Subgroups — C₂×#20243105 #26213678

Order 12032.

2398 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	C₂²×#13093983	C₂	1	2	yes	maximal
2	#20243105	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688886	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	—	2	4	no	—
6	C₂×#13093983	—	2	8	no	—
7	C₂×#13093983	C₄	1	16	yes	complemented
8	C₂×#13093983	C₂²	1	8	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
11	C₂²×C₄×Dic₄₇	—	2	2	no	—
12	#15688886	C₂²	1	16	yes	complemented
13	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	—	2	8	no	—
15	C₂³×C₈×C₄₇	C₄	1	8	yes	complemented
16	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	4	yes	—
18	C₂³×Dic₄₇	—	2	4	no	—
19	#13093983	—	2	32	no	—
20	#13093983	—	2	16	no	—
21	#13093983	C₂×C₄	1	32	yes	complemented
22	C₂×C₄⋊Dic₄₇	—	2	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	8	no	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	16	yes	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
26	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
27	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄×Dic₄₇	—	2	16	no	—
29	C₂×C₄×Dic₄₇	—	2	8	no	—
30	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	8	yes	—
36	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₇	—	2	32	no	—
40	C₂²×C₈×C₄₇	—	2	8	no	—
41	C₂²×C₈×C₄₇	—	2	16	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	C₂⁴×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	8	yes	—
47	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
49	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₄²	1	16	yes	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	16	yes	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	16	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	32	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	16	no	—
69	C₂²×Dic₄₇	—	2	8	no	—
70	C₂²×Dic₄₇	—	4	16	no	—
71	C₂×C₈×C₄₇	—	2	16	no	—
72	C₂×C₈×C₄₇	—	2	32	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²⋊Dic₄₇	—	2	32	no	—
76	C₄⋊Dic₄₇	—	2	16	no	—
77	C₄⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	C₂²×C₄	1	16	yes	—
79	C₄×Dic₄₇	—	2	32	no	—
80	C₂³×C₄₇	—	2	8	no	—
81	C₂³×C₄₇	—	2	32	no	—
82	C₂³×C₄₇	—	2	16	no	—
83	C₂³×C₄₇	#95	1	16	yes	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₄₇	#135	1	8	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#95	1	32	yes	—
91	C₂×C₄×C₄₇	#95	1	16	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
93	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×Dic₄₇	—	2	16	no	—
107	C₂×Dic₄₇	—	4	16	no	—
108	C₂×Dic₄₇	—	4	32	no	—
109	C₈×C₄₇	—	2	8	no	—
110	C₂×#1735	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	16	no	—
112	C₂²×C₄₇	—	2	8	no	—
113	C₂²×C₄₇	—	2	32	no	—
114	C₂²×C₄₇	C₂×#95	1	8	yes	—
115	C₂²×C₄₇	C₂×#95	1	4	yes	—
116	C₂²×C₄₇	#416	1	16	yes	—
117	C₂²×C₄₇	#427	1	8	yes	—
118	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#135	1	4	yes	—
121	C₄×C₄₇	C₂×#95	1	8	yes	—
122	C₄×C₄₇	C₂×#95	1	4	yes	—
123	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
126	Dic₄₇	—	4	16	no	—
127	C₂⁴×C₈	D₄₇	1	1	yes	characteristic
128	C₂²×#117	—	47	2	no	—
129	#1735	—	47	8	no	—
130	C₂×#340	—	47	4	no	—
131	C₂×C₄₇	—	2	8	no	—
132	C₂×C₄₇	#1735	1	8	yes	complemented
133	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#416	1	4	yes	—
135	C₂×C₄₇	C₂×#427	1	2	yes	—
136	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
137	C₂³×C₈	—	2	8	no	—
138	C₂³×C₈	Dic₄₇	1	8	yes	complemented
139	C₂³×C₈	C₂×D₄₇	1	2	yes	—
140	C₂³×C₈	C₂×D₄₇	1	4	yes	—
141	C₂×#117	—	47	16	no	—
142	C₂×#117	—	47	8	no	—
143	C₂×#117	—	94	4	no	—
144	C₂×#117	—	94	8	no	—
145	C₂²×C₄⋊C₄	—	47	4	no	—
146	C₂²×C₂²⋊C₄	—	94	2	no	—
147	C₂²×C₄²	—	94	2	no	—
148	#340	—	47	16	no	—
149	C₄₇	C₂×#1735	1	1	yes	characteristic, complemented, Sylow
150	C₂⁵	D₁₈₈	1	1	yes	characteristic
151	C₂³×C₄	—	94	4	no	—
152	C₂³×C₄	Dic₉₄	1	4	yes	—
153	C₂³×C₄	C₄×D₄₇	1	8	yes	—
154	C₂³×C₄	D₁₈₈	1	2	yes	—
155	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
157	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
158	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
160	C₂²×C₈	—	2	32	no	—
161	C₂²×C₈	—	2	8	no	—
162	C₂²×C₈	—	2	16	no	—
163	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
164	C₂²×C₈	C₂×Dic₄₇	1	16	yes	complemented
165	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
166	#117	—	47	32	no	—
167	#117	—	94	32	no	—
168	#117	—	94	16	no	—
169	C₂×C₄⋊C₄	—	47	16	no	—
170	C₂×C₄⋊C₄	—	47	8	no	—
171	C₂×C₄⋊C₄	—	94	4	no	—
172	C₂×C₄⋊C₄	—	94	8	no	—
173	C₂×C₂²⋊C₄	—	94	16	no	—
174	C₂×C₂²⋊C₄	—	94	8	no	—
175	C₂×C₄²	—	94	16	no	—
176	C₂×C₄²	—	94	8	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
179	C₂⁴	(C₂×D₄₇)⋊C₄	1	8	yes	—
180	C₂⁴	C₂×D₁₈₈	1	2	yes	—
181	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂²×C₄	—	94	16	no	—
183	C₂²×C₄	—	94	8	no	—
184	C₂²×C₄	—	188	16	no	—
185	C₂²×C₄	C₄×Dic₄₇	1	16	yes	—
186	C₂²×C₄	Dic₄₇⋊C₄	1	32	yes	—
187	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
188	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
189	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
190	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	16	yes	—
191	C₂²×C₄	C₂²⋊Dic₄₇	1	16	yes	—
192	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
193	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
196	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	2	16	no	—
204	C₂×C₈	—	2	32	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
207	C₄⋊C₄	—	47	16	no	—
208	C₄⋊C₄	—	94	16	no	—
209	C₄⋊C₄	—	94	8	no	—
210	C₂²⋊C₄	—	94	32	no	—
211	C₄²	—	94	32	no	—
212	C₂³	—	2	32	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	2	16	no	—
215	C₂³	C₄₇⋊#95	1	16	yes	—
216	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	8	yes	—
217	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
218	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
219	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
220	C₂³	#13094046	1	8	yes	—
221	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
222	C₂×C₄	—	94	16	no	—
223	C₂×C₄	—	188	32	no	—
224	C₂×C₄	—	188	16	no	—
225	C₂×C₄	C₄₇⋊#95	1	32	yes	—
226	C₂×C₄	C₄₇⋊#95	1	16	yes	—
227	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
228	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
229	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	8	yes	—
230	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
233	C₂×C₄	C₂×Dic₄₇⋊C₄	1	16	yes	—
234	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
237	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
238	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
240	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	32	no	—
244	C₂²	—	2	16	no	—
245	C₂²	C₂×C₄₇⋊#95	1	8	yes	—
246	C₂²	C₂×C₄₇⋊#95	1	4	yes	—
247	C₂²	#15689120	1	16	yes	—
248	C₂²	#15689161	1	8	yes	—
249	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
250	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
251	C₂²	C₂×#13094046	1	4	yes	—
252	C₄	—	188	16	no	—
253	C₄	C₂×C₄₇⋊#95	1	8	yes	—
254	C₄	C₂×C₄₇⋊#95	1	4	yes	—
255	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
256	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
257	C₄	C₂²×Dic₄₇⋊C₄	1	2	yes	—
258	C₂	—	2	8	no	—
259	C₂	#20243105	1	8	yes	direct product factor
260	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
261	C₂	C₂×#15689120	1	4	yes	—
262	C₂	C₂×#15689161	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂⁴×C₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂⁴×C₈	D₄₇
hypocenter	C₄₇	C₂×#1735
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. #20243105	C₂	direct (N × H)
3. C₂×#15688886	C₂	semidirect
7. C₂×#13093983	C₄	semidirect
12. #15688886	C₂²	semidirect
15. C₂³×C₈×C₄₇	C₄	semidirect
21. #13093983	C₂×C₄	semidirect
43. C₂²×C₈×C₄₇	C₂×C₄	semidirect
132. C₂×C₄₇	#1735	semidirect
138. C₂³×C₈	Dic₄₇	semidirect
149. C₄₇	C₂×#1735	semidirect
164. C₂²×C₈	C₂×Dic₄₇	semidirect
259. C₂	#20243105	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×#13093983	2
#20243105	2
C₂×#15688886	2
C₂⁴×C₈×C₄₇	2
C₂×#1735	47

Subgroups — C2×#20243105 #26213678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243105 #26213678