Subgroups — C₂×#20243550 #26216683

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093957	C₂	1	2	yes	maximal
2	C₂²×#13093955	C₂	1	2	yes	maximal
3	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
5	#20243550	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×#13093957	—	2	8	no	—
11	C₂×#13093957	C₂²	1	8	yes	—
12	C₂×#13093955	—	2	8	no	—
13	C₂×#13093955	C₂²	1	8	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
16	#15688989	—	2	2	no	—
17	#15688989	C₂²	1	4	yes	—
18	C₂×C₄₇⋊#95	—	2	2	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#382	—	2	2	no	—
22	C₄₇×#382	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	—	2	2	no	—
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
29	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
30	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	#13093957	—	2	16	no	—
38	#13093955	—	2	16	no	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₄₇⋊#95	—	2	8	no	—
41	C₂³×C₄×C₄₇	—	2	2	no	—
42	C₂³×C₄×C₄₇	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	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	Q₈	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#95×C₄₇	—	2	8	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	4	8	no	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	4	4	no	—
70	C₂²×Dic₄₇	—	4	8	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
76	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
81	C₂×C₄₇⋊C₈	C₄⋊C₄	1	16	yes	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
85	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#127	1	2	yes	—
90	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
91	C₂³×C₄₇	#135	1	8	yes	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	16	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	—	8	8	no	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	#127	1	2	yes	—
103	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	8	no	—
114	C₂×Dic₄₇	—	8	8	no	—
115	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
116	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×#1846	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	#384	1	4	yes	—
119	C₂²×C₄₇	#427	1	8	yes	—
120	C₂²×C₄₇	#495	1	8	yes	—
121	C₂²×C₄₇	#499	1	8	yes	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#135	1	4	yes	—
126	C₄×C₄₇	—	2	8	no	—
127	C₄×C₄₇	—	8	4	no	—
128	C₄×C₄₇	#384	1	4	yes	—
129	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	Dic₄₇	—	8	4	no	—
134	C₂²×#107	—	47	2	no	—
135	C₂²×#106	—	47	2	no	—
136	C₂²×C₄×C₈	—	47	1	no	—
137	C₂×#382	—	47	1	no	—
138	C₂×#382	D₄₇	1	1	yes	characteristic
139	#1846	—	47	8	no	—
140	C₂×C₄₇	#1846	1	8	yes	complemented
141	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
142	C₂×C₄₇	C₂×#427	1	2	yes	—
143	C₂×C₄₇	C₂×#495	1	2	yes	—
144	C₂×C₄₇	C₂×#499	1	2	yes	—
145	C₂³×C₈	—	47	1	no	—
146	C₂³×C₈	—	47	1	no	—
147	C₂²×C₄⋊C₄	—	47	2	no	—
148	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
149	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	C₂×#107	—	47	8	no	—
151	C₂×#107	—	94	8	no	—
152	C₂×#106	—	47	8	no	—
153	C₂×#106	—	94	8	no	—
154	C₂×C₄×C₈	—	47	8	no	—
155	C₂×C₄×C₈	—	47	4	no	—
156	#382	—	2	2	no	—
157	#382	—	47	4	no	—
158	#382	—	94	2	no	—
159	#382	C₂×D₄₇	1	4	yes	—
160	C₂×#95	—	2	2	no	—
161	C₂×#95	—	94	2	no	—
162	C₄₇	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
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	4	no	—
168	C₂²×C₈	—	47	4	no	—
169	C₂²×C₈	—	47	4	no	—
170	C₂²×C₈	—	47	4	no	—
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	—	47	2	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	47	4	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	—	94	4	no	—
181	C₂×C₄⋊C₄	—	94	2	no	—
182	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
183	C₂×C₄²	Dic₉₄	1	4	yes	—
184	C₂×C₄²	C₄×D₄₇	1	4	yes	—
185	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
187	#107	—	94	16	no	—
188	#106	—	94	16	no	—
189	C₄×C₈	—	47	16	no	—
190	#95	—	2	8	no	—
191	#95	—	94	8	no	—
192	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	94	4	no	—
198	C₂²×C₄	—	188	4	no	—
199	C₂²×C₄	—	188	8	no	—
200	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
201	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
203	C₂²×C₄	D₄×D₄₇	1	2	yes	—
204	C₂²×C₄	D₄×D₄₇	1	2	yes	—
205	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
206	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	C₂×C₈	—	47	16	no	—
211	C₂×C₈	—	47	16	no	—
212	C₂×C₈	—	47	4	no	—
213	C₂×C₈	—	47	4	no	—
214	C₂×C₈	—	47	4	no	—
215	C₂×C₈	—	47	4	no	—
216	C₂×C₈	—	47	4	no	—
217	C₂×C₈	—	47	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	94	4	no	—
222	C₄⋊C₄	—	94	4	no	—
223	C₄⋊C₄	—	188	4	no	—
224	C₄²	Dic₄₇⋊C₄	1	8	yes	—
225	C₄²	C₂×Dic₉₄	1	4	yes	—
226	C₄²	C₂×C₄×D₄₇	1	4	yes	—
227	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
228	C₂³	#13094020	1	2	yes	—
229	C₂³	#13094022	1	2	yes	—
230	C₂³	#13094044	1	8	yes	—
231	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	8	8	no	—
237	C₂×C₄	—	376	8	no	—
238	C₂×C₄	—	188	8	no	—
239	C₂×C₄	—	188	4	no	—
240	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
241	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
242	C₂×C₄	C₂×Dic₄₇⋊C₄	1	8	yes	—
243	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
244	C₂×C₄	#13094020	1	2	yes	—
245	C₂×C₄	#13094022	1	2	yes	—
246	C₂×C₄	#13094023	1	4	yes	—
247	C₂×C₄	#13094023	1	4	yes	—
248	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
254	C₈	—	47	8	no	—
255	C₈	—	47	8	no	—
256	C₂²	#15688992	1	4	yes	—
257	C₂²	#15689159	1	8	yes	—
258	C₂²	#15689505	1	8	yes	—
259	C₂²	#15689522	1	8	yes	—
260	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
261	C₂²	C₂×#13094020	1	1	yes	characteristic
262	C₂²	C₂×#13094022	1	1	yes	characteristic
263	C₂²	C₂×#13094044	1	4	yes	—
264	C₄	—	2	8	no	—
265	C₄	—	8	4	no	—
266	C₄	—	376	4	no	—
267	C₄	#15688992	1	4	yes	—
268	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
269	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
270	C₄	C₂×#13094023	1	1	yes	characteristic
271	C₄	C₂×#13094023	1	1	yes	characteristic
272	C₂	#20243550	1	8	yes	direct product factor
273	C₂	C₂×#15688992	1	1	yes	characteristic
274	C₂	C₂×#15689159	1	2	yes	—
275	C₂	C₂×#15689505	1	2	yes	—
276	C₂	C₂×#15689522	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×#1846
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
5. #20243550	C₂	direct (N × H)
14. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
39. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
140. C₂×C₄₇	#1846	semidirect
162. C₄₇	C₂×#1846	semidirect
272. C₂	#20243550	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×#13093957	2
C₂²×#13093955	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688989	2
#20243550	2
C₂×C₄₇×#382	2
C₂×#1846	47

Subgroups — C2×#20243550 #26216683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243550 #26216683