Subgroups — C₂×#20248417 #26388032

Order 12032.

3182 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	C₂²×#13094023	C₂	1	4	yes	maximal
2	#20248417	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689043	C₂	1	4	yes	maximal
4	C₂×#15689019	C₂	1	4	yes	maximal
5	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
6	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	4	yes	—
8	C₂×#13094023	—	2	32	no	—
9	C₂×#13094023	C₂²	1	32	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×Dic₄₇⋊C₄	C₂²	1	8	yes	—
13	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
14	#15689043	C₂²	1	32	yes	—
15	#15689019	C₂²	1	32	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	16	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#384	C₂²	1	8	yes	—
23	C₂²×Dic₉₄	—	2	32	no	—
24	C₂²×Dic₉₄	—	4	8	no	—
25	C₂²×Dic₉₄	C₂³	1	16	yes	—
26	C₂³×Dic₄₇	C₂³	1	4	yes	—
27	C₂³×Dic₄₇	C₂³	1	4	yes	—
28	#13094023	—	2	128	no	—
29	C₂×C₄⋊Dic₄₇	D₄	1	16	yes	—
30	C₂×C₄⋊Dic₄₇	Q₈	1	16	yes	complemented
31	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×Dic₄₇⋊C₄	—	2	32	no	—
34	C₂×Dic₄₇⋊C₄	C₂³	1	32	yes	—
35	C₂×C₄×Dic₄₇	—	2	16	no	—
36	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
37	C₄₇⋊#95	C₂³	1	16	yes	—
38	C₄₇⋊#95	C₂³	1	16	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
43	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
46	C₂×C₄²×C₄₇	Q₈	1	8	yes	complemented
47	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
48	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	16	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
60	C₂²×Dic₄₇	—	2	8	no	—
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	C₂×D₄	1	16	yes	—
65	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
66	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
67	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
68	C₂×Dic₉₄	—	2	64	no	—
69	C₂×Dic₉₄	—	4	16	no	—
70	C₂×Dic₉₄	—	4	64	no	—
71	C₄⋊C₄×C₄₇	—	2	32	no	—
72	C₄⋊C₄×C₄₇	—	4	8	no	—
73	C₄²×C₄₇	C₂×Q₈	1	16	yes	complemented
74	C₄⋊Dic₄₇	C₂×D₄	1	32	yes	—
75	C₄⋊Dic₄₇	C₂×Q₈	1	32	yes	complemented
76	Dic₄₇⋊C₄	—	2	64	no	—
77	C₄×Dic₄₇	—	2	32	no	—
78	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
79	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
80	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
81	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
82	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
83	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	16	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	4	16	no	—
90	C₂×C₄×C₄₇	C₄⋊Q₈	1	32	yes	complemented
91	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
92	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
93	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
94	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
95	C₂×Dic₄₇	—	2	32	no	—
96	C₂×Dic₄₇	—	2	32	no	—
97	C₂×Dic₄₇	—	2	8	no	—
98	C₂×Dic₄₇	—	2	8	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	2	8	no	—
101	C₂×Dic₄₇	C₂²×D₄	1	16	yes	—
102	C₂×Dic₄₇	C₂²×Q₈	1	16	yes	—
103	Dic₉₄	—	4	64	no	—
104	C₂×#2551	—	47	1	no	maximal, Sylow
105	C₂²×C₄₇	C₂×C₄⋊Q₈	1	8	yes	—
106	C₂²×C₄₇	#544	1	4	yes	—
107	C₂²×C₄₇	D₄²	1	4	yes	—
108	C₂²×C₄₇	D₄×Q₈	1	8	yes	—
109	C₂²×C₄₇	Q₈²	1	4	yes	—
110	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
111	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
112	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₄₇	—	2	4	no	—
116	C₄×C₄₇	—	4	8	no	—
117	C₄×C₄₇	C₂×C₄⋊Q₈	1	16	yes	complemented
118	Dic₄₇	—	2	16	no	—
119	Dic₄₇	—	2	16	no	—
120	C₂²×C₄⋊Q₈	—	47	4	no	—
121	#2551	—	47	16	no	—
122	C₂×#395	—	47	4	no	—
123	C₂×#389	—	47	4	no	—
124	C₂×#384	D₄₇	1	1	yes	characteristic
125	C₂×C₄×C₄⋊C₄	—	47	2	no	—
126	C₂×C₄₇	#2551	1	8	yes	complemented
127	C₂×C₄₇	C₂²×C₄⋊Q₈	1	2	yes	—
128	C₂×C₄₇	C₂×#544	1	1	yes	characteristic
129	C₂×C₄₇	C₂×D₄²	1	1	yes	characteristic
130	C₂×C₄₇	C₂×D₄×Q₈	1	2	yes	—
131	C₂×C₄₇	C₂×Q₈²	1	1	yes	characteristic
132	C₂³×Q₈	—	47	4	no	—
133	C₂×C₄⋊Q₈	—	47	32	no	—
134	C₂×C₄⋊Q₈	—	94	32	no	—
135	C₂²×C₄⋊C₄	—	47	2	no	—
136	C₂²×C₄⋊C₄	—	47	2	no	—
137	C₂²×C₄⋊C₄	—	47	8	no	—
138	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
139	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
140	C₂²×C₄²	—	47	4	no	—
141	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
142	#395	—	47	32	no	—
143	#389	—	47	32	no	—
144	#384	C₂×D₄₇	1	8	yes	—
145	C₄×C₄⋊C₄	—	47	16	no	—
146	C₂×#95	—	47	4	no	—
147	C₂×#95	—	47	4	no	—
148	C₄₇	C₂×#2551	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	47	16	no	—
150	C₂²×Q₈	—	94	32	no	—
151	C₂²×Q₈	—	188	8	no	—
152	C₂³×C₄	—	47	4	no	—
153	C₂³×C₄	—	47	4	no	—
154	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
155	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
156	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
157	C₄⋊Q₈	—	94	128	no	—
158	C₂×C₄⋊C₄	—	2	16	no	—
159	C₂×C₄⋊C₄	—	4	4	no	—
160	C₂×C₄⋊C₄	—	47	16	no	—
161	C₂×C₄⋊C₄	—	47	16	no	—
162	C₂×C₄⋊C₄	—	47	8	no	—
163	C₂×C₄⋊C₄	—	47	8	no	—
164	C₂×C₄⋊C₄	—	47	32	no	—
165	C₂×C₄⋊C₄	—	94	32	no	—
166	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
167	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
168	C₂×C₄²	—	47	16	no	—
169	C₂×C₄²	—	94	16	no	—
170	C₂×C₄²	Dic₉₄	1	8	yes	complemented
171	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
172	#95	—	47	16	no	—
173	#95	—	47	16	no	—
174	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
175	C₂×Q₈	—	94	64	no	—
176	C₂×Q₈	—	188	16	no	—
177	C₂×Q₈	—	188	64	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	47	16	no	—
183	C₂²×C₄	—	47	16	no	—
184	C₂²×C₄	—	47	8	no	—
185	C₂²×C₄	—	47	8	no	—
186	C₂²×C₄	—	94	8	no	—
187	C₂²×C₄	—	94	8	no	—
188	C₂²×C₄	—	94	8	no	—
189	C₂²×C₄	—	94	8	no	—
190	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
191	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
192	C₂²×C₄	D₄×D₄₇	1	8	yes	—
193	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
197	C₄⋊C₄	—	2	32	no	—
198	C₄⋊C₄	—	4	8	no	—
199	C₄⋊C₄	—	47	32	no	—
200	C₄⋊C₄	—	47	32	no	—
201	C₄⋊C₄	—	94	64	no	—
202	C₄²	—	94	32	no	—
203	C₄²	C₂×Dic₉₄	1	16	yes	complemented
204	C₂³	C₂×D₄×D₄₇	1	4	yes	—
205	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
206	C₂³	C₂²×Dic₉₄	1	2	yes	—
207	C₂³	#13094069	1	2	yes	—
208	C₂³	#13094072	1	2	yes	—
209	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
210	Q₈	—	188	64	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	16	no	—
217	C₂×C₄	—	47	16	no	—
218	C₂×C₄	—	47	16	no	—
219	C₂×C₄	—	94	32	no	—
220	C₂×C₄	—	94	32	no	—
221	C₂×C₄	—	94	8	no	—
222	C₂×C₄	—	94	8	no	—
223	C₂×C₄	—	94	8	no	—
224	C₂×C₄	—	94	8	no	—
225	C₂×C₄	#13094023	1	32	yes	complemented
226	C₂×C₄	C₂×D₄×D₄₇	1	8	yes	—
227	C₂×C₄	C₂×Q₈×D₄₇	1	8	yes	—
228	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
229	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
230	C₂²	C₂×#13094023	1	8	yes	—
231	C₂²	#15689754	1	4	yes	—
232	C₂²	#15689758	1	4	yes	—
233	C₂²	#15689788	1	8	yes	—
234	C₂²	#15689853	1	4	yes	—
235	C₂²	C₂²×D₄×D₄₇	1	2	yes	—
236	C₂²	C₂²×Q₈×D₄₇	1	2	yes	—
237	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
238	C₂²	C₂×#13094069	1	1	yes	characteristic
239	C₂²	C₂×#13094072	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	4	8	no	—
242	C₄	—	94	16	no	—
243	C₄	—	94	16	no	—
244	C₄	C₂×#13094023	1	16	yes	complemented
245	C₂	#20248417	1	8	yes	direct product factor
246	C₂	C₂²×#13094023	1	2	yes	—
247	C₂	C₂×#15689754	1	1	yes	characteristic
248	C₂	C₂×#15689758	1	1	yes	characteristic
249	C₂	C₂×#15689788	1	2	yes	—
250	C₂	C₂×#15689853	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	C₂×#2551
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #20248417	C₂	direct (N × H)
30. C₂×C₄⋊Dic₄₇	Q₈	semidirect
46. C₂×C₄²×C₄₇	Q₈	semidirect
73. C₄²×C₄₇	C₂×Q₈	semidirect
75. C₄⋊Dic₄₇	C₂×Q₈	semidirect
90. C₂×C₄×C₄₇	C₄⋊Q₈	semidirect
117. C₄×C₄₇	C₂×C₄⋊Q₈	semidirect
126. C₂×C₄₇	#2551	semidirect
148. C₄₇	C₂×#2551	semidirect
170. C₂×C₄²	Dic₉₄	semidirect
203. C₄²	C₂×Dic₉₄	semidirect
225. C₂×C₄	#13094023	semidirect
244. C₄	C₂×#13094023	semidirect
245. C₂	#20248417	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13094023	2
#20248417	2
C₂×#15689043	2
C₂×#15689019	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂×C₄₇×#384	2
C₂×#2551	47

Subgroups — C2×#20248417 #26388032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20248417 #26388032