Subgroups — #26122086

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₂×C₄₇⋊#338	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15689147	—	2	4	no	—
6	#15689147	—	2	2	no	—
7	#15689147	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	—
9	C₄₇⋊#338	—	2	8	no	—
10	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₄₇	—	2	2	no	—
13	C₄×M₁₆×C₄₇	—	2	2	no	—
14	C₂×#97×C₄₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	—	2	4	no	—
18	C₂×#10625329	—	2	4	no	—
19	C₂×#10625329	—	2	8	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₄₇	—	2	4	no	—
26	C₂×M₁₆×C₄₇	—	2	4	no	—
27	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
28	C₂×M₁₆×C₄₇	Q₈	1	4	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	#97×C₄₇	—	2	4	no	—
37	C₄×C₈×C₄₇	—	2	2	no	—
38	C₄×C₈×C₄₇	—	2	2	no	—
39	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	M₁₆×C₄₇	—	2	8	no	—
53	M₁₆×C₄₇	—	2	8	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₄²×C₄₇	C₄²	1	2	yes	—
58	C₄²×C₄₇	C₄²	1	2	yes	—
59	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
64	#10625329	—	4	8	no	—
65	#10625329	—	4	4	no	—
66	#10625329	—	4	4	no	—
67	C₂×C₄₇⋊C₈	—	2	8	no	—
68	C₂×C₄₇⋊C₈	—	2	16	no	—
69	C₂³×C₄₇	—	4	2	no	—
70	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₄₇	#125	1	2	yes	—
73	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	—	2	8	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	4	4	no	—
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	#95	1	2	yes	—
79	C₂×C₄×C₄₇	#95	1	4	yes	—
80	C₂×C₄×C₄₇	#95	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	#117	1	2	yes	—
86	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
88	C₂×C₄×C₄₇	#121	1	4	yes	—
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₄₇	#124	1	2	yes	—
91	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₈×C₄₇	—	2	8	no	—
94	C₄₇⋊C₈	—	4	8	no	—
95	C₄⋊#338	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	4	2	no	—
99	C₂²×C₄₇	—	4	2	no	—
100	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
101	C₂²×C₄₇	#382	1	1	yes	characteristic
102	C₂²×C₄₇	#384	1	1	yes	characteristic
103	C₂²×C₄₇	#389	1	4	yes	—
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	#338	1	4	yes	complemented
107	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
108	C₄×C₄₇	#380	1	2	yes	—
109	C₄×C₄₇	#390	1	2	yes	—
110	C₄×C₄₇	#391	1	2	yes	—
111	C₂×#423	—	47	2	no	—
112	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
113	C₂×#338	—	47	4	no	—
114	C₂×C₄₇	—	2	1	no	—
115	C₂×C₄₇	—	2	1	no	—
116	C₂×C₄₇	—	2	4	no	—
117	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#338	1	2	yes	—
119	C₂²×M₁₆	—	47	4	no	—
120	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
121	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
122	#423	—	94	4	no	—
123	#423	—	94	2	no	—
124	#423	—	94	2	no	—
125	C₂×C₄⋊C₈	—	47	8	no	—
126	C₄×M₁₆	—	2	2	no	—
127	C₄×M₁₆	—	2	2	no	—
128	C₂×#97	Dic₄₇	1	2	yes	—
129	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
130	#338	—	94	8	no	—
131	C₄₇	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	Dic₉₄	1	2	yes	—
133	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	94	8	no	—
137	C₂×M₁₆	—	94	4	no	—
138	C₂×M₁₆	—	94	8	no	—
139	C₂×M₁₆	—	94	4	no	—
140	C₂×M₁₆	Dic₉₄	1	4	yes	—
141	C₂×M₁₆	D₁₈₈	1	4	yes	—
142	C₂²×C₈	—	47	8	no	—
143	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
144	C₂×C₄²	C₄×D₄₇	1	4	yes	—
145	C₂×C₄²	D₁₈₈	1	2	yes	—
146	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
150	C₄⋊C₈	—	94	8	no	—
151	C₄⋊C₈	—	94	8	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
158	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
159	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
160	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
161	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
163	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
165	C₂²×C₄	#10625350	1	4	yes	—
166	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	8	no	—
170	M₁₆	—	188	8	no	—
171	M₁₆	—	188	4	no	—
172	M₁₆	—	188	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	94	8	no	—
176	C₂×C₈	—	94	16	no	—
177	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
178	C₄²	C₄×Dic₄₇	1	2	yes	—
179	C₄²	C₄×Dic₄₇	1	2	yes	—
180	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
181	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
182	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
187	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
188	C₂³	#13094016	1	2	yes	—
189	C₂³	#13094024	1	2	yes	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₄₇⋊#95	1	2	yes	—
195	C₂×C₄	C₄₇⋊#95	1	4	yes	—
196	C₂×C₄	C₄₇⋊#95	1	2	yes	—
197	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
201	C₂×C₄	#13093983	1	2	yes	—
202	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
203	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
204	C₂×C₄	#13093998	1	4	yes	—
205	C₂×C₄	#13094003	1	4	yes	—
206	C₂×C₄	#13094012	1	2	yes	—
207	C₂×C₄	#13094021	1	1	yes	characteristic
208	C₂×C₄	#13094024	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	188	8	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
216	C₂²	#15688989	1	1	yes	characteristic
217	C₂²	#15688993	1	1	yes	characteristic
218	C₂²	#15689021	1	4	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	C₄₇⋊#338	1	4	yes	complemented
222	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
223	C₄	#15688983	1	2	yes	—
224	C₄	#15689025	1	2	yes	—
225	C₄	#15689029	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	#20241618	1	1	yes	characteristic
230	C₂	C₂×C₄₇⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₄⋊#338
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
106. C₄×C₄₇	#338	semidirect
131. C₄₇	C₄⋊#338	semidirect
221. C₄	C₄₇⋊#338	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×C₄₇⋊#338	2
C₂×C₄×M₁₆×C₄₇	2
C₄⋊#338	47