Subgroups — C₂×#20242144 #26185276

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	#20242144	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688874	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	2	no	—
6	#15689147	—	2	4	no	—
7	#15689147	C₂²	1	8	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688874	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	4	yes	—
18	C₂×#10625329	—	2	8	no	—
19	C₂×#10625329	—	4	8	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	—	2	8	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
39	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	4	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	2	16	no	—
61	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
62	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
72	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
74	#10625329	—	4	16	no	—
75	C₂×C₄₇⋊C₈	—	2	8	no	—
76	C₂×C₄₇⋊C₈	—	4	8	no	—
77	C₂³×C₄₇	—	2	2	no	—
78	C₂³×C₄₇	—	2	2	no	—
79	C₂³×C₄₇	#106	1	4	yes	—
80	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	#106	1	8	yes	—
89	C₂×C₄×C₄₇	#106	1	4	yes	—
90	C₂×C₄×C₄₇	#108	1	4	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×SD₁₆	1	8	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₈×C₄₇	—	2	4	no	—
104	C₈×C₄₇	C₂×M₁₆	1	8	yes	complemented
105	C₄₇⋊C₈	—	4	8	no	—
106	C₂×#1566	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	2	8	no	—
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	#423	1	2	yes	—
111	C₂²×C₄₇	C₂×#106	1	4	yes	—
112	C₂²×C₄₇	C₂×#106	1	2	yes	—
113	C₂²×C₄₇	C₂×#108	1	2	yes	—
114	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	#423	1	2	yes	—
119	C₄×C₄₇	C₂×#106	1	2	yes	—
120	C₄×C₄₇	C₂×#106	1	4	yes	—
121	C₄×C₄₇	C₂×#108	1	4	yes	—
122	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
123	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
124	C₂×#423	—	47	2	no	—
125	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
126	#1566	—	47	8	no	—
127	C₂×#334	—	47	4	no	—
128	C₂×C₄₇	—	2	4	no	—
129	C₂×C₄₇	#1566	1	4	yes	complemented
130	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
131	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
132	C₂×C₄₇	C₂²×#108	1	1	yes	characteristic
133	C₂²×M₁₆	—	94	2	no	—
134	C₂³×C₈	Dic₄₇	1	2	yes	—
135	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
136	#423	—	47	8	no	—
137	#423	—	94	4	no	—
138	C₂×C₄⋊C₈	—	47	4	no	—
139	C₂×C₄⋊C₈	—	94	2	no	—
140	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
141	C₂×C₄×C₈	Dic₄₇	1	4	yes	—
142	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
145	#334	—	47	16	no	—
146	C₄₇	C₂×#1566	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
148	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
150	C₂×M₁₆	—	94	8	no	—
151	C₂×M₁₆	—	188	8	no	—
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	—	94	4	no	—
154	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
155	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
156	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
157	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
158	C₂×C₄²	Dic₉₄	1	2	yes	—
159	C₂×C₄²	D₁₈₈	1	2	yes	—
160	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
162	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₄⋊C₈	—	47	8	no	—
166	C₄⋊C₈	—	94	4	no	—
167	C₄⋊C₈	—	94	8	no	—
168	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
169	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
170	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
171	C₄×C₈	C₂²×D₄₇	1	4	yes	—
172	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	#10625325	1	8	yes	—
176	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
181	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
184	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
185	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
188	M₁₆	—	188	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	—	94	8	no	—
192	C₂×C₈	—	188	8	no	—
193	C₂×C₈	#10625329	1	8	yes	complemented
194	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
195	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
196	C₄²	C₄⋊Dic₄₇	1	2	yes	—
197	C₄²	C₄⋊Dic₄₇	1	2	yes	—
198	C₄²	C₄⋊Dic₄₇	1	2	yes	—
199	C₄²	C₂×Dic₉₄	1	2	yes	—
200	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
201	C₄²	C₂×D₁₈₈	1	2	yes	—
202	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
203	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₄²	C₂²×Dic₄₇	1	2	yes	—
205	C₄²	C₂³×D₄₇	1	1	yes	characteristic
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	#13093956	1	4	yes	—
209	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
210	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	#13093956	1	8	yes	—
218	C₂×C₄	#13093956	1	4	yes	—
219	C₂×C₄	#13093960	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
226	C₂×C₄	C₂×#10625329	1	4	yes	—
227	C₂×C₄	C₂×#10625325	1	4	yes	—
228	C₂×C₄	C₂×#10625325	1	8	yes	—
229	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
231	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	188	8	no	—
234	C₈	C₂×#10625329	1	8	yes	complemented
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	8	no	—
238	C₂²	#15689147	1	2	yes	—
239	C₂²	C₂×#13093956	1	4	yes	—
240	C₂²	C₂×#13093956	1	2	yes	—
241	C₂²	C₂×#13093960	1	2	yes	—
242	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	#15689147	1	2	yes	—
247	C₄	C₂×#13093956	1	4	yes	—
248	C₄	C₂×#13093956	1	2	yes	—
249	C₄	C₂×#13093960	1	4	yes	—
250	C₄	C₂²×#10625329	1	2	yes	—
251	C₄	C₂²×#10625325	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	#20242144	1	4	yes	direct product factor
254	C₂	C₂×#15689147	1	1	yes	characteristic
255	C₂	C₂²×#13093956	1	1	yes	characteristic
256	C₂	C₂²×#13093960	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#1566
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20242144	C₂	direct (N × H)
3. C₂×#15688874	C₂	semidirect
10. #15688874	C₂²	semidirect
61. C₂×C₈×C₄₇	M₁₆	semidirect
104. C₈×C₄₇	C₂×M₁₆	semidirect
129. C₂×C₄₇	#1566	semidirect
146. C₄₇	C₂×#1566	semidirect
193. C₂×C₈	#10625329	semidirect
234. C₈	C₂×#10625329	semidirect
253. C₂	#20242144	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×#15689147	2
#20242144	2
C₂×#15688874	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1566	47

Subgroups — C2×#20242144 #26185276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242144 #26185276