Subgroups — #26226630

Order 12032.

914 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×#15689180	C₂	1	2	yes	maximal
2	#20244112	C₂	1	8	yes	maximal, complemented
3	#20243368	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	8	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×#13093935	C₂²	1	4	yes	—
10	#15688878	C₂²	1	8	yes	—
11	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
16	C₂×#95×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#95×C₄₇	C₄	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
22	#13093935	—	2	8	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₄₇	—	4	4	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	4	yes	—
34	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
37	#95×C₄₇	C₂×C₄	1	4	yes	—
38	#95×C₄₇	C₂×C₄	1	4	yes	—
39	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	2	4	no	—
41	C₂²×C₄×C₄₇	—	2	2	no	—
42	C₂²×C₄×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	—	4	2	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	—	4	2	no	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄×C₄₇	—	2	16	no	—
61	C₄²×C₄₇	—	2	8	no	—
62	C₄²×C₄₇	—	4	4	no	—
63	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
64	C₄²×C₄₇	C₂×D₄	1	4	yes	—
65	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
66	C₂×C₄₇⋊C₈	—	2	8	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	2	8	no	—
69	C₂³×C₄₇	—	2	1	no	—
70	C₂³×C₄₇	—	2	1	no	—
71	C₂³×C₄₇	—	2	2	no	—
72	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	4	8	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
97	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
98	C₄₇⋊C₈	—	4	8	no	—
99	#21955	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	#423	1	4	yes	—
105	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₇	#518	1	1	yes	characteristic
107	C₂²×C₄₇	#519	1	1	yes	characteristic
108	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
109	C₂²×C₄₇	#568	1	2	yes	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	2	4	no	—
113	C₄×C₄₇	—	2	8	no	—
114	C₄×C₄₇	—	4	4	no	—
115	C₄×C₄₇	#537	1	2	yes	—
116	C₄×C₄₇	#544	1	2	yes	—
117	C₂×#432	—	47	2	no	—
118	C₂×C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
119	#1940	—	47	8	no	—
120	#1797	—	47	4	no	—
121	C₂×C₄₇	—	2	4	no	—
122	C₂×C₄₇	#2316	1	1	yes	characteristic
123	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
124	C₂×C₄₇	#2905	1	1	yes	characteristic
125	C₂×C₄₇	#2973	1	2	yes	—
126	C₂×C₄₇	#2983	1	2	yes	—
127	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
128	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
129	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
130	#432	—	94	8	no	—
131	C₂×C₄⋊C₈	—	47	4	no	—
132	C₂×C₄⋊C₈	—	47	8	no	—
133	C₂×C₂²⋊C₈	—	47	4	no	—
134	C₂×#97	—	47	4	no	—
135	C₄×C₄⋊C₄	Dic₄₇	1	4	yes	—
136	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
137	C₂×#95	Dic₄₇	1	1	yes	characteristic
138	C₂×#95	Dic₄₇	1	1	yes	characteristic
139	#336	—	47	8	no	—
140	C₄₇	#21955	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
142	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
143	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
144	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂²×C₈	—	47	8	no	—
147	C₂×C₄⋊C₄	—	2	8	no	—
148	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
149	C₂×C₄²	—	4	4	no	—
150	C₂×C₄²	Dic₉₄	1	4	yes	—
151	C₂×C₄²	D₁₈₈	1	4	yes	—
152	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
153	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
156	C₄⋊C₈	—	94	8	no	—
157	C₄⋊C₈	—	188	8	no	—
158	C₂²⋊C₈	—	94	8	no	—
159	#97	—	94	8	no	—
160	#95	C₂×Dic₄₇	1	4	yes	—
161	#95	C₂×Dic₄₇	1	4	yes	—
162	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	#10625329	1	8	yes	—
173	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
174	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
175	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
176	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	C₂×C₈	—	94	8	no	—
184	C₂×C₈	—	94	8	no	—
185	C₂×C₈	—	94	8	no	—
186	C₄⋊C₄	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	4	4	no	—
189	C₄²	C₄⋊Dic₄₇	1	8	yes	—
190	C₄²	C₂×Dic₉₄	1	4	yes	—
191	C₄²	C₂×D₁₈₈	1	4	yes	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
196	C₂³	C₂×#10625329	1	4	yes	—
197	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
215	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
216	C₂×C₄	C₂×#10625329	1	8	yes	—
217	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
219	C₂×C₄	#13094069	1	2	yes	—
220	C₂×C₄	#13094072	1	2	yes	—
221	C₈	—	188	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#15689147	1	4	yes	—
227	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
228	C₂²	#15689598	1	1	yes	characteristic
229	C₂²	#15689601	1	1	yes	characteristic
230	C₂²	C₂²×#10625329	1	2	yes	—
231	C₂²	#15689890	1	2	yes	—
232	C₄	—	2	8	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	4	no	—
237	C₄	#15689704	1	2	yes	—
238	C₄	#15689754	1	2	yes	—
239	C₂	—	2	4	no	—
240	C₂	#20246006	1	1	yes	characteristic
241	C₂	C₂×#15689147	1	1	yes	characteristic
242	C₂	#20251952	1	1	yes	characteristic
243	C₂	#20252417	1	2	yes	—
244	C₂	#20252473	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄×C₄⋊C₄×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#21955
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20244112	C₂	semidirect
140. C₄₇	#21955	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
#20244112	2
#20243368	2
C₂×C₄×C₄⋊C₄×C₄₇	2
#21955	47