Subgroups — C₂×#20243524 #26216502

Order 12032.

870 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	C₂×#15689146	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243524	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688887	C₂	1	2	yes	maximal
6	C₂×C₄₇×#341	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15689146	—	2	4	no	—
11	#15689146	C₂²	1	8	yes	—
12	C₂×Dic₄₇⋊C₈	—	2	4	no	—
13	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	#15688888	C₂²	1	4	yes	—
18	#15688887	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#341	C₂²	1	4	yes	—
22	C₂³×Dic₄₇	Q₈	1	2	yes	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂³	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	Dic₄₇⋊C₈	—	2	16	no	—
40	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
41	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	4	no	—
45	C₂×M₁₆×C₄₇	—	2	4	no	—
46	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₄₇	—	2	4	no	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	#41	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
63	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	4	4	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	4	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
77	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
80	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	#125	1	4	yes	—
94	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#117	1	2	yes	—
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	#117	1	2	yes	—
100	C₂×C₄×C₄₇	#117	1	2	yes	—
101	C₂×C₄×C₄₇	#124	1	4	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#125	1	4	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	#108	1	8	yes	—
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	C₈×C₄₇	—	4	4	no	—
116	C₄₇⋊C₈	—	2	4	no	—
117	C₄₇⋊C₈	—	4	4	no	—
118	C₂×#1837	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	8	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	#382	1	2	yes	—
123	C₂²×C₄₇	#382	1	2	yes	—
124	C₂²×C₄₇	C₂×#108	1	4	yes	—
125	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#125	1	2	yes	—
127	C₄×C₄₇	#382	1	2	yes	—
128	C₄×C₄₇	#382	1	2	yes	—
129	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#124	1	2	yes	—
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	C₂×#108	1	8	yes	—
134	C₂×#423	—	47	2	no	—
135	C₂²×C₄×C₈	—	47	1	no	—
136	#1837	—	47	8	no	—
137	C₂×#341	—	47	2	no	—
138	C₂×#341	—	47	1	no	—
139	C₂×#341	D₄₇	1	1	yes	characteristic
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	—	2	2	no	—
142	C₂×C₄₇	#1837	1	4	yes	complemented
143	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
144	C₂×C₄₇	C₂²×#108	1	2	yes	—
145	C₂²×M₁₆	—	47	2	no	—
146	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
147	C₂³×C₈	—	47	1	no	—
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂²×C₄²	—	47	1	no	—
150	#423	—	47	8	no	—
151	#423	—	94	4	no	—
152	C₂×C₄⋊C₈	—	94	4	no	—
153	C₂×C₄×C₈	—	47	2	no	—
154	C₂×C₄×C₈	—	47	4	no	—
155	C₂×C₄×C₈	—	47	2	no	—
156	C₂×C₄×C₈	—	47	4	no	—
157	#341	—	47	8	no	—
158	#341	—	47	4	no	—
159	#341	C₂×D₄₇	1	4	yes	—
160	C₄₇	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	47	2	no	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	47	4	no	—
167	C₂×M₁₆	—	94	2	no	—
168	C₂×M₁₆	—	94	4	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	47	2	no	—
174	C₂²×C₈	—	47	2	no	—
175	C₂²×C₈	—	47	2	no	—
176	C₂²×C₈	—	94	4	no	—
177	C₂²×C₈	—	94	2	no	—
178	C₂²×C₈	C₄×D₄₇	1	2	yes	—
179	C₂²×C₈	C₄×D₄₇	1	2	yes	—
180	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
181	C₂×C₄²	—	47	1	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₄⋊C₈	—	94	16	no	—
189	C₄×C₈	—	47	8	no	—
190	C₄×C₈	—	47	8	no	—
191	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	47	4	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	94	8	no	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	#10625350	1	4	yes	—
200	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	#10625352	1	4	yes	—
203	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	4	no	—
208	M₁₆	—	94	4	no	—
209	M₁₆	—	94	4	no	—
210	M₁₆	—	188	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	47	4	no	—
217	C₂×C₈	—	94	4	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	—	188	4	no	—
222	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
223	C₄²	—	47	2	no	—
224	C₄²	—	47	4	no	—
225	C₄²	—	47	2	no	—
226	C₄²	—	47	2	no	—
227	C₄²	—	47	2	no	—
228	C₄²	—	47	4	no	—
229	C₂³	—	2	4	no	—
230	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
231	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
232	C₂³	#13094014	1	4	yes	—
233	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
234	C₂×C₄	—	47	4	no	—
235	C₂×C₄	—	47	8	no	—
236	C₂×C₄	—	47	4	no	—
237	C₂×C₄	—	94	8	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	—	94	4	no	—
240	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄	#13093980	1	2	yes	—
243	C₂×C₄	#13093980	1	2	yes	—
244	C₂×C₄	#13093981	1	2	yes	—
245	C₂×C₄	#13093981	1	2	yes	—
246	C₂×C₄	#13094010	1	4	yes	—
247	C₂×C₄	#13094010	1	4	yes	—
248	C₂×C₄	#13094014	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625352	1	2	yes	—
252	C₂×C₄	C₂×#10625350	1	2	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	4	4	no	—
255	C₈	—	94	4	no	—
256	C₈	—	188	4	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	#15688987	1	2	yes	—
261	C₂²	#15688987	1	2	yes	—
262	C₂²	#108×D₄₇	1	4	yes	—
263	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
264	C₂²	C₂×#13094014	1	2	yes	—
265	C₄	—	47	8	no	—
266	C₄	—	94	4	no	—
267	C₄	#15688987	1	2	yes	—
268	C₄	#15688987	1	2	yes	—
269	C₄	C₂×#13093980	1	1	yes	characteristic
270	C₄	C₂×#13093981	1	1	yes	characteristic
271	C₄	C₂×#13094010	1	2	yes	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	#20243524	1	4	yes	direct product factor
275	C₂	C₂×#15688987	1	1	yes	characteristic
276	C₂	C₂×#108×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#341	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#341	D₄₇
hypocenter	C₄₇	C₂×#1837
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
3. #20243524	C₂	direct (N × H)
142. C₂×C₄₇	#1837	semidirect
160. C₄₇	C₂×#1837	semidirect
274. C₂	#20243524	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×#15689146	2
C₂²×C₈×Dic₄₇	2
#20243524	2
C₂×#15688888	2
C₂×#15688887	2
C₂×C₄₇×#341	2
C₂×#1837	47

Subgroups — C2×#20243524 #26216502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243524 #26216502