Subgroups — #26374507

Order 12032.

3598 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₄₇	C₂	1	2	yes	maximal
2	#20245753	C₂	1	4	yes	maximal, complemented
3	#20245749	C₂	1	12	yes	maximal, complemented
4	#117×Dic₄₇	C₂	1	6	yes	maximal, complemented
5	C₄×#13093983	C₂	1	3	yes	maximal, complemented
6	C₂×#15688969	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#41×Dic₄₇	C₄	1	32	yes	complemented
9	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
10	C₂×D₄×Dic₄₇	C₂²	1	12	yes	—
11	C₂×#13093983	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
13	C₄⋊C₄×Dic₄₇	C₂²	1	24	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	12	yes	—
15	C₂²⋊C₄×Dic₄₇	C₂²	1	24	yes	—
16	C₄×C₂²⋊Dic₄₇	C₂²	1	12	yes	—
17	#15688969	C₂²	1	1	yes	characteristic
18	#15688969	C₂²	1	3	yes	—
19	#15688969	C₂²	1	6	yes	—
20	#15688969	C₂²	1	12	yes	—
21	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
22	C₄²×Dic₄₇	C₂²	1	6	yes	—
23	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×#41×C₄₇	C₄	1	16	yes	complemented
25	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₄₇	C₂²	1	6	yes	—
27	C₂×#117×C₄₇	C₂²	1	3	yes	—
28	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
29	C₂³×Dic₄₇	C₂³	1	12	yes	—
30	Q₈×Dic₄₇	C₂×C₄	1	32	yes	complemented
31	D₄×Dic₄₇	C₂×C₄	1	96	yes	complemented
32	#13093983	C₂×C₄	1	48	yes	complemented
33	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	12	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	48	yes	complemented
36	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
40	C₄₇⋊#95	C₂³	1	4	yes	—
41	C₄₇⋊#95	C₂³	1	12	yes	—
42	C₄₇⋊#95	C₂³	1	24	yes	—
43	C₂×#41×C₄₇	C₂×C₄	1	16	yes	—
44	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₄₇	C₂³	1	3	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
49	C₄×Q₈×C₄₇	C₂×C₄	1	16	yes	complemented
50	C₄×D₄×C₄₇	C₂×C₄	1	48	yes	complemented
51	#117×C₄₇	C₂×C₄	1	24	yes	complemented
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂³	1	6	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	24	yes	complemented
55	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
58	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
59	#41×C₄₇	C₄²	1	64	yes	complemented
60	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₄₇	C₂²×C₄	1	24	yes	—
63	C₂×D₄×C₄₇	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₄₇	—	4	12	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	4	24	no	—
73	C₂²×Dic₄₇	C₂²×C₄	1	48	yes	—
74	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
75	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
76	C₄⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
77	C₂²⋊C₄×C₄₇	C₂²×C₄	1	24	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	12	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	24	yes	—
81	C₂²⋊Dic₄₇	C₂²×C₄	1	48	yes	—
82	C₄⋊Dic₄₇	C₂²×C₄	1	48	yes	—
83	C₄×Dic₄₇	—	4	24	no	—
84	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
85	C₄×Dic₄₇	C₂²×C₄	1	24	yes	—
86	C₄×Dic₄₇	C₂²×C₄	1	48	yes	—
87	C₂³×C₄₇	—	2	12	no	—
88	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
89	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
90	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
91	Q₈×C₄₇	C₂×C₄²	1	16	yes	—
92	D₄×C₄₇	C₂×C₄²	1	48	yes	—
93	C₂×C₄×C₄₇	—	2	12	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	4	24	no	—
96	C₂×C₄×C₄₇	C₂×C₄²	1	48	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
101	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
103	C₂×Dic₄₇	—	2	24	no	—
104	C₂×Dic₄₇	—	2	8	no	—
105	C₂×Dic₄₇	—	4	48	no	—
106	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
107	C₂×Dic₄₇	C₂³×C₄	1	24	yes	—
108	#33999	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	24	no	—
110	C₂²×C₄₇	—	2	12	no	—
111	C₂²×C₄₇	C₂²×C₄²	1	12	yes	—
112	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
113	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₂²×C₄₇	#585	1	4	yes	—
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	2	12	no	—
117	C₄×C₄₇	C₂²×C₄²	1	4	yes	—
118	C₄×C₄₇	C₂²×C₄²	1	12	yes	—
119	Dic₄₇	—	2	8	no	—
120	Dic₄₇	—	2	24	no	—
121	C₂×C₄×#41	—	47	2	no	—
122	C₂×C₄×#41	D₄₇	1	1	yes	characteristic
123	#2276	—	47	4	no	—
124	#2275	—	47	12	no	—
125	C₄×#117	—	47	3	no	—
126	C₄×#117	—	47	6	no	—
127	C₂×#376	—	47	3	no	—
128	C₂×C₄₇	—	2	12	no	—
129	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
130	C₂×C₄₇	#3428	1	4	yes	—
131	C₂×C₄₇	#3428	1	2	yes	—
132	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
133	C₄×#41	—	47	32	no	—
134	C₄×#41	Dic₄₇	1	16	yes	complemented
135	C₂×C₄×Q₈	—	47	4	no	—
136	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
137	C₂×C₄×D₄	—	47	12	no	—
138	C₂×C₄×D₄	C₂×D₄₇	1	6	yes	—
139	C₂×#117	—	47	6	no	—
140	C₂×#117	C₂×D₄₇	1	3	yes	—
141	C₂²×C₄²	—	47	6	no	—
142	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
143	C₄×C₄⋊C₄	—	47	24	no	—
144	C₄×C₄⋊C₄	—	47	12	no	—
145	C₄×C₂²⋊C₄	—	47	12	no	—
146	C₄×C₂²⋊C₄	—	47	24	no	—
147	#376	—	47	1	no	—
148	#376	—	47	3	no	—
149	#376	—	47	6	no	—
150	#376	—	47	12	no	—
151	C₂×#95	—	47	12	no	—
152	C₄³	—	47	6	no	—
153	C₄₇	#33999	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	C₄×D₄₇	1	16	yes	—
155	C₂×#41	C₂×Dic₄₇	1	8	yes	—
156	C₂²×Q₈	C₂²×D₄₇	1	1	yes	characteristic
157	C₂²×D₄	C₂²×D₄₇	1	3	yes	—
158	C₂³×C₄	—	47	12	no	—
159	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
160	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
161	C₄×Q₈	—	47	32	no	—
162	C₄×Q₈	C₂×Dic₄₇	1	16	yes	complemented
163	C₄×D₄	—	47	96	no	—
164	C₄×D₄	C₂×Dic₄₇	1	48	yes	complemented
165	#117	—	47	48	no	—
166	#117	C₂×Dic₄₇	1	24	yes	complemented
167	C₂×C₄⋊C₄	—	47	12	no	—
168	C₂×C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
169	C₂×C₂²⋊C₄	—	47	12	no	—
170	C₂×C₂²⋊C₄	C₂²×D₄₇	1	6	yes	—
171	C₂×C₄²	—	47	48	no	—
172	C₂×C₄²	—	47	12	no	—
173	C₂×C₄²	—	47	6	no	—
174	C₂×C₄²	—	47	2	no	—
175	C₂×C₄²	—	47	24	no	—
176	C₂×C₄²	C₂×Dic₄₇	1	24	yes	complemented
177	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
178	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
179	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
180	#95	—	47	4	no	—
181	#95	—	47	24	no	—
182	#95	—	47	12	no	—
183	C₂⁴	C₂³×D₄₇	1	3	yes	—
184	#41	C₄×Dic₄₇	1	64	yes	complemented
185	C₂×Q₈	C₂×C₄×D₄₇	1	8	yes	—
186	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
187	C₂×D₄	C₂×C₄×D₄₇	1	24	yes	—
188	C₂×D₄	C₂²×Dic₄₇	1	12	yes	—
189	C₂²×C₄	—	4	12	no	—
190	C₂²×C₄	—	47	4	no	—
191	C₂²×C₄	—	47	12	no	—
192	C₂²×C₄	—	47	48	no	—
193	C₂²×C₄	—	188	24	no	—
194	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
195	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
196	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
197	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
199	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
201	C₄⋊C₄	—	47	48	no	—
202	C₄⋊C₄	C₂²×Dic₄₇	1	24	yes	—
203	C₂²⋊C₄	—	47	48	no	—
204	C₂²⋊C₄	C₂²×Dic₄₇	1	24	yes	—
205	C₄²	—	47	8	no	—
206	C₄²	—	47	24	no	—
207	C₄²	—	47	48	no	—
208	C₄²	—	188	24	no	—
209	C₄²	C₂²×Dic₄₇	1	4	yes	—
210	C₄²	C₂²×Dic₄₇	1	12	yes	—
211	C₄²	C₂²×Dic₄₇	1	24	yes	—
212	C₂³	—	2	12	no	—
213	C₂³	C₂²×C₄×D₄₇	1	12	yes	—
214	C₂³	C₂³×Dic₄₇	1	6	yes	—
215	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
216	Q₈	C₂×C₄×Dic₄₇	1	16	yes	—
217	D₄	C₂×C₄×Dic₄₇	1	48	yes	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	24	no	—
221	C₂×C₄	—	47	8	no	—
222	C₂×C₄	—	47	24	no	—
223	C₂×C₄	—	94	24	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	—	188	48	no	—
226	C₂×C₄	C₂×C₄×Dic₄₇	1	48	yes	—
227	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
229	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
231	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
233	C₂²	—	2	24	no	—
234	C₂²	—	2	12	no	—
235	C₂²	C₂²×C₄×Dic₄₇	1	12	yes	—
236	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
237	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
238	C₂²	#15689959	1	4	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	2	12	no	—
241	C₄	—	94	8	no	—
242	C₄	—	94	24	no	—
243	C₄	C₂²×C₄×Dic₄₇	1	4	yes	—
244	C₄	C₂²×C₄×Dic₄₇	1	12	yes	—
245	C₂	—	2	12	no	—
246	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
247	C₂	#20256587	1	4	yes	—
248	C₂	#20256591	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄×#41×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂²×#41×C₄₇	C₂²
hypercenter	C₂×C₄×#41	D₄₇
hypocenter	C₄₇	#33999
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #20245753	C₂	semidirect
3. #20245749	C₂	semidirect
4. #117×Dic₄₇	C₂	semidirect
5. C₄×#13093983	C₂	semidirect
6. C₂×#15688969	C₂	semidirect
8. #41×Dic₄₇	C₄	semidirect
24. C₄×#41×C₄₇	C₄	semidirect
30. Q₈×Dic₄₇	C₂×C₄	semidirect
31. D₄×Dic₄₇	C₂×C₄	semidirect
32. #13093983	C₂×C₄	semidirect
35. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
49. C₄×Q₈×C₄₇	C₂×C₄	semidirect
50. C₄×D₄×C₄₇	C₂×C₄	semidirect
51. #117×C₄₇	C₂×C₄	semidirect
54. C₂×C₄²×C₄₇	C₂×C₄	semidirect
59. #41×C₄₇	C₄²	semidirect
134. C₄×#41	Dic₄₇	semidirect
153. C₄₇	#33999	semidirect
162. C₄×Q₈	C₂×Dic₄₇	semidirect
164. C₄×D₄	C₂×Dic₄₇	semidirect
166. #117	C₂×Dic₄₇	semidirect
176. C₂×C₄²	C₂×Dic₄₇	semidirect
184. #41	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×Dic₄₇	2
#20245753	2
#20245749	2
#117×Dic₄₇	2
C₄×#13093983	2
C₂×#15688969	2
C₂×C₄×#41×C₄₇	2
#33999	47