Subgroups — C₂×#20242782 #26198634

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689333	C₂	1	2	yes	maximal
3	#20242782	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688868	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625337	—	2	2	no	—
8	C₂×#13094024	C₂²	1	2	yes	—
9	#15689528	—	2	2	no	—
10	#15689528	C₂²	1	4	yes	—
11	#15689333	—	2	4	no	—
12	#15689333	C₂²	1	8	yes	—
13	C₂×#13093958	—	2	1	no	—
14	C₂×#13093958	—	2	1	no	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093947	—	2	2	no	—
17	#15688876	C₂²	1	4	yes	—
18	#15688868	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₈×Q₈×C₄₇	—	2	2	no	—
21	C₈×Q₈×C₄₇	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	—	2	1	no	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂×#10625337	—	2	8	no	—
28	C₂×#10625337	—	4	8	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13094024	—	2	2	no	—
31	#13094024	—	2	2	no	—
32	#13094024	—	2	2	no	—
33	#13094024	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	#13093958	—	2	4	no	—
37	#13093958	—	2	4	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
40	#13093947	—	2	8	no	—
41	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₄₇	—	2	1	no	—
46	C₄×Q₈×C₄₇	—	2	1	no	—
47	C₄×Q₈×C₄₇	D₄	1	2	yes	—
48	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	4	no	—
54	C₄⋊C₈×C₄₇	—	2	1	no	—
55	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
56	C₄×C₈×C₄₇	—	2	4	no	—
57	C₄×C₈×C₄₇	D₄	1	2	yes	—
58	C₄×C₈×C₄₇	C₂³	1	2	yes	—
59	C₂×Q₈×C₄₇	—	2	1	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	D₈	1	4	yes	—
62	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	1	no	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂×Dic₉₄	—	2	4	no	—
70	C₂×Dic₉₄	—	4	4	no	—
71	C₂×Dic₉₄	—	4	4	no	—
72	C₂×Dic₉₄	—	4	2	no	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	—	2	1	no	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₇	#41	1	1	yes	characteristic
88	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
89	#10625337	—	4	16	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	4	4	no	—
96	C₂³×C₄₇	#121	1	1	yes	characteristic
97	Q₈×C₄₇	—	2	2	no	—
98	Q₈×C₄₇	—	2	4	no	—
99	Q₈×C₄₇	C₂×D₈	1	4	yes	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₄₇	#135	1	2	yes	—
111	C₂×C₄×C₄₇	#137	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	8	4	no	—
117	Dic₉₄	—	4	4	no	—
118	Dic₉₄	—	4	4	no	—
119	Dic₉₄	—	8	4	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	—	4	2	no	—
122	C₈×C₄₇	C₂×Q₁₆	1	4	yes	complemented
123	C₄₇⋊C₈	—	4	4	no	—
124	C₂×#1674	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	#462	1	2	yes	—
126	C₂²×C₄₇	#466	1	2	yes	—
127	C₂²×C₄₇	#473	1	2	yes	—
128	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	2	1	no	—
132	C₄×C₄₇	—	4	2	no	—
133	C₄×C₄₇	#462	1	2	yes	—
134	C₄×C₄₇	#466	1	2	yes	—
135	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
137	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
139	Dic₄₇	—	8	4	no	—
140	C₂×#500	—	47	1	no	—
141	C₂×#462	—	47	2	no	—
142	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
143	#1674	—	47	8	no	—
144	C₂×#335	—	47	1	no	—
145	C₂×#332	—	47	2	no	—
146	C₂×C₄₇	#1674	1	4	yes	complemented
147	C₂×C₄₇	C₂×#462	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
150	C₂²×Q₁₆	—	94	2	no	—
151	C₂×C₄⋊Q₈	—	47	2	no	—
152	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
153	#500	—	47	4	no	—
154	#500	—	94	2	no	—
155	#462	—	47	8	no	—
156	#462	—	94	4	no	—
157	C₈×Q₈	—	2	2	no	—
158	C₈×Q₈	C₂×D₄₇	1	4	yes	—
159	C₂×#107	—	94	1	no	—
160	C₂×#107	—	94	1	no	—
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#103	—	94	2	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
167	#335	—	47	4	no	—
168	#332	—	47	8	no	—
169	C₄₇	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	94	2	no	—
171	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
172	C₂×Q₁₆	—	94	8	no	—
173	C₂×Q₁₆	—	188	8	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	94	2	no	—
176	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	47	4	no	—
179	C₄⋊Q₈	—	94	2	no	—
180	C₄⋊Q₈	—	94	2	no	—
181	C₄⋊Q₈	—	94	2	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	D₁₈₈	1	2	yes	—
185	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	94	2	no	—
188	C₂×C₄⋊C₄	—	94	2	no	—
189	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#107	—	94	4	no	—
193	#107	—	94	4	no	—
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	47	4	no	—
197	C₄⋊C₈	—	94	2	no	—
198	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
199	#103	—	94	8	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
202	C₄×C₈	C₂²×D₄₇	1	2	yes	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	94	4	no	—
206	C₂×Q₈	—	188	4	no	—
207	C₂×Q₈	—	188	4	no	—
208	C₂×Q₈	—	188	2	no	—
209	C₂×Q₈	D₃₇₆	1	4	yes	—
210	C₂×Q₈	C₂×D₁₈₈	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	188	2	no	—
214	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
215	C₂²×C₄	#10625350	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	Q₁₆	—	188	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	188	4	no	—
223	C₂×C₈	#10625337	1	4	yes	complemented
224	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₄⋊C₄	—	188	2	no	—
232	C₄⋊C₄	—	188	2	no	—
233	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
237	C₄²	#10625350	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₄²	C₂³×D₄₇	1	1	yes	characteristic
240	C₂³	#13093998	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	2	no	—
243	Q₈	—	376	4	no	—
244	Q₈	—	188	4	no	—
245	Q₈	—	188	4	no	—
246	Q₈	C₂×D₃₇₆	1	4	yes	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	376	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	#13093998	1	1	yes	characteristic
255	C₂×C₄	#13093998	1	1	yes	characteristic
256	C₂×C₄	#13093998	1	1	yes	characteristic
257	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
258	C₂×C₄	C₂×#10625337	1	2	yes	—
259	C₂×C₄	#13094046	1	2	yes	—
260	C₂×C₄	#13094054	1	2	yes	—
261	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
263	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	188	4	no	—
267	C₈	C₂×#10625337	1	4	yes	complemented
268	C₂²	#15689333	1	2	yes	—
269	C₂²	#15689359	1	2	yes	—
270	C₂²	#15689398	1	2	yes	—
271	C₂²	C₂×#13093998	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	376	4	no	—
277	C₄	#15689333	1	2	yes	—
278	C₄	#15689359	1	2	yes	—
279	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
280	C₄	C₂²×#10625337	1	1	yes	characteristic
281	C₄	C₂×#13094046	1	1	yes	characteristic
282	C₄	C₂×#13094054	1	1	yes	characteristic
283	C₂	#20242782	1	4	yes	direct product factor
284	C₂	C₂×#15689333	1	1	yes	characteristic
285	C₂	C₂×#15689359	1	1	yes	characteristic
286	C₂	C₂×#15689398	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	C₂×#1674
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20242782	C₂	direct (N × H)
76. C₂×C₈×C₄₇	Q₁₆	semidirect
122. C₈×C₄₇	C₂×Q₁₆	semidirect
146. C₂×C₄₇	#1674	semidirect
169. C₄₇	C₂×#1674	semidirect
223. C₂×C₈	#10625337	semidirect
267. C₈	C₂×#10625337	semidirect
283. C₂	#20242782	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×#15689528	2
C₂×#15689333	2
#20242782	2
C₂×#15688876	2
C₂×#15688868	2
C₂×C₈×Q₈×C₄₇	2
C₂×#1674	47

Subgroups — C2×#20242782 #26198634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242782 #26198634