Subgroups — C₂×#20245108 #26369170

Order 12032.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	#20245108	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688927	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	1	no	—
7	C₂×#13093964	—	2	4	no	—
8	C₂×#13093964	C₂²	1	4	yes	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	#15689161	—	2	2	no	—
11	#15689161	C₂²	1	4	yes	—
12	C₂×#13093958	C₂²	1	2	yes	—
13	C₂×#13093956	—	2	1	no	—
14	#15688927	C₂²	1	16	yes	complemented
15	C₂³×Q₈×C₄₇	—	2	1	no	—
16	C₂²×Q₁₆×C₄₇	—	2	1	no	—
17	C₂²×Q₁₆×C₄₇	—	2	2	no	—
18	C₂²×Q₁₆×C₄₇	—	2	4	no	—
19	C₂²×Q₁₆×C₄₇	C₄	1	8	yes	complemented
20	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	#13093983	—	2	4	no	—
24	#13093983	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
27	C₂×C₄×Dic₄₇	—	4	1	no	—
28	#13093964	—	2	8	no	—
29	C₂×C₄₇⋊C₁₆	—	2	4	no	—
30	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
31	#13093958	—	2	2	no	—
32	#13093958	C₂³	1	4	yes	—
33	#13093956	—	2	4	no	—
34	C₂²×Q₈×C₄₇	—	2	8	no	—
35	C₂²×Q₈×C₄₇	—	2	2	no	—
36	C₂²×Q₈×C₄₇	—	2	4	no	—
37	C₂²×Q₈×C₄₇	—	4	1	no	—
38	C₂²×Q₈×C₄₇	—	4	2	no	—
39	C₂²×Q₈×C₄₇	—	4	4	no	—
40	C₂³×C₄×C₄₇	—	4	1	no	—
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₄₇	—	2	16	no	—
43	C₂×Q₁₆×C₄₇	—	2	8	no	—
44	C₂×Q₁₆×C₄₇	—	2	4	no	—
45	C₂×Q₁₆×C₄₇	—	2	2	no	—
46	C₂×Q₁₆×C₄₇	—	2	4	no	—
47	C₂×Q₁₆×C₄₇	—	2	8	no	—
48	C₂×Q₁₆×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	complemented
50	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	2	8	no	—
59	C₂×Q₈×C₄₇	—	2	16	no	—
60	C₂×Q₈×C₄₇	—	2	4	no	—
61	C₂×Q₈×C₄₇	—	4	16	no	—
62	C₂×Q₈×C₄₇	—	4	8	no	—
63	C₂×Q₈×C₄₇	—	4	4	no	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂×Q₈×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	—	4	8	no	—
68	C₂²×C₄×C₄₇	—	4	4	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	D₈	1	2	yes	—
73	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	Q₁₆×C₄₇	—	2	8	no	—
78	Q₁₆×C₄₇	—	2	16	no	—
79	Q₁₆×C₄₇	—	2	4	no	—
80	Q₁₆×C₄₇	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊Dic₄₇	—	4	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄×Dic₄₇	—	4	4	no	—
94	C₄₇⋊C₁₆	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	#102	1	4	yes	—
97	C₂³×C₄₇	C₂×D₈	1	2	yes	—
98	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₄₇	—	2	8	no	—
100	Q₈×C₄₇	—	4	4	no	—
101	Q₈×C₄₇	—	4	8	no	—
102	Q₈×C₄₇	—	4	16	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	4	16	no	—
105	C₂×C₄×C₄₇	—	4	8	no	—
106	C₂×C₄×C₄₇	#102	1	8	yes	—
107	C₂×C₄×C₄₇	#102	1	4	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	8	4	no	—
117	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
119	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×#2141	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	C₂×#102	1	4	yes	—
125	C₂²×C₄₇	C₂×#102	1	2	yes	—
126	C₂²×C₄₇	#510	1	4	yes	—
127	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₄₇	—	4	8	no	—
129	C₄×C₄₇	C₂×#102	1	4	yes	—
130	C₄×C₄₇	C₂×#102	1	2	yes	—
131	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
133	Dic₄₇	—	8	4	no	—
134	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
135	C₂²×M₃₂	—	47	1	no	—
136	C₂×#427	—	47	1	no	—
137	#2141	—	47	8	no	—
138	C₂×#358	—	47	4	no	—
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	#2141	1	4	yes	complemented
141	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
142	C₂×C₄₇	C₂×#510	1	2	yes	—
143	C₂³×Q₈	—	2	1	no	—
144	C₂²×Q₁₆	—	2	1	no	—
145	C₂²×Q₁₆	—	2	2	no	—
146	C₂²×Q₁₆	—	2	4	no	—
147	C₂²×Q₁₆	Dic₄₇	1	8	yes	complemented
148	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
149	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
150	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#117	—	94	1	no	—
152	C₂×M₃₂	—	47	4	no	—
153	C₂×M₃₂	—	94	4	no	—
154	C₂²×C₁₆	—	47	2	no	—
155	#427	—	47	4	no	—
156	#427	—	94	2	no	—
157	C₂×#107	—	47	2	no	—
158	C₂×#106	—	94	1	no	—
159	#358	—	47	16	no	—
160	C₄₇	C₂×#2141	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂²×Q₈	—	4	2	no	—
165	C₂²×Q₈	—	4	1	no	—
166	C₂²×Q₈	—	4	4	no	—
167	C₂³×C₄	—	4	1	no	—
168	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×Q₁₆	—	2	16	no	—
170	C₂×Q₁₆	—	2	8	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	C₂×Dic₄₇	1	8	yes	—
176	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	complemented
177	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
178	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
182	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
184	#117	—	94	4	no	—
185	#117	—	188	2	no	—
186	C₂×C₄⋊C₄	—	94	2	no	—
187	C₂×C₂²⋊C₄	—	188	1	no	—
188	C₂×C₄²	—	188	1	no	—
189	M₃₂	—	94	8	no	—
190	C₂×C₁₆	—	47	4	no	—
191	C₂×C₁₆	—	94	4	no	—
192	#107	—	47	4	no	—
193	#107	—	94	2	no	—
194	#106	—	94	4	no	—
195	C₂⁴	#10625334	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	16	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	8	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	188	2	no	—
209	C₂²×C₄	#10625334	1	2	yes	—
210	C₂²×C₄	#10625334	1	1	yes	characteristic
211	C₂²×C₄	#10625335	1	4	yes	—
212	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	Q₁₆	—	2	16	no	—
217	Q₁₆	—	2	4	no	—
218	Q₁₆	C₂²×Dic₄₇	1	8	yes	—
219	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
220	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
221	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
222	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
227	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
228	C₄⋊C₄	—	94	4	no	—
229	C₄⋊C₄	—	188	2	no	—
230	C₂²⋊C₄	—	188	4	no	—
231	C₄²	—	188	4	no	—
232	C₁₆	—	94	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	#13093945	1	4	yes	—
235	C₂³	C₂×#10625334	1	2	yes	—
236	C₂³	C₂×#10625334	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	8	no	—
239	Q₈	—	4	16	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	376	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	#13093945	1	8	yes	—
247	C₂×C₄	#13093945	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
249	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
250	C₂×C₄	C₂×#10625334	1	2	yes	—
251	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625335	1	4	yes	—
253	C₂×C₄	C₂×#10625335	1	2	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₈	C₂×C₂²⋊Dic₄₇	1	2	yes	—
256	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
257	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
258	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	C₂×#13093945	1	4	yes	—
262	C₂²	C₂×#13093945	1	2	yes	—
263	C₂²	#15689564	1	4	yes	—
264	C₂²	C₂²×#10625334	1	1	yes	characteristic
265	C₄	—	4	8	no	—
266	C₄	—	376	4	no	—
267	C₄	C₂×#13093945	1	4	yes	—
268	C₄	C₂×#13093945	1	2	yes	—
269	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
270	C₄	C₂²×#10625335	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#20245108	1	4	yes	direct product factor
273	C₂	C₂²×#13093945	1	1	yes	characteristic
274	C₂	C₂×#15689564	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2141
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20245108	C₂	direct (N × H)
4. C₂×#15688927	C₂	semidirect
14. #15688927	C₂²	semidirect
19. C₂²×Q₁₆×C₄₇	C₄	semidirect
49. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
140. C₂×C₄₇	#2141	semidirect
147. C₂²×Q₁₆	Dic₄₇	semidirect
160. C₄₇	C₂×#2141	semidirect
176. C₂×Q₁₆	C₂×Dic₄₇	semidirect
272. C₂	#20245108	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×#13093964	2
C₂×#15689161	2
#20245108	2
C₂×#15688927	2
C₂³×Q₁₆×C₄₇	2
C₂×#2141	47

Subgroups — C2×#20245108 #26369170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245108 #26369170