Subgroups — C₂×#20246006 #26374892

Order 12032.

3502 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#20246006	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689022	C₂	1	8	yes	maximal, complemented
4	C₂×#15688983	C₂	1	4	yes	maximal
5	C₂³×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	16	no	—
7	C₂×#13093983	C₂²	1	16	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	8	yes	—
10	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
12	#15689022	C₂²	1	64	yes	complemented
13	#15688983	C₂²	1	32	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
15	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₄₇	C₄	1	16	yes	complemented
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	8	yes	—
20	C₂³×Dic₄₇	C₂³	1	8	yes	—
21	#13093983	—	2	64	no	—
22	C₂×C₄⋊Dic₄₇	—	2	16	no	—
23	C₂×C₄⋊Dic₄₇	—	4	16	no	—
24	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	32	yes	—
26	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
27	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
28	C₂×C₄×Dic₄₇	—	2	16	no	—
29	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
30	C₄₇⋊#95	C₂³	1	32	yes	—
31	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	—	4	8	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₄₇	D₄	1	8	yes	—
36	C₂³×C₄×C₄₇	Q₈	1	8	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	64	yes	complemented
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
44	C₂⁴×C₄₇	—	2	4	no	—
45	C₂⁴×C₄₇	—	2	4	no	—
46	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
48	C₂⁴×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
50	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	16	no	—
52	C₂²×C₄×C₄₇	—	4	32	no	—
53	C₂²×C₄×C₄₇	—	4	16	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	64	yes	complemented
55	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	32	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	16	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	16	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
65	C₂²×Dic₄₇	—	2	16	no	—
66	C₂²×Dic₄₇	—	2	16	no	—
67	C₂²×Dic₄₇	—	2	16	no	—
68	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
69	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
70	C₂²⋊Dic₄₇	—	2	32	no	—
71	C₄⋊Dic₄₇	—	2	32	no	—
72	C₄⋊Dic₄₇	—	4	32	no	—
73	C₄×Dic₄₇	—	2	32	no	—
74	C₂³×C₄₇	—	2	8	no	—
75	C₂³×C₄₇	—	2	8	no	—
76	C₂³×C₄₇	—	2	16	no	—
77	C₂³×C₄₇	—	2	16	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	—	2	4	no	—
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
81	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
82	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
83	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
84	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
85	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
86	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
87	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
88	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
89	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	32	no	—
91	C₂×C₄×C₄₇	—	2	16	no	—
92	C₂×C₄×C₄₇	—	4	32	no	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	32	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	64	yes	complemented
95	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
98	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
99	C₂×Dic₄₇	—	2	16	no	—
100	C₂×Dic₄₇	—	2	16	no	—
101	C₂×Dic₄₇	—	2	16	no	—
102	C₂×Dic₄₇	—	4	32	no	—
103	C₂×#2316	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	32	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	—	2	8	no	—
108	C₂²×C₄₇	—	2	8	no	—
109	C₂²×C₄₇	C₂²×C₄⋊C₄	1	8	yes	—
110	C₂²×C₄₇	C₂²×C₄⋊C₄	1	4	yes	—
111	C₂²×C₄₇	#518	1	4	yes	—
112	C₂²×C₄₇	#519	1	4	yes	—
113	C₂²×C₄₇	#537	1	8	yes	—
114	C₂²×C₄₇	#544	1	8	yes	—
115	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
116	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
120	C₄×C₄₇	—	2	16	no	—
121	C₄×C₄₇	C₂²×C₄⋊C₄	1	16	yes	—
122	Dic₄₇	—	4	16	no	—
123	C₂²×#117	—	47	2	no	—
124	C₂³×C₄⋊C₄	D₄₇	1	1	yes	characteristic
125	#2316	—	47	16	no	—
126	C₂×#389	—	47	8	no	—
127	C₂×#380	—	47	4	no	—
128	C₂×C₄₇	—	2	8	no	—
129	C₂×C₄₇	#2316	1	8	yes	complemented
130	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#518	1	1	yes	characteristic
132	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#537	1	2	yes	—
134	C₂×C₄₇	C₂×#544	1	2	yes	—
135	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
136	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
137	C₂×#117	—	47	16	no	—
138	C₂×#117	—	94	16	no	—
139	C₂²×C₄⋊C₄	—	47	4	no	—
140	C₂²×C₄⋊C₄	—	47	8	no	—
141	C₂²×C₄⋊C₄	Dic₄₇	1	16	yes	complemented
142	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
143	C₂²×C₄⋊C₄	C₂×D₄₇	1	8	yes	—
144	C₂²×C₂²⋊C₄	—	47	4	no	—
145	C₂²×C₄²	—	47	4	no	—
146	#389	—	47	64	no	—
147	#380	—	47	32	no	—
148	C₂×#95	—	47	8	no	—
149	C₄₇	C₂×#2316	1	1	yes	characteristic, complemented, Sylow
150	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
151	C₂³×C₄	—	4	8	no	—
152	C₂³×C₄	—	47	8	no	—
153	C₂³×C₄	Dic₉₄	1	8	yes	—
154	C₂³×C₄	D₁₈₈	1	8	yes	—
155	C₂³×C₄	C₂×Dic₄₇	1	16	yes	—
156	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
157	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
158	C₂³×C₄	C₂²×D₄₇	1	8	yes	—
159	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
160	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
161	#117	—	94	64	no	—
162	C₂×C₄⋊C₄	—	47	16	no	—
163	C₂×C₄⋊C₄	—	47	32	no	—
164	C₂×C₄⋊C₄	—	94	16	no	—
165	C₂×C₄⋊C₄	—	188	16	no	—
166	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₄₇	1	64	yes	complemented
168	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
169	C₂×C₂²⋊C₄	—	47	16	no	—
170	C₂×C₂²⋊C₄	—	94	16	no	—
171	C₂×C₄²	—	47	16	no	—
172	C₂×C₄²	—	94	16	no	—
173	#95	—	47	32	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	C₂×Dic₉₄	1	4	yes	—
177	C₂⁴	C₂×D₁₈₈	1	4	yes	—
178	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
179	C₂⁴	C₂³×D₄₇	1	2	yes	—
180	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	16	no	—
182	C₂²×C₄	—	4	32	no	—
183	C₂²×C₄	—	4	16	no	—
184	C₂²×C₄	—	47	16	no	—
185	C₂²×C₄	—	94	16	no	—
186	C₂²×C₄	—	94	16	no	—
187	C₂²×C₄	—	94	16	no	—
188	C₂²×C₄	C₄⋊Dic₄₇	1	64	yes	complemented
189	C₂²×C₄	C₂×Dic₉₄	1	8	yes	—
190	C₂²×C₄	C₂×Dic₉₄	1	16	yes	—
191	C₂²×C₄	C₂×D₁₈₈	1	8	yes	—
192	C₂²×C₄	C₂×D₁₈₈	1	16	yes	—
193	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
194	C₂²×C₄	C₂²×Dic₄₇	1	32	yes	—
195	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
196	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
197	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
199	C₄⋊C₄	—	94	32	no	—
200	C₄⋊C₄	—	188	32	no	—
201	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
202	C₂²⋊C₄	—	94	32	no	—
203	C₄²	—	94	32	no	—
204	C₂³	—	2	16	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	16	no	—
210	C₂³	C₂×C₄⋊Dic₄₇	1	16	yes	—
211	C₂³	C₂³×Dic₄₇	1	4	yes	—
212	C₂³	C₂³×Dic₄₇	1	2	yes	—
213	C₂³	C₂²×D₁₈₈	1	4	yes	—
214	C₂³	C₂²×D₁₈₈	1	2	yes	—
215	C₂³	C₂²×Dic₉₄	1	4	yes	—
216	C₂³	C₂²×Dic₉₄	1	2	yes	—
217	C₂³	#13094069	1	4	yes	—
218	C₂³	#13094072	1	4	yes	—
219	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	4	32	no	—
223	C₂×C₄	—	94	16	no	—
224	C₂×C₄	—	94	16	no	—
225	C₂×C₄	—	94	16	no	—
226	C₂×C₄	—	188	32	no	—
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	32	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₄₇	1	64	yes	complemented
229	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
231	C₂×C₄	C₂²×D₁₈₈	1	8	yes	—
232	C₂×C₄	C₂²×Dic₉₄	1	8	yes	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	32	no	—
238	C₂²	C₂²×C₄⋊Dic₄₇	1	8	yes	—
239	C₂²	C₂²×C₄⋊Dic₄₇	1	4	yes	—
240	C₂²	#15689598	1	4	yes	—
241	C₂²	#15689601	1	4	yes	—
242	C₂²	#15689704	1	8	yes	—
243	C₂²	#15689754	1	8	yes	—
244	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
245	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
246	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
247	C₂²	C₂×#13094069	1	2	yes	—
248	C₂²	C₂×#13094072	1	2	yes	—
249	C₄	—	2	16	no	—
250	C₄	—	188	16	no	—
251	C₄	C₂²×C₄⋊Dic₄₇	1	16	yes	—
252	C₂	—	2	8	no	—
253	C₂	#20246006	1	8	yes	direct product factor
254	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂	C₂×#15689598	1	1	yes	characteristic
256	C₂	C₂×#15689601	1	1	yes	characteristic
257	C₂	C₂×#15689704	1	2	yes	—
258	C₂	C₂×#15689754	1	2	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×#2316
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
2. #20246006	C₂	direct (N × H)
3. C₂×#15689022	C₂	semidirect
12. #15689022	C₂²	semidirect
17. C₂²×C₄⋊C₄×C₄₇	C₄	semidirect
42. C₂×C₄⋊C₄×C₄₇	C₂×C₄	semidirect
54. C₂²×C₄×C₄₇	C₄⋊C₄	semidirect
94. C₂×C₄×C₄₇	C₂×C₄⋊C₄	semidirect
129. C₂×C₄₇	#2316	semidirect
141. C₂²×C₄⋊C₄	Dic₄₇	semidirect
149. C₄₇	C₂×#2316	semidirect
167. C₂×C₄⋊C₄	C₂×Dic₄₇	semidirect
188. C₂²×C₄	C₄⋊Dic₄₇	semidirect
228. C₂×C₄	C₂×C₄⋊Dic₄₇	semidirect
253. C₂	#20246006	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
#20246006	2
C₂×#15689022	2
C₂×#15688983	2
C₂³×C₄⋊C₄×C₄₇	2
C₂×#2316	47

Subgroups — C2×#20246006 #26374892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246006 #26374892