Subgroups — C₂×#20243517 #26216464

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688988	C₂	1	2	yes	maximal
3	#20243517	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688885	C₂	1	2	yes	maximal
6	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
11	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
12	#15688988	—	2	4	no	—
13	#15688988	C₂²	1	8	yes	—
14	C₂×C₄₇⋊#95	—	2	2	no	—
15	C₂×C₄₇⋊#95	—	2	2	no	—
16	#15688886	C₂²	1	4	yes	—
17	#15688885	C₂²	1	8	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#340	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
23	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
35	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
36	C₄₇⋊#95	—	2	8	no	—
37	C₄₇⋊#95	—	2	8	no	—
38	C₂³×C₄×C₄₇	—	2	2	no	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
47	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	4	4	no	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₄₇	#41	1	4	yes	—
56	C₂²×C₄×C₄₇	#41	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	2	no	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	2	2	no	—
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₂²×Dic₄₇	—	4	4	no	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
68	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	4	4	no	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	2	4	no	—
77	C₄⋊Dic₄₇	—	4	4	no	—
78	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
79	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
80	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₇	#124	1	4	yes	—
86	C₂³×C₄₇	#135	1	2	yes	—
87	C₂³×C₄₇	#135	1	4	yes	—
88	C₂³×C₄₇	#135	1	2	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	4	no	—
93	C₂×C₄×C₄₇	—	8	8	no	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	#117	1	4	yes	—
96	C₂×C₄×C₄₇	#117	1	4	yes	—
97	C₂×C₄×C₄₇	#124	1	4	yes	—
98	C₂×C₄×C₄₇	#125	1	8	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	2	8	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	8	8	no	—
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₈×C₄₇	—	2	4	no	—
115	C₄₇⋊C₈	—	2	4	no	—
116	C₂×#1835	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	#382	1	4	yes	—
118	C₂²×C₄₇	#427	1	4	yes	—
119	C₂²×C₄₇	#427	1	4	yes	—
120	C₂²×C₄₇	#488	1	8	yes	—
121	C₂²×C₄₇	#488	1	8	yes	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#124	1	2	yes	—
124	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#135	1	2	yes	—
126	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
127	C₄×C₄₇	—	8	4	no	—
128	C₄×C₄₇	#382	1	4	yes	—
129	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#125	1	2	yes	—
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	8	4	no	—
135	C₂²×C₄×C₈	—	47	1	no	—
136	C₂×#382	—	47	2	no	—
137	#1835	—	47	8	no	—
138	C₂×#340	—	47	1	no	—
139	C₂×#340	—	47	2	no	—
140	C₂×#340	D₄₇	1	1	yes	characteristic
141	C₂×C₄₇	#1835	1	8	yes	complemented
142	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#488	1	2	yes	—
146	C₂×C₄₇	C₂×#488	1	2	yes	—
147	C₂³×C₈	—	47	1	no	—
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	—	47	2	no	—
150	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
151	C₂²×C₄²	—	47	1	no	—
152	C₂×C₄×C₈	—	47	8	no	—
153	C₂×C₄×C₈	—	47	4	no	—
154	#382	—	47	8	no	—
155	#382	—	94	4	no	—
156	C₂×#95	—	94	2	no	—
157	C₂×#95	—	94	2	no	—
158	#340	—	47	4	no	—
159	#340	—	47	8	no	—
160	#340	C₂×D₄₇	1	4	yes	—
161	C₄₇	C₂×#1835	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	—	47	1	no	—
164	C₂³×C₄	—	47	1	no	—
165	C₂³×C₄	—	94	2	no	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	47	4	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	94	4	no	—
171	C₂²×C₈	C₄×D₄₇	1	4	yes	—
172	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄⋊C₄	—	94	4	no	—
178	C₂×C₄⋊C₄	—	94	4	no	—
179	C₂×C₄⋊C₄	—	94	2	no	—
180	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
181	C₂×C₄²	—	47	4	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	4	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₄×C₈	—	47	16	no	—
186	#95	—	94	8	no	—
187	#95	—	94	8	no	—
188	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	47	4	no	—
194	C₂²×C₄	—	47	4	no	—
195	C₂²×C₄	—	47	2	no	—
196	C₂²×C₄	—	47	2	no	—
197	C₂²×C₄	—	94	2	no	—
198	C₂²×C₄	—	94	4	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	—	188	4	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	—	188	4	no	—
204	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
205	C₂²×C₄	#10625350	1	4	yes	—
206	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	#10625352	1	4	yes	—
208	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	47	4	no	—
213	C₂×C₈	—	94	8	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	94	4	no	—
220	C₄⋊C₄	—	94	4	no	—
221	C₄⋊C₄	—	188	4	no	—
222	C₄²	—	47	8	no	—
223	C₄²	—	47	4	no	—
224	C₄²	—	47	4	no	—
225	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
226	C₂³	#13094010	1	4	yes	—
227	C₂³	#13094043	1	2	yes	—
228	C₂³	#13094045	1	4	yes	—
229	C₂³	#13094047	1	2	yes	—
230	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	8	8	no	—
235	C₂×C₄	—	47	4	no	—
236	C₂×C₄	—	47	4	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	4	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	8	no	—
241	C₂×C₄	—	94	8	no	—
242	C₂×C₄	—	376	8	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
247	C₂×C₄	#13093980	1	4	yes	—
248	C₂×C₄	#13093981	1	4	yes	—
249	C₂×C₄	#13094010	1	4	yes	—
250	C₂×C₄	#13094014	1	8	yes	—
251	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625352	1	2	yes	—
254	C₂×C₄	C₂×#10625350	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	94	4	no	—
257	C₂²	#15688987	1	4	yes	—
258	C₂²	#15689158	1	4	yes	—
259	C₂²	#15689160	1	4	yes	—
260	C₂²	#15689476	1	8	yes	—
261	C₂²	#15689479	1	8	yes	—
262	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
263	C₂²	C₂×#13094010	1	2	yes	—
264	C₂²	C₂×#13094043	1	1	yes	characteristic
265	C₂²	C₂×#13094045	1	2	yes	—
266	C₂²	C₂×#13094047	1	1	yes	characteristic
267	C₄	—	8	4	no	—
268	C₄	—	94	4	no	—
269	C₄	—	94	4	no	—
270	C₄	—	376	4	no	—
271	C₄	#15688987	1	4	yes	—
272	C₄	C₂×#13093980	1	1	yes	characteristic
273	C₄	C₂×#13093981	1	1	yes	characteristic
274	C₄	C₂×#13094014	1	2	yes	—
275	C₂	#20243517	1	8	yes	direct product factor
276	C₂	C₂×#15688987	1	1	yes	characteristic
277	C₂	C₂×#15689158	1	1	yes	characteristic
278	C₂	C₂×#15689160	1	1	yes	characteristic
279	C₂	C₂×#15689476	1	2	yes	—
280	C₂	C₂×#15689479	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	C₂×#1835
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
3. #20243517	C₂	direct (N × H)
10. C₂×C₈×Dic₄₇	C₄	semidirect
35. C₈×Dic₄₇	C₂×C₄	semidirect
141. C₂×C₄₇	#1835	semidirect
161. C₄₇	C₂×#1835	semidirect
275. C₂	#20243517	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×C₈×Dic₄₇	2
C₂×#15688988	2
#20243517	2
C₂×#15688886	2
C₂×#15688885	2
C₂×C₄₇×#340	2
C₂×#1835	47

Subgroups — C2×#20243517 #26216464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243517 #26216464