Subgroups — C₂×#20245991 #26374866

Order 12032.

2894 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245991	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689009	C₂	1	8	yes	maximal, complemented
3	C₂×#15688986	C₂	1	2	yes	maximal
4	C₂×#15688983	C₂	1	2	yes	maximal
5	C₂×C₄×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689009	C₂²	1	64	yes	complemented
11	#15688986	C₂²	1	16	yes	—
12	#15688983	C₂²	1	16	yes	—
13	C₄×C₂²⋊Dic₄₇	C₂²	1	16	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	C₄	1	16	yes	complemented
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	8	yes	—
23	C₂³×Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₄⋊Dic₄₇	—	4	8	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
26	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
27	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
28	C₂×C₄×Dic₄₇	—	4	8	no	—
29	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
30	C₄₇⋊#95	C₂³	1	16	yes	—
31	C₄₇⋊#95	C₂³	1	16	yes	—
32	C₄₇⋊#95	C₂³	1	32	yes	—
33	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	—	2	8	no	—
35	C₂³×C₄×C₄₇	—	4	4	no	—
36	C₂³×C₄×C₄₇	—	4	4	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	64	yes	complemented
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
49	C₂⁴×C₄₇	—	2	4	no	—
50	C₂⁴×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
52	C₂⁴×C₄₇	#41	1	8	yes	—
53	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
54	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	32	no	—
56	C₂²×C₄×C₄₇	—	2	16	no	—
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	2	8	no	—
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	—	4	16	no	—
61	C₂²×C₄×C₄₇	—	4	16	no	—
62	C₂²×C₄×C₄₇	—	4	8	no	—
63	C₂²×C₄×C₄₇	—	4	8	no	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
73	C₂²×Dic₄₇	—	2	16	no	—
74	C₂²×Dic₄₇	—	2	16	no	—
75	C₂²×Dic₄₇	—	2	16	no	—
76	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
77	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
78	C₂²⋊Dic₄₇	—	2	32	no	—
79	C₄⋊Dic₄₇	—	4	16	no	—
80	C₄×Dic₄₇	—	4	16	no	—
81	C₂³×C₄₇	—	2	8	no	—
82	C₂³×C₄₇	—	2	8	no	—
83	C₂³×C₄₇	—	2	32	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	#117	1	16	yes	—
87	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
88	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
89	C₂³×C₄₇	C₂×#41	1	8	yes	—
90	C₂³×C₄₇	C₂×#41	1	4	yes	—
91	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
92	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
93	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	32	no	—
95	C₂×C₄×C₄₇	—	2	16	no	—
96	C₂×C₄×C₄₇	—	2	16	no	—
97	C₂×C₄×C₄₇	—	2	16	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	4	16	no	—
102	C₂×C₄×C₄₇	—	4	16	no	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
106	C₂×Dic₄₇	—	2	16	no	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	2	16	no	—
109	C₂×Dic₄₇	—	4	32	no	—
110	C₂×#2313	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	32	no	—
112	C₂²×C₄₇	—	2	4	no	—
113	C₂²×C₄₇	—	2	4	no	—
114	C₂²×C₄₇	—	2	16	no	—
115	C₂²×C₄₇	C₂×#117	1	8	yes	—
116	C₂²×C₄₇	C₂×#117	1	4	yes	—
117	C₂²×C₄₇	#518	1	4	yes	—
118	C₂²×C₄₇	#519	1	4	yes	—
119	C₂²×C₄₇	#539	1	16	yes	—
120	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂²×#41	1	2	yes	—
122	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
123	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
124	C₄×C₄₇	—	2	8	no	—
125	C₄×C₄₇	—	2	8	no	—
126	C₄×C₄₇	—	2	8	no	—
127	Dic₄₇	—	4	16	no	—
128	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic
129	#2313	—	47	16	no	—
130	C₂×#387	—	47	8	no	—
131	C₂×#381	—	47	2	no	—
132	C₂×#380	—	47	2	no	—
133	C₂×C₄×C₂²⋊C₄	—	47	2	no	—
134	C₂×C₄₇	—	2	8	no	—
135	C₂×C₄₇	#2313	1	8	yes	complemented
136	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#539	1	4	yes	—
140	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
141	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
142	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	—	47	4	no	—
144	C₂²×C₄⋊C₄	Dic₄₇	1	16	yes	complemented
145	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
146	C₂²×C₄⋊C₄	C₂×D₄₇	1	8	yes	—
147	C₂²×C₂²⋊C₄	—	47	4	no	—
148	C₂²×C₄²	—	47	4	no	—
149	#387	—	47	64	no	—
150	#381	—	47	16	no	—
151	#380	—	47	16	no	—
152	C₄×C₂²⋊C₄	—	47	16	no	—
153	C₂×#95	—	47	4	no	—
154	C₂×#95	—	47	4	no	—
155	C₂×#95	—	47	8	no	—
156	C₄₇	C₂×#2313	1	1	yes	characteristic, complemented, Sylow
157	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	4	4	no	—
160	C₂³×C₄	—	4	4	no	—
161	C₂³×C₄	—	47	8	no	—
162	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
163	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
164	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
165	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
168	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
171	C₂×C₄⋊C₄	—	47	16	no	—
172	C₂×C₄⋊C₄	—	188	8	no	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₄₇	1	64	yes	complemented
175	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
176	C₂×C₂²⋊C₄	—	47	16	no	—
177	C₂×C₂²⋊C₄	—	94	16	no	—
178	C₂×C₄²	—	47	16	no	—
179	C₂×C₄²	—	188	8	no	—
180	#95	—	47	16	no	—
181	#95	—	47	16	no	—
182	#95	—	47	32	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	—	2	4	no	—
185	C₂⁴	#10625350	1	8	yes	—
186	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
187	C₂⁴	C₂³×D₄₇	1	2	yes	—
188	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	—	2	32	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	4	16	no	—
195	C₂²×C₄	—	4	16	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	47	16	no	—
199	C₂²×C₄	—	94	16	no	—
200	C₂²×C₄	—	94	16	no	—
201	C₂²×C₄	—	94	16	no	—
202	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
203	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
204	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
205	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
208	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
210	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
211	C₄⋊C₄	—	188	16	no	—
212	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
213	C₂²⋊C₄	—	94	32	no	—
214	C₄²	—	188	16	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	2	8	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	32	no	—
220	C₂³	#13093983	1	16	yes	—
221	C₂³	C₂³×Dic₄₇	1	4	yes	—
222	C₂³	C₂³×Dic₄₇	1	2	yes	—
223	C₂³	C₂×#10625350	1	8	yes	—
224	C₂³	C₂×#10625350	1	4	yes	—
225	C₂³	#13094069	1	4	yes	—
226	C₂³	#13094072	1	4	yes	—
227	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	16	no	—
236	C₂×C₄	—	4	16	no	—
237	C₂×C₄	—	94	16	no	—
238	C₂×C₄	—	94	16	no	—
239	C₂×C₄	—	94	16	no	—
240	C₂×C₄	—	188	32	no	—
241	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
242	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
243	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	16	no	—
247	C₂²	—	2	32	no	—
248	C₂²	C₂×#13093983	1	8	yes	—
249	C₂²	C₂×#13093983	1	4	yes	—
250	C₂²	#15689598	1	4	yes	—
251	C₂²	#15689601	1	4	yes	—
252	C₂²	#15689720	1	16	yes	—
253	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
254	C₂²	C₂²×#10625350	1	2	yes	—
255	C₂²	C₂×#13094069	1	2	yes	—
256	C₂²	C₂×#13094072	1	2	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	2	8	no	—
259	C₄	—	2	8	no	—
260	C₄	—	188	16	no	—
261	C₂	—	2	8	no	—
262	C₂	#20245991	1	8	yes	direct product factor
263	C₂	C₂²×#13093983	1	1	yes	characteristic
264	C₂	C₂×#15689598	1	1	yes	characteristic
265	C₂	C₂×#15689601	1	1	yes	characteristic
266	C₂	C₂×#15689720	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₄₇
hypocenter	C₄₇	C₂×#2313
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20245991	C₂	direct (N × H)
2. C₂×#15689009	C₂	semidirect
10. #15689009	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₄₇	C₄	semidirect
47. C₂×C₄⋊C₄×C₄₇	C₂×C₄	semidirect
135. C₂×C₄₇	#2313	semidirect
144. C₂²×C₄⋊C₄	Dic₄₇	semidirect
156. C₄₇	C₂×#2313	semidirect
174. C₂×C₄⋊C₄	C₂×Dic₄₇	semidirect
262. C₂	#20245991	direct (N × H)

Maximal subgroups

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

subgroup	index
#20245991	2
C₂×#15689009	2
C₂×#15688986	2
C₂×#15688983	2
C₂×C₄×C₂²⋊Dic₄₇	2
C₂³×C₄⋊C₄×C₄₇	2
C₂×#2313	47

Subgroups — C2×#20245991 #26374866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245991 #26374866