Subgroups — C₂×#20244360 #26220848

Order 12032.

1422 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	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20244360	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₉₄	—	2	1	no	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	2	no	—
9	C₂×#13094024	—	2	4	no	—
10	C₂×#13094024	C₄	1	8	yes	complemented
11	C₂×#13094024	C₂²	1	4	yes	—
12	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×#13093935	—	2	4	no	—
15	C₂×#13093935	C₂²	1	4	yes	—
16	#15688885	C₂²	1	16	yes	—
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#382	—	2	2	no	—
20	C₄₇×#382	C₂²	1	4	yes	—
21	C₂×#95×C₄₇	—	2	1	no	—
22	C₂×#95×C₄₇	—	2	1	no	—
23	C₂²×Dic₉₄	—	2	8	no	—
24	C₂²×Dic₉₄	—	2	4	no	—
25	C₂²×Dic₉₄	—	4	8	no	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	#13094024	—	2	16	no	—
32	#13094024	—	2	8	no	—
33	#13094024	C₂×C₄	1	16	yes	complemented
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄⋊Dic₄₇	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	#13093935	—	2	8	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#95×C₄₇	—	2	4	no	—
53	#95×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	4	8	no	—
74	C₂×Dic₉₄	—	2	16	no	—
75	C₂×Dic₉₄	—	4	16	no	—
76	C₂×Dic₉₄	—	4	32	no	—
77	C₄⋊C₄×C₄₇	—	2	8	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
80	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
81	C₄²×C₄₇	C₂×D₄	1	4	yes	—
82	C₄⋊Dic₄₇	—	2	16	no	—
83	C₄⋊Dic₄₇	—	2	8	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
86	C₂×C₄₇⋊C₈	—	2	16	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
91	C₂³×C₄₇	#124	1	2	yes	—
92	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
94	C₂³×C₄₇	#137	1	4	yes	—
95	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	8	8	no	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
109	C₂×C₄×C₄₇	#124	1	2	yes	—
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×Dic₄₇	—	2	8	no	—
116	C₂×Dic₄₇	—	4	8	no	—
117	C₂×Dic₄₇	—	4	16	no	—
118	Dic₉₄	—	4	32	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	C₂×#1993	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	#391	1	4	yes	—
122	C₂²×C₄₇	#418	1	4	yes	—
123	C₂²×C₄₇	#419	1	4	yes	—
124	C₂²×C₄₇	#489	1	8	yes	—
125	C₂²×C₄₇	#501	1	8	yes	—
126	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₂²×C₄₇	C₂×#137	1	2	yes	—
131	C₄×C₄₇	—	2	4	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	8	4	no	—
134	C₄×C₄₇	#391	1	4	yes	—
135	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
137	Dic₄₇	—	4	8	no	—
138	C₂²×C₄⋊Q₈	—	47	1	no	—
139	C₂²×#97	—	47	1	no	—
140	C₂×#382	D₄₇	1	1	yes	characteristic
141	#1993	—	47	8	no	—
142	C₂×#340	—	47	4	no	—
143	C₂×C₄₇	#1993	1	8	yes	complemented
144	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#418	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#419	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#489	1	2	yes	—
148	C₂×C₄₇	C₂×#501	1	2	yes	—
149	C₂³×Q₈	—	94	1	no	—
150	C₂³×C₈	—	47	2	no	—
151	C₂×C₄⋊Q₈	—	47	8	no	—
152	C₂×C₄⋊Q₈	—	47	4	no	—
153	C₂×C₄⋊Q₈	—	94	4	no	—
154	C₂×C₄⋊Q₈	—	94	2	no	—
155	C₂²×C₄⋊C₄	—	47	2	no	—
156	C₂²×C₄⋊C₄	—	94	1	no	—
157	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#97	—	47	4	no	—
160	C₂×#97	—	94	4	no	—
161	#382	—	2	2	no	—
162	#382	C₂×D₄₇	1	4	yes	—
163	C₂×#95	—	2	1	no	—
164	C₂×#95	—	2	1	no	—
165	#340	—	47	16	no	—
166	C₄₇	C₂×#1993	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	94	8	no	—
168	C₂²×Q₈	—	94	4	no	—
169	C₂²×Q₈	—	188	8	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	94	2	no	—
172	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂²×C₈	—	47	8	no	—
176	C₂²×C₈	—	47	4	no	—
177	C₂²×C₈	—	94	4	no	—
178	C₂²×C₈	—	94	4	no	—
179	C₄⋊Q₈	—	47	16	no	—
180	C₄⋊Q₈	—	94	16	no	—
181	C₄⋊Q₈	—	94	8	no	—
182	C₂×C₄⋊C₄	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	47	8	no	—
185	C₂×C₄⋊C₄	—	47	4	no	—
186	C₂×C₄⋊C₄	—	94	8	no	—
187	C₂×C₄⋊C₄	—	94	4	no	—
188	C₂×C₄⋊C₄	—	94	2	no	—
189	C₂×C₄⋊C₄	—	94	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
191	C₂×C₄²	C₄×D₄₇	1	4	yes	—
192	C₂×C₄²	D₁₈₈	1	4	yes	—
193	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
194	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
195	#97	—	94	8	no	—
196	#95	—	2	4	no	—
197	#95	—	2	4	no	—
198	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂×Q₈	—	94	16	no	—
200	C₂×Q₈	—	188	32	no	—
201	C₂×Q₈	—	188	16	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	94	8	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	188	8	no	—
211	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
212	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
215	C₂²×C₄	#10625352	1	4	yes	—
216	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	47	8	no	—
222	C₂×C₈	—	94	16	no	—
223	C₂×C₈	—	94	4	no	—
224	C₂×C₈	—	94	4	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	47	8	no	—
228	C₄⋊C₄	—	94	8	no	—
229	C₄⋊C₄	—	94	16	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
232	C₄²	C₂×C₄×D₄₇	1	4	yes	—
233	C₄²	C₂×D₁₈₈	1	4	yes	—
234	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
235	C₂³	#13094011	1	2	yes	—
236	C₂³	#13094022	1	2	yes	—
237	C₂³	#13094049	1	4	yes	—
238	C₂³	#13094054	1	4	yes	—
239	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
240	Q₈	—	188	32	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	8	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	188	16	no	—
251	C₂×C₄	—	188	8	no	—
252	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
253	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
254	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₄	#13093989	1	8	yes	—
256	C₂×C₄	#13094004	1	8	yes	—
257	C₂×C₄	#13094011	1	2	yes	—
258	C₂×C₄	#13094022	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×#10625352	1	2	yes	—
263	C₈	—	94	8	no	—
264	C₂²	#15689028	1	4	yes	—
265	C₂²	#15689127	1	4	yes	—
266	C₂²	#15689131	1	4	yes	—
267	C₂²	#15689482	1	8	yes	—
268	C₂²	#15689533	1	8	yes	—
269	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
270	C₂²	C₂×#13094011	1	1	yes	characteristic
271	C₂²	C₂×#13094022	1	1	yes	characteristic
272	C₂²	C₂×#13094049	1	2	yes	—
273	C₂²	C₂×#13094054	1	2	yes	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	8	4	no	—
277	C₄	—	188	8	no	—
278	C₄	#15689028	1	4	yes	—
279	C₄	C₂×#13093989	1	2	yes	—
280	C₄	C₂×#13094004	1	2	yes	—
281	C₂	#20244360	1	8	yes	direct product factor
282	C₂	C₂×#15689028	1	1	yes	characteristic
283	C₂	C₂×#15689127	1	1	yes	characteristic
284	C₂	C₂×#15689131	1	1	yes	characteristic
285	C₂	C₂×#15689482	1	2	yes	—
286	C₂	C₂×#15689533	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#382	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#382	D₄₇
hypocenter	C₄₇	C₂×#1993
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
3. #20244360	C₂	direct (N × H)
10. C₂×#13094024	C₄	semidirect
29. C₂²×C₄₇⋊C₈	Q₈	semidirect
33. #13094024	C₂×C₄	semidirect
89. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
143. C₂×C₄₇	#1993	semidirect
166. C₄₇	C₂×#1993	semidirect
281. C₂	#20244360	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×#13094024	2
C₂²×#13093935	2
#20244360	2
C₂×#15688885	2
C₂×C₄₇×#382	2
C₂×#1993	47

Subgroups — C2×#20244360 #26220848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244360 #26220848