Subgroups — #26213701

Order 12032.

1978 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689890	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13094031	C₂	1	1	yes	maximal, characteristic
3	#20243114	C₂	1	6	yes	maximal, complemented
4	C₂×#15688887	C₂	1	6	yes	maximal, complemented
5	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic
6	#15689890	—	2	4	no	—
7	#15689890	C₄	1	8	yes	—
8	C₂×#13094031	—	2	4	no	—
9	C₂×#13094031	—	2	4	no	—
10	C₂×#13094031	—	2	4	no	—
11	C₂×#13094031	C₄	1	8	yes	—
12	C₂²×#10625329	C₂²	1	6	yes	—
13	C₂²×#10625329	C₂²	1	3	yes	—
14	C₂³×C₄₇⋊C₈	C₂²	1	3	yes	—
15	#15688887	C₂²	1	4	yes	—
16	#15688887	C₂²	1	12	yes	—
17	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#568	—	2	4	no	—
19	C₄₇×#568	C₄	1	8	yes	—
20	C₂×#131×C₄₇	—	2	4	no	—
21	C₂²×M₁₆×C₄₇	C₂²	1	6	yes	—
22	#13094031	—	2	32	no	—
23	#13094031	—	2	32	no	—
24	C₂×#10625329	—	2	6	no	—
25	C₂×#10625329	—	2	6	no	—
26	C₂×#10625329	—	2	6	no	—
27	C₂×#10625329	—	2	12	no	—
28	C₂×#10625329	C₂×C₄	1	24	yes	—
29	C₂×#10625329	C₂×C₄	1	12	yes	—
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	6	no	—
32	C₂²×C₄₇⋊C₈	—	2	6	no	—
33	C₂²×C₄₇⋊C₈	—	2	6	no	—
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	12	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
37	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×#41×C₄₇	D₄	1	4	yes	—
41	C₂×#41×C₄₇	D₄	1	4	yes	—
42	C₂×#41×C₄₇	D₄	1	4	yes	—
43	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
44	C₂²×D₄×C₄₇	Q₈	1	3	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
46	#131×C₄₇	—	2	32	no	—
47	C₂×M₁₆×C₄₇	—	2	6	no	—
48	C₂×M₁₆×C₄₇	—	2	12	no	—
49	C₂×M₁₆×C₄₇	C₂×C₄	1	24	yes	—
50	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₇	—	2	6	no	—
52	C₂⁴×C₄₇	C₂×Q₈	1	3	yes	—
53	#41×C₄₇	C₄²	1	16	yes	—
54	#41×C₄₇	C₂²⋊C₄	1	16	yes	—
55	#41×C₄₇	C₂²⋊C₄	1	16	yes	—
56	#41×C₄₇	C₂²⋊C₄	1	16	yes	—
57	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
58	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
60	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
61	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
62	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₄₇	—	2	24	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₄₇	—	2	24	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	#10625329	—	2	24	no	—
77	#10625329	—	2	24	no	—
78	#10625329	—	2	4	no	—
79	#10625329	C₂²×C₄	1	8	yes	—
80	C₂×C₄₇⋊C₈	—	2	24	no	—
81	C₂×C₄₇⋊C₈	—	2	24	no	—
82	C₂×C₄₇⋊C₈	—	2	1	no	—
83	C₂×C₄₇⋊C₈	—	2	1	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	3	no	—
86	C₂×C₄₇⋊C₈	—	2	3	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	6	yes	—
89	C₂³×C₄₇	—	4	6	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
93	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
94	Q₈×C₄₇	#95	1	16	yes	—
95	D₄×C₄₇	#95	1	48	yes	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₄₇	—	2	8	no	—
110	C₄₇⋊C₈	—	2	2	no	—
111	C₄₇⋊C₈	—	2	8	no	—
112	C₄₇⋊C₈	—	2	6	no	—
113	#20351	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	12	no	—
116	C₂²×C₄₇	—	4	6	no	—
117	C₂²×C₄₇	—	4	6	no	—
118	C₂²×C₄₇	C₂×#95	1	12	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#95	1	12	yes	—
123	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×#568	—	47	1	no	—
128	C₂×#568	D₄₇	1	1	yes	characteristic
129	C₂²×#131	—	47	1	no	—
130	#1738	—	47	6	no	—
131	C₂×#341	—	47	6	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	12	no	—
135	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
136	C₂×C₄₇	#2916	1	2	yes	—
137	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
138	#568	—	2	4	no	—
139	#568	—	47	8	no	—
140	#568	—	94	4	no	—
141	#568	Dic₄₇	1	8	yes	—
142	C₂×#131	—	2	4	no	—
143	C₂×#131	—	47	8	no	—
144	C₂×#131	—	94	4	no	—
145	C₂×#131	—	94	4	no	—
146	C₂×#131	—	94	4	no	—
147	C₂²×M₁₆	—	47	3	no	—
148	C₂²×M₁₆	—	47	6	no	—
149	C₂²×M₁₆	C₂×D₄₇	1	6	yes	—
150	C₂³×C₈	—	47	3	no	—
151	#341	—	47	4	no	—
152	#341	—	47	12	no	—
153	C₄₇	#20351	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	C₄×D₄₇	1	4	yes	—
155	C₂×#41	C₄×D₄₇	1	4	yes	—
156	C₂×#41	D₁₈₈	1	4	yes	—
157	C₂×#41	C₂×Dic₄₇	1	4	yes	—
158	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
159	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
160	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
161	C₂²×D₄	Dic₉₄	1	3	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
163	#131	—	2	32	no	—
164	#131	—	94	32	no	—
165	#131	—	94	32	no	—
166	C₂×M₁₆	—	2	6	no	—
167	C₂×M₁₆	—	2	12	no	—
168	C₂×M₁₆	—	47	12	no	—
169	C₂×M₁₆	—	47	24	no	—
170	C₂×M₁₆	—	47	4	no	—
171	C₂×M₁₆	—	94	6	no	—
172	C₂×M₁₆	—	94	12	no	—
173	C₂×M₁₆	—	94	6	no	—
174	C₂×M₁₆	—	94	6	no	—
175	C₂×M₁₆	C₂×Dic₄₇	1	24	yes	—
176	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
177	C₂²×C₈	—	2	6	no	—
178	C₂²×C₈	—	47	12	no	—
179	C₂²×C₈	—	47	1	no	—
180	C₂²×C₈	—	47	3	no	—
181	C₂²×C₈	—	94	6	no	—
182	C₂²×C₈	—	94	6	no	—
183	C₂²×C₈	—	94	6	no	—
184	C₂⁴	C₂×Dic₉₄	1	3	yes	—
185	#41	C₄×Dic₄₇	1	16	yes	—
186	#41	(C₂×D₄₇)⋊C₄	1	16	yes	—
187	#41	(C₂×D₄₇)⋊C₄	1	16	yes	—
188	#41	C₂²⋊Dic₄₇	1	16	yes	—
189	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
190	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
191	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
192	C₂×D₄	Dic₄₇⋊C₄	1	12	yes	—
193	C₂×D₄	Dic₄₇⋊C₄	1	12	yes	—
194	C₂×D₄	C₄⋊Dic₄₇	1	12	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
198	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	M₁₆	—	2	24	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	47	8	no	—
206	M₁₆	—	94	24	no	—
207	M₁₆	—	94	24	no	—
208	M₁₆	—	94	4	no	—
209	M₁₆	C₂²×Dic₄₇	1	8	yes	—
210	C₂×C₈	—	2	24	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	47	2	no	—
213	C₂×C₈	—	47	6	no	—
214	C₂×C₈	—	94	24	no	—
215	C₂×C₈	—	94	24	no	—
216	C₂×C₈	—	94	1	no	—
217	C₂×C₈	—	94	3	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	94	1	no	—
220	C₂×C₈	—	94	3	no	—
221	C₂³	—	4	6	no	—
222	C₂³	C₂×Dic₄₇⋊C₄	1	6	yes	—
223	C₂³	C₂×Dic₄₇⋊C₄	1	6	yes	—
224	C₂³	C₂×C₄⋊Dic₄₇	1	6	yes	—
225	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
226	Q₈	C₄₇⋊#95	1	16	yes	—
227	D₄	C₄₇⋊#95	1	48	yes	—
228	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
229	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	12	yes	—
230	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	12	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
232	C₂×C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
233	C₂×C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
235	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₈	—	94	2	no	—
243	C₈	—	94	6	no	—
244	C₈	—	94	8	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	12	no	—
247	C₂²	—	4	6	no	—
248	C₂²	—	4	6	no	—
249	C₂²	C₂×C₄₇⋊#95	1	12	yes	—
250	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
251	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
252	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
253	C₄	C₂×C₄₇⋊#95	1	12	yes	—
254	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
255	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	12	no	—
261	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
262	C₂	#20251997	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#568	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#568	D₄₇
hypocenter	C₄₇	#20351
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #20243114	C₂	semidirect
4. C₂×#15688887	C₂	semidirect
153. C₄₇	#20351	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
C₂²×#13094031	2
#20243114	2
C₂×#15688887	2
C₂×C₄₇×#568	2
#20351	47