₈	—	2	8	no	—
168	C₄⋊C₈	—	94	32	no	—
169	C₄⋊C₈	—	94	8	no	—
170	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	47	4	no	—
173	C₂²×C₄	—	47	16	no	—
174	C₂²×C₄	—	47	4	no	—
175	C₂²×C₄	—	94	16	no	—
176	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
177	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
181	C₂²×C₄	D₄×D₄₇	1	4	yes	—
182	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	M₁₆	—	4	8	no	—
186	M₁₆	—	188	8	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	94	8	no	—
190	C₂×C₈	—	94	4	no	—
191	C₄²	—	47	8	no	—
192	C₄²	—	47	8	no	—
193	C₄²	—	47	16	no	—
194	C₄²	—	47	8	no	—
195	C₄²	—	47	8	no	—
196	C₄²	—	94	16	no	—
197	C₄²	—	94	8	no	—
198	C₄²	Dic₄₇⋊C₄	1	4	yes	—
199	C₄²	Dic₄₇⋊C₄	1	4	yes	—
200	C₄²	C₂×Dic₉₄	1	4	yes	—
201	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	C₄⋊C₄×D₄₇	1	4	yes	—
204	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
205	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	47	8	no	—
209	C₂×C₄	—	47	16	no	—
210	C₂×C₄	—	47	8	no	—
211	C₂×C₄	—	94	32	no	—
212	C₂×C₄	—	94	8	no	—
213	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
214	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
215	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
217	C₂×C₄	#13094020	1	4	yes	—
218	C₂×C₄	#13094022	1	4	yes	—
219	C₂×C₄	#13094023	1	8	yes	—
220	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
221	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
222	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
224	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
225	C₈	—	4	4	no	—
226	C₈	—	188	4	no	—
227	C₂²	—	2	8	no	—
228	C₂²	—	2	4	no	—
229	C₂²	#15688992	1	4	yes	—
230	C₂²	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
231	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
232	C₂²	C₂×M₁₆×D₄₇	1	4	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	47	16	no	—
235	C₄	—	94	8	no	—
236	C₄	#15688992	1	4	yes	—
237	C₄	#15689146	1	8	yes	complemented
238	C₄	C₂×#13094020	1	1	yes	characteristic
239	C₄	C₂×#13094022	1	1	yes	characteristic
240	C₄	C₂×#13094023	1	2	yes	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#15688992	1	1	yes	characteristic
243	C₂	#423×D₄₇	1	4	yes	—
244	C₂	C₂×#15689146	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#423	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20791
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20243539	C₂	semidirect
73. C₄×Dic₄₇	M₁₆	semidirect
113. C₄×C₄₇	#423	semidirect
118. Dic₄₇	#423	semidirect
143. C₄₇	#20791	semidirect
237. C₄	#15689146	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
C₂×#15689146	2
C₂×C₄²×Dic₄₇	2
#20243539	2
C₂×C₄₇×#423	2
#20791	47