Subgroups — C₂×#20241352 #26126013

Order 12032.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
3	#20241352	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#15688876	—	2	1	no	—
10	#15688876	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄×Q₁₆×C₄₇	—	2	1	no	—
14	C₄×Q₁₆×C₄₇	—	2	1	no	—
15	C₄×Q₁₆×C₄₇	C₄	1	2	yes	—
16	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
18	C₂×#103×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	—	2	4	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
25	C₂²×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₁₆×C₄₇	—	2	1	no	—
27	C₂×Q₁₆×C₄₇	—	2	2	no	—
28	C₂×Q₁₆×C₄₇	C₈	1	4	yes	complemented
29	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₄₇	—	2	2	no	—
33	C₄×Q₈×C₄₇	—	2	2	no	—
34	C₄×Q₈×C₄₇	—	4	1	no	—
35	C₄×Q₈×C₄₇	—	4	1	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
38	C₂×C₄²×C₄₇	—	4	1	no	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	#107×C₄₇	—	2	1	no	—
41	#107×C₄₇	C₂×C₄	1	2	yes	—
42	#103×C₄₇	—	2	2	no	—
43	#103×C₄₇	—	2	2	no	—
44	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	—	2	2	no	—
50	C₂×Q₈×C₄₇	—	4	1	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	—	4	1	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
57	Q₁₆×C₄₇	—	2	4	no	—
58	Q₁₆×C₄₇	—	2	2	no	—
59	Q₁₆×C₄₇	C₂×C₈	1	4	yes	complemented
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₄⋊C₄×C₄₇	—	4	1	no	—
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	—	4	2	no	—
72	C₄²×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₄₇	D₈	1	1	yes	characteristic
75	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	8	2	no	—
79	C₄₇⋊C₁₆	—	2	4	no	—
80	C₂³×C₄₇	#102	1	1	yes	characteristic
81	Q₈×C₄₇	—	2	4	no	—
82	Q₈×C₄₇	—	4	4	no	—
83	Q₈×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	2	no	—
88	C₂×C₄×C₄₇	—	8	2	no	—
89	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#104	1	2	yes	—
94	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
95	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₄₇	—	2	1	no	—
100	C₈×C₄₇	—	2	1	no	—
101	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
102	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
103	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
104	C₄₇⋊C₈	—	8	2	no	—
105	C₂×#1332	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	D₄⋊C₈	1	2	yes	—
107	C₂²×C₄₇	#358	1	2	yes	—
108	C₂²×C₄₇	#359	1	2	yes	—
109	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
110	C₄×C₄₇	—	2	1	no	—
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	4	4	no	—
113	C₄×C₄₇	—	8	2	no	—
114	C₄×C₄₇	D₄⋊C₈	1	2	yes	—
115	C₄×C₄₇	#358	1	2	yes	—
116	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
118	C₄×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
119	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
120	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
121	C₂×C₄×C₁₆	—	47	1	no	—
122	C₂×#335	—	47	1	no	—
123	#1332	—	47	4	no	—
124	C₂×C₄₇	#1332	1	4	yes	complemented
125	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
127	C₂×C₄₇	C₂×#359	1	1	yes	characteristic
128	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	2	1	no	—
130	C₂²×C₁₆	—	94	1	no	—
131	C₄×Q₁₆	—	2	1	no	—
132	C₄×Q₁₆	—	2	1	no	—
133	C₄×Q₁₆	Dic₄₇	1	2	yes	—
134	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
135	C₂×#107	Dic₄₇	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	94	1	no	—
137	C₂×#103	—	2	1	no	—
138	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
139	C₄×C₁₆	—	47	2	no	—
140	C₄×C₁₆	—	94	1	no	—
141	#335	—	47	2	no	—
142	#335	—	94	1	no	—
143	C₄₇	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	2	1	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	C₄₇⋊C₈	1	4	yes	complemented
148	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
149	C₂²×C₈	—	188	1	no	—
150	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₄×Q₈	—	4	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	4	1	no	—
158	C₂×C₄²	—	4	1	no	—
159	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂×C₁₆	—	94	4	no	—
161	C₂×C₁₆	—	94	2	no	—
162	#107	—	2	1	no	—
163	#107	C₂×Dic₄₇	1	2	yes	—
164	C₄⋊C₈	—	94	2	no	—
165	C₄⋊C₈	—	188	1	no	—
166	#103	—	2	2	no	—
167	#103	—	2	2	no	—
168	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	#10625334	1	1	yes	characteristic
179	C₂²×C₄	#10625335	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	2	2	no	—
183	Q₁₆	C₂×C₄₇⋊C₈	1	4	yes	complemented
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	376	2	no	—
186	C₂×C₈	—	188	2	no	—
187	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
188	C₂×C₈	#10625329	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	#10625334	1	1	yes	characteristic
200	C₄²	#10625335	1	1	yes	characteristic
201	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₁₆	—	94	4	no	—
204	C₂³	#13093945	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
214	C₂×C₄	#13093945	1	1	yes	characteristic
215	C₂×C₄	#13093945	1	1	yes	characteristic
216	C₂×C₄	#13093945	1	1	yes	characteristic
217	C₂×C₄	#13093951	1	2	yes	—
218	C₂×C₄	#13093967	1	2	yes	—
219	C₂×C₄	#13093970	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
222	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	376	2	no	—
226	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
227	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
228	C₈	C₂×#10625329	1	1	yes	characteristic
229	C₂²	D₄⋊(C₄₇⋊C₈)	1	2	yes	—
230	C₂²	#15688927	1	2	yes	—
231	C₂²	#15688930	1	2	yes	—
232	C₂²	C₂×#13093945	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	D₄⋊(C₄₇⋊C₈)	1	2	yes	—
238	C₄	#15688927	1	2	yes	—
239	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
240	C₄	C₂×#13093951	1	1	yes	characteristic
241	C₄	C₂×#13093967	1	1	yes	characteristic
242	C₄	C₂×#13093970	1	1	yes	characteristic
243	C₂	#20241352	1	4	yes	direct product factor
244	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
245	C₂	C₂×#15688927	1	1	yes	characteristic
246	C₂	C₂×#15688930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#10625334
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#1332
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20241352	C₂	direct (N × H)
28. C₂×Q₁₆×C₄₇	C₈	semidirect
59. Q₁₆×C₄₇	C₂×C₈	semidirect
124. C₂×C₄₇	#1332	semidirect
143. C₄₇	C₂×#1332	semidirect
147. C₂×Q₁₆	C₄₇⋊C₈	semidirect
183. Q₁₆	C₂×C₄₇⋊C₈	semidirect
243. C₂	#20241352	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
C₂×#15688876	2
#20241352	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#1332	47

Subgroups — C2×#20241352 #26126013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241352 #26126013