Subgroups — C₂×#20245569 #26371036

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689359	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245569	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688949	C₂	1	2	yes	maximal, complemented
5	C₂×#15688925	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
8	C₂×#13093998	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
10	#15689359	—	2	2	no	—
11	#15689359	C₂²	1	4	yes	—
12	C₂×#13093958	—	2	2	no	—
13	C₂×#13093958	—	2	1	no	—
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093958	C₂²	1	2	yes	—
16	C₂×#13093958	C₂²	1	4	yes	complemented
17	C₂×#13093946	—	2	1	no	—
18	#15688949	C₂²	1	8	yes	complemented
19	#15688925	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
22	C₄₇×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×D₁₈₈	—	2	1	no	—
24	C₂²×C₂²⋊D₄₇	—	4	1	no	—
25	C₂³×Dic₄₇	—	4	1	no	—
26	C₂×D₃₇₆	—	2	4	no	—
27	C₂×D₃₇₆	—	2	1	no	—
28	C₂×D₃₇₆	C₂³	1	2	yes	—
29	#13093998	—	2	4	no	—
30	#13093998	—	4	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	4	1	no	—
35	C₂×C₄⋊Dic₄₇	—	4	2	no	—
36	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
37	#13093958	—	2	1	no	—
38	#13093958	—	2	2	no	—
39	#13093958	—	2	4	no	—
40	#13093958	C₂³	1	2	yes	—
41	#13093958	C₂³	1	4	yes	—
42	#13093946	—	2	4	no	—
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	—	2	4	no	—
50	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
51	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
52	C₂³×D₄₇	—	4	1	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	D₈	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₄₇	—	4	4	no	—
60	C₂×C₂²⋊D₄₇	—	8	4	no	—
61	C₂²×Dic₄₇	—	4	1	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	2	no	—
64	C₂²×Dic₄₇	—	8	4	no	—
65	C₂×D₁₈₈	—	2	2	no	—
66	C₂×D₁₈₈	—	4	2	no	—
67	C₂×D₁₈₈	—	4	2	no	—
68	C₂×D₁₈₈	—	4	1	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	#41	1	4	yes	—
75	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
76	(C₂×D₄₇)⋊C₄	—	4	4	no	—
77	C₄⋊Dic₄₇	—	2	2	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄⋊Dic₄₇	—	4	1	no	—
82	D₃₇₆	—	2	4	no	—
83	D₃₇₆	—	4	2	no	—
84	C₁₆×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	D₁₆	1	4	yes	—
88	C₂³×C₄₇	C₂×D₈	1	2	yes	—
89	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₄₇	—	4	2	no	—
91	C₂²×D₄₇	—	8	2	no	—
92	C₂²×D₄₇	—	8	2	no	—
93	C₂²×D₄₇	—	8	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	#123	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#137	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂²⋊D₄₇	—	8	8	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	8	8	no	—
107	C₂×Dic₄₇	—	8	4	no	—
108	C₂×Dic₄₇	—	8	2	no	—
109	D₁₈₈	—	4	2	no	—
110	D₁₈₈	—	4	2	no	—
111	D₁₈₈	—	8	2	no	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	#123	1	2	yes	—
114	C₈×C₄₇	C₂×#41	1	2	yes	—
115	C₂×#2232	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	8	no	—
119	C₂²×C₄₇	#480	1	2	yes	—
120	C₂²×C₄₇	C₂×D₁₆	1	4	yes	—
121	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
122	C₂²×C₄₇	#510	1	2	yes	—
123	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
124	C₂×D₄₇	—	8	2	no	—
125	C₂×D₄₇	—	8	2	no	—
126	C₂×D₄₇	—	8	2	no	—
127	C₂×D₄₇	—	16	4	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	#480	1	2	yes	—
130	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
132	Dic₄₇	—	8	4	no	—
133	Dic₄₇	—	8	2	no	—
134	C₂×#466	—	47	1	no	—
135	C₂²×#107	—	47	1	no	—
136	#2232	—	47	8	no	—
137	C₂×#366	—	47	2	no	—
138	C₂×#357	—	47	2	no	—
139	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
140	C₂×C₄₇	—	2	4	no	—
141	C₂×C₄₇	#2232	1	4	yes	complemented
142	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
143	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
145	D₄₇	—	16	2	no	—
146	C₂²×D₈	—	47	1	no	—
147	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#121	—	94	1	no	—
149	C₂²×C₄⋊C₄	—	94	1	no	—
150	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
151	#466	—	47	4	no	—
152	#466	—	94	2	no	—
153	C₂×#107	—	47	2	no	—
154	C₂×#107	—	47	4	no	—
155	C₂×#107	—	47	1	no	—
156	C₂×#107	—	94	1	no	—
157	C₂×#107	—	94	2	no	—
158	C₂×#102	—	94	1	no	—
159	#366	—	47	8	no	—
160	#357	—	47	8	no	—
161	C₂²⋊C₁₆	C₂×D₄₇	1	4	yes	complemented
162	C₄₇	C₂×#2232	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	—	94	1	no	—
164	C₂²×D₄	—	188	1	no	—
165	C₂³×C₄	—	188	1	no	—
166	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
167	C₂×D₈	—	47	2	no	—
168	C₂×D₈	—	94	1	no	—
169	C₂×D₈	—	94	4	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
172	C₂²×C₈	D₁₈₈	1	2	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#121	—	94	4	no	—
176	#121	—	188	2	no	—
177	C₂×C₄⋊C₄	—	94	4	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄⋊C₄	—	94	1	no	—
180	C₂×C₄⋊C₄	—	188	2	no	—
181	C₂×C₄⋊C₄	—	188	1	no	—
182	C₂×C₂²⋊C₄	—	188	1	no	—
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
185	#107	—	47	4	no	—
186	#107	—	47	2	no	—
187	#107	—	94	4	no	—
188	#107	—	94	2	no	—
189	#107	—	94	1	no	—
190	#102	—	94	4	no	—
191	C₂⁴	—	188	1	no	—
192	C₂⁴	D₃₇₆	1	1	yes	characteristic
193	C₂×D₄	—	94	2	no	—
194	C₂×D₄	—	376	4	no	—
195	C₂×D₄	—	188	4	no	—
196	C₂×D₄	—	188	2	no	—
197	C₂×D₄	—	188	1	no	—
198	C₂×D₄	—	188	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	376	4	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	—	188	1	no	—
204	C₂²×C₄	—	188	2	no	—
205	C₂²×C₄	D₃₇₆	1	2	yes	—
206	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
208	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
209	D₈	—	94	4	no	—
210	D₈	—	188	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
215	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
216	C₂×C₈	#10625352	1	4	yes	—
217	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
218	C₄⋊C₄	—	94	2	no	—
219	C₄⋊C₄	—	94	4	no	—
220	C₄⋊C₄	—	188	4	no	—
221	C₄⋊C₄	—	188	2	no	—
222	C₄⋊C₄	—	188	1	no	—
223	C₂²⋊C₄	—	188	4	no	—
224	C₁₆	—	2	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	376	2	no	—
228	C₂³	—	376	2	no	—
229	C₂³	—	376	2	no	—
230	C₂³	—	188	2	no	—
231	C₂³	D₇₅₂	1	4	yes	—
232	C₂³	C₂×D₃₇₆	1	2	yes	—
233	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
234	D₄	—	376	8	no	—
235	D₄	—	376	2	no	—
236	D₄	—	188	2	no	—
237	D₄	—	188	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	376	8	no	—
243	C₂×C₄	—	376	4	no	—
244	C₂×C₄	—	376	2	no	—
245	C₂×C₄	—	188	2	no	—
246	C₂×C₄	—	188	4	no	—
247	C₂×C₄	#13094008	1	2	yes	—
248	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
249	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
250	C₂×C₄	#13094057	1	2	yes	—
251	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	#13094008	1	2	yes	—
254	C₈	C₂×#10625352	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	376	2	no	—
259	C₂²	—	376	2	no	—
260	C₂²	—	376	2	no	—
261	C₂²	—	752	4	no	—
262	C₂²	#15689437	1	2	yes	—
263	C₂²	C₂×D₇₅₂	1	4	yes	—
264	C₂²	C₂×D₇₅₂	1	2	yes	—
265	C₂²	#15689566	1	2	yes	—
266	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	376	4	no	—
269	C₄	—	376	2	no	—
270	C₄	#15689437	1	2	yes	—
271	C₄	C₂×#13094008	1	1	yes	characteristic
272	C₄	C₂×#13094057	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	752	2	no	—
275	C₂	#20245569	1	4	yes	direct product factor
276	C₂	C₂×#15689437	1	1	yes	characteristic
277	C₂	C₂²×D₇₅₂	1	1	yes	characteristic
278	C₂	C₂×#15689566	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂×#15689359	C₂
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2232
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×#13093958	C₂	semidirect
3. #20245569	C₂	direct (N × H)
4. C₂×#15688949	C₂	semidirect
5. C₂×#15688925	C₂	semidirect
6. C₂×C₄₇×C₂²⋊C₁₆	C₂	semidirect
16. C₂×#13093958	C₂²	semidirect
18. #15688949	C₂²	semidirect
19. #15688925	C₂²	semidirect
22. C₄₇×C₂²⋊C₁₆	C₂²	semidirect
139. C₂×C₂²⋊C₁₆	D₄₇	semidirect
141. C₂×C₄₇	#2232	semidirect
161. C₂²⋊C₁₆	C₂×D₄₇	semidirect
162. C₄₇	C₂×#2232	semidirect
275. C₂	#20245569	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×#15689359	2
C₂²×#13093958	2
#20245569	2
C₂×#15688949	2
C₂×#15688925	2
C₂×C₄₇×C₂²⋊C₁₆	2
C₂×#2232	47

Subgroups — C2×#20245569 #26371036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245569 #26371036