Subgroups — #26121721

Order 12032.

1180 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689097	C₂	1	2	yes	maximal
2	#20243099	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093983	C₂²	1	4	yes	—
5	#15689097	—	2	4	no	—
6	#15689097	C₄	1	8	yes	complemented
7	C₂×#13093939	—	2	4	no	—
8	#15688884	—	2	8	no	—
9	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#568	—	2	8	no	—
11	C₂×#131×C₄₇	—	2	4	no	—
12	C₂²×M₁₆×C₄₇	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
14	#13093983	—	2	4	no	—
15	#13093983	—	2	4	no	—
16	#13093983	—	2	4	no	—
17	#13093983	C₂×C₄	1	8	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
20	C₂×C₄×Dic₄₇	—	2	4	no	—
21	#13093939	—	2	16	no	—
22	C₂×#41×C₄₇	—	2	4	no	—
23	C₂×#41×C₄₇	—	2	4	no	—
24	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
25	C₂×#41×C₄₇	D₄	1	4	yes	—
26	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
27	C₂²×D₄×C₄₇	D₄	1	2	yes	—
28	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	#131×C₄₇	—	2	32	no	—
32	C₂×M₁₆×C₄₇	—	2	4	no	—
33	C₂×M₁₆×C₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	—	2	4	no	—
35	C₂×M₁₆×C₄₇	—	2	16	no	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
38	C₂²×C₈×C₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
42	#41×C₄₇	—	2	8	no	—
43	#41×C₄₇	—	2	8	no	—
44	#41×C₄₇	—	2	8	no	—
45	#41×C₄₇	—	2	8	no	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂×D₄×C₄₇	—	2	8	no	—
49	C₂×D₄×C₄₇	—	2	4	no	—
50	C₂×D₄×C₄₇	C₂²⋊C₄	1	8	yes	—
51	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	M₁₆×C₄₇	—	2	16	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	8	no	—
67	M₁₆×C₄₇	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₄₇	—	2	16	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	8	no	—
71	C₂²⋊Dic₄₇	—	2	8	no	—
72	C₂²⋊Dic₄₇	—	4	4	no	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	4	4	no	—
75	C₄×Dic₄₇	—	2	8	no	—
76	C₄×Dic₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	—	8	1	no	—
79	C₂³×C₄₇	—	8	2	no	—
80	C₂³×C₄₇	#95	1	4	yes	—
81	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₄₇	#121	1	4	yes	—
84	Q₈×C₄₇	—	2	8	no	—
85	D₄×C₄₇	—	2	16	no	—
86	D₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	#95	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₇	#117	1	2	yes	—
97	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₄₇	#120	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
101	C₂×C₄×C₄₇	#123	1	4	yes	—
102	C₂×Dic₄₇	—	4	8	no	—
103	C₂×Dic₄₇	—	8	8	no	—
104	C₈×C₄₇	—	2	8	no	—
105	#10766	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	4	4	no	—
108	C₂²×C₄₇	—	4	1	no	—
109	C₂²×C₄₇	—	4	2	no	—
110	C₂²×C₄₇	—	4	4	no	—
111	C₂²×C₄₇	—	8	4	no	—
112	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
113	C₂²×C₄₇	#380	1	2	yes	—
114	C₂²×C₄₇	#386	1	4	yes	—
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
117	C₄×C₄₇	#379	1	1	yes	characteristic
118	C₄×C₄₇	#381	1	1	yes	characteristic
119	C₄×C₄₇	#385	1	4	yes	—
120	Dic₄₇	—	8	8	no	—
121	C₂×#568	D₄₇	1	1	yes	characteristic
122	C₂×#410	—	47	2	no	—
123	#1733	—	47	4	no	—
124	C₂×C₄₇	—	2	4	no	—
125	C₂×C₄₇	—	2	1	no	—
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	—	4	4	no	—
128	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
129	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
130	#568	—	2	8	no	—
131	C₂×#131	—	2	4	no	—
132	C₂²×M₁₆	Dic₄₇	1	4	yes	complemented
133	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
134	C₂×#117	—	47	4	no	—
135	#410	—	47	8	no	—
136	#410	—	94	4	no	—
137	C₂×#99	—	94	4	no	—
138	#339	—	94	8	no	—
139	C₄₇	#10766	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	2	4	no	—
141	C₂×#41	—	2	4	no	—
142	C₂×#41	C₄×D₄₇	1	4	yes	—
143	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
144	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
145	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
146	C₂²×D₄	D₁₈₈	1	2	yes	—
147	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
148	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
149	#131	—	2	32	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	16	no	—
154	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
155	C₂×M₁₆	C₂²⋊D₄₇	1	8	yes	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	#117	—	47	8	no	—
159	#117	—	94	4	no	—
160	#117	—	94	4	no	—
161	#117	—	94	4	no	—
162	C₂×C₄⋊C₄	—	94	4	no	—
163	C₂×C₂²⋊C₄	—	94	4	no	—
164	C₂×C₄²	—	94	4	no	—
165	#99	—	94	16	no	—
166	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
167	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
168	#41	—	2	8	no	—
169	#41	—	2	8	no	—
170	#41	—	2	8	no	—
171	#41	—	2	8	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
174	C₂×D₄	—	2	8	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	Dic₄₇⋊C₄	1	4	yes	—
177	C₂×D₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	188	4	no	—
181	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
182	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
184	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
186	C₂²×C₄	#10625350	1	4	yes	—
187	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
190	M₁₆	—	2	16	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	C₂²⋊Dic₄₇	1	8	yes	—
194	C₂×C₈	—	2	16	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₄⋊C₄	—	94	8	no	—
198	C₄⋊C₄	—	188	4	no	—
199	C₂²⋊C₄	—	94	8	no	—
200	C₂²⋊C₄	—	188	4	no	—
201	C₄²	—	94	8	no	—
202	C₄²	—	188	4	no	—
203	C₂³	—	2	4	no	—
204	C₂³	—	8	1	no	—
205	C₂³	—	8	2	no	—
206	C₂³	C₄₇⋊#95	1	4	yes	—
207	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
208	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
209	C₂³	#13093998	1	4	yes	—
210	Q₈	—	2	8	no	—
211	D₄	—	2	16	no	—
212	D₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	376	8	no	—
218	C₂×C₄	—	188	8	no	—
219	C₂×C₄	C₄₇⋊#95	1	4	yes	—
220	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
224	C₂×C₄	#13093983	1	2	yes	—
225	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
226	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
227	C₂×C₄	#13093993	1	4	yes	—
228	C₂×C₄	#13094003	1	4	yes	—
229	C₂×C₄	#13094007	1	4	yes	—
230	C₈	—	2	8	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
238	C₂²	#15688983	1	2	yes	—
239	C₂²	#15689002	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	376	8	no	—
242	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
243	C₄	#15688979	1	1	yes	characteristic
244	C₄	#15688986	1	1	yes	characteristic
245	C₄	#15688996	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	4	no	—
250	C₂	#20241611	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15688979
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#568	C₂
socle	C₂×C₄₇	C₂²⋊#95
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#568	D₄₇
hypocenter	C₄₇	#10766
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
2. #20243099	C₂	semidirect
6. #15689097	C₄	semidirect
12. C₂²×M₁₆×C₄₇	C₄	semidirect
132. C₂²×M₁₆	Dic₄₇	semidirect
139. C₄₇	#10766	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689097	2
#20243099	2
C₂×C₄₇×#568	2
#10766	47