Subgroups — C₂×#20244209 #26219928

Order 12032.

1598 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093948	C₂	1	4	yes	maximal
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
4	#20244209	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×#13093948	—	2	16	no	—
9	C₂×#13093948	C₂²	1	16	yes	—
10	C₂×#13093935	—	2	4	no	—
11	C₂×#13093935	C₂²	1	4	yes	—
12	#15688989	—	2	2	no	—
13	#15688989	C₂²	1	4	yes	—
14	C₂×C₄₇⋊#95	—	2	2	no	—
15	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	#13093948	—	2	32	no	—
31	#13093935	—	2	8	no	—
32	C₄₇⋊#95	—	2	8	no	—
33	C₂²×Q₈×C₄₇	—	2	4	no	—
34	C₂²×Q₈×C₄₇	—	2	2	no	—
35	C₂²×Q₈×C₄₇	—	2	8	no	—
36	C₂²×Q₈×C₄₇	—	2	8	no	—
37	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
38	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₇	—	2	2	no	—
40	C₂³×C₄×C₄₇	D₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₄₇	—	2	16	no	—
43	C₄⋊Q₈×C₄₇	—	2	8	no	—
44	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	D₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₄₇	—	2	8	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	—	2	32	no	—
55	C₂×Q₈×C₄₇	—	2	8	no	—
56	C₂×Q₈×C₄₇	—	2	8	no	—
57	C₂×Q₈×C₄₇	—	4	16	no	—
58	C₂×Q₈×C₄₇	—	4	4	no	—
59	C₂×Q₈×C₄₇	—	4	4	no	—
60	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₄₇	#41	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	8	no	—
77	C₄⋊C₄×C₄₇	—	2	32	no	—
78	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
80	C₄²×C₄₇	C₂×D₄	1	4	yes	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	16	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
87	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂³×C₄₇	#124	1	2	yes	—
89	C₂³×C₄₇	#127	1	2	yes	—
90	C₂³×C₄₇	#137	1	8	yes	—
91	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₄₇	—	2	16	no	—
93	Q₈×C₄₇	—	4	8	no	—
94	Q₈×C₄₇	—	4	16	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	16	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	16	no	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
104	C₂×C₄×C₄₇	#121	1	8	yes	—
105	C₂×C₄×C₄₇	#124	1	2	yes	—
106	C₂×C₄×C₄₇	#127	1	2	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	8	no	—
113	C₂×Dic₄₇	—	8	8	no	—
114	C₄₇⋊C₈	—	2	8	no	—
115	C₂×#1960	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	#390	1	4	yes	—
117	C₂²×C₄₇	#419	1	8	yes	—
118	C₂²×C₄₇	#491	1	8	yes	—
119	C₂²×C₄₇	#497	1	8	yes	—
120	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#137	1	4	yes	—
124	C₄×C₄₇	—	2	8	no	—
125	C₄×C₄₇	—	4	8	no	—
126	C₄×C₄₇	#390	1	4	yes	—
127	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
128	C₄×C₄₇	C₂×#121	1	2	yes	—
129	Dic₄₇	—	8	4	no	—
130	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
131	C₂²×#103	—	47	4	no	—
132	C₂²×#97	—	47	1	no	—
133	C₂×#382	—	47	1	no	—
134	#1960	—	47	8	no	—
135	C₂×C₄₇	#1960	1	8	yes	complemented
136	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#419	1	2	yes	—
138	C₂×C₄₇	C₂×#491	1	2	yes	—
139	C₂×C₄₇	C₂×#497	1	2	yes	—
140	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
141	C₂³×C₈	—	47	2	no	—
142	C₂×C₄⋊Q₈	—	2	2	no	—
143	C₂×C₄⋊Q₈	—	2	4	no	—
144	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
145	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
146	C₂²×C₄⋊C₄	—	2	2	no	—
147	C₂²×C₄⋊C₄	—	47	2	no	—
148	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#103	—	47	16	no	—
150	C₂×#103	—	94	16	no	—
151	C₂×#97	—	47	4	no	—
152	C₂×#97	—	94	4	no	—
153	#382	—	47	4	no	—
154	#382	—	94	2	no	—
155	C₂×#95	—	94	2	no	—
156	C₄₇	C₂×#1960	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×Q₈	—	2	8	no	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
162	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	—	94	2	no	—
165	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	—	47	8	no	—
168	C₂²×C₈	—	47	4	no	—
169	C₂²×C₈	—	94	4	no	—
170	C₂²×C₈	—	94	4	no	—
171	C₄⋊Q₈	—	2	16	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	complemented
174	C₂×C₄⋊C₄	—	2	16	no	—
175	C₂×C₄⋊C₄	—	2	8	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄⋊C₄	—	94	8	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
183	#103	—	94	32	no	—
184	#97	—	94	8	no	—
185	#95	—	94	8	no	—
186	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×Q₈	—	2	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	32	no	—
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	4	16	no	—
193	C₂×Q₈	—	4	4	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	—	188	8	no	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
205	C₂²×C₄	D₄×D₄₇	1	4	yes	—
206	C₂²×C₄	#10625352	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
211	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	94	16	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	94	4	no	—
216	C₄⋊C₄	—	2	32	no	—
217	C₄⋊C₄	—	94	8	no	—
218	C₄⋊C₄	—	188	4	no	—
219	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
220	C₄²	C₂²×Dic₄₇	1	4	yes	—
221	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
222	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
223	C₂³	#13094011	1	2	yes	—
224	C₂³	#13094020	1	2	yes	—
225	C₂³	#13094055	1	8	yes	—
226	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
227	Q₈	—	2	16	no	—
228	Q₈	—	4	16	no	—
229	Q₈	—	4	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	4	16	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	376	8	no	—
237	C₂×C₄	—	188	8	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
241	C₂×C₄	D₄×Dic₄₇	1	8	yes	—
242	C₂×C₄	#13093995	1	8	yes	—
243	C₂×C₄	#13094011	1	2	yes	—
244	C₂×C₄	#13094020	1	2	yes	—
245	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625352	1	2	yes	—
249	C₈	—	94	8	no	—
250	C₂²	#15689026	1	4	yes	—
251	C₂²	#15689132	1	8	yes	—
252	C₂²	#15689491	1	8	yes	—
253	C₂²	#15689515	1	8	yes	—
254	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂²	C₂×#13094011	1	1	yes	characteristic
256	C₂²	C₂×#13094020	1	1	yes	characteristic
257	C₂²	C₂×#13094055	1	4	yes	—
258	C₄	—	2	8	no	—
259	C₄	—	4	8	no	—
260	C₄	—	376	4	no	—
261	C₄	#15689026	1	4	yes	—
262	C₄	C₂×D₄×Dic₄₇	1	2	yes	—
263	C₄	C₂×#13093995	1	2	yes	—
264	C₂	#20244209	1	8	yes	direct product factor
265	C₂	C₂×#15689026	1	1	yes	characteristic
266	C₂	C₂×#15689132	1	2	yes	—
267	C₂	C₂×#15689491	1	2	yes	—
268	C₂	C₂×#15689515	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#1960
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #20244209	C₂	direct (N × H)
18. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
44. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
135. C₂×C₄₇	#1960	semidirect
144. C₂×C₄⋊Q₈	Dic₄₇	semidirect
156. C₄₇	C₂×#1960	semidirect
173. C₄⋊Q₈	C₂×Dic₄₇	semidirect
264. C₂	#20244209	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×#13093948	2
C₂²×#13093935	2
C₂×#15688989	2
#20244209	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#1960	47

Subgroups — C2×#20244209 #26219928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244209 #26219928