Subgroups — C₂×#20243289 #26214373

Order 12032.

1086 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×#13093959	C₂	1	2	yes	maximal
2	C₂×M₁₆×Dic₄₇	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243289	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
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	C₂×#13093959	C₄	1	16	yes	complemented
11	C₂×#13093959	C₂²	1	8	yes	—
12	M₁₆×Dic₄₇	—	2	4	no	—
13	M₁₆×Dic₄₇	C₂²	1	8	yes	—
14	C₂×#13093934	—	2	2	no	—
15	C₂×C₈×Dic₄₇	—	2	2	no	—
16	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
17	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
19	#15688888	C₂²	1	4	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#341	C₂²	1	4	yes	—
23	C₂³×Dic₄₇	Q₈	1	2	yes	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂×C₄	1	8	yes	—
27	C₂×#10625329	C₂³	1	4	yes	—
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	C₂×C₄	1	8	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	#13093959	C₂×C₄	1	32	yes	complemented
41	#13093934	—	2	8	no	—
42	C₈×Dic₄₇	—	2	8	no	—
43	C₈×Dic₄₇	C₂×C₄	1	16	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	4	no	—
47	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
48	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
68	M₁₆×C₄₇	—	2	8	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
82	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
83	#10625329	—	2	8	no	—
84	#10625329	—	2	4	no	—
85	#10625329	C₂²×C₄	1	8	yes	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
89	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
96	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	#117	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	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	C₂³×C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	#108	1	8	yes	complemented
114	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₈×C₄₇	—	2	4	no	—
117	C₈×C₄₇	—	2	8	no	—
118	C₄₇⋊C₈	—	2	8	no	—
119	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
120	C₂×C₄×#108	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
126	C₂²×C₄₇	C₂×#108	1	4	yes	—
127	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
129	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
130	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
131	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
134	Dic₄₇	—	2	4	no	—
135	Dic₄₇	C₂×#108	1	8	yes	complemented
136	C₂²×#108	—	47	2	no	—
137	C₂×C₄×M₁₆	—	47	2	no	—
138	C₂²×C₄×C₈	—	47	1	no	—
139	C₄×#108	—	47	8	no	—
140	C₂×#341	—	47	1	no	—
141	C₂×#341	D₄₇	1	1	yes	characteristic
142	C₂×C₄₇	—	2	2	no	—
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	C₄×#108	1	4	yes	complemented
145	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄₇	C₂²×#108	1	2	yes	—
147	C₂²×M₁₆	—	47	2	no	—
148	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
149	C₂³×C₈	—	47	1	no	—
150	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂²×C₄²	—	47	1	no	—
152	C₂×#108	—	47	16	no	—
153	C₂×#108	—	47	8	no	—
154	C₄×M₁₆	—	47	8	no	—
155	C₄×M₁₆	—	94	4	no	—
156	C₂×#97	—	94	2	no	—
157	C₂×C₄×C₈	—	47	4	no	—
158	C₂×C₄×C₈	—	47	4	no	—
159	C₂×C₄×C₈	—	47	4	no	—
160	C₂×C₄×C₈	—	94	2	no	—
161	#341	—	47	4	no	—
162	#341	C₂×D₄₇	1	4	yes	—
163	C₄₇	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	—	47	2	no	—
165	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	47	8	no	—
169	C₂×M₁₆	—	47	4	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	4	no	—
172	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
173	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	47	2	no	—
177	C₂²×C₈	—	47	2	no	—
178	C₂²×C₈	—	47	8	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	C₄×D₄₇	1	2	yes	—
182	C₂²×C₈	C₄×D₄₇	1	2	yes	—
183	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
184	C₂×C₄²	—	47	2	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	1	no	—
188	C₂×C₄²	—	47	1	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	47	2	no	—
191	#108	—	47	32	no	—
192	#97	—	94	8	no	—
193	C₄×C₈	—	47	16	no	—
194	C₄×C₈	—	94	8	no	—
195	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	—	47	2	no	—
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	94	8	no	—
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
204	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	#10625352	1	4	yes	—
207	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
209	M₁₆	—	2	8	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	47	8	no	—
212	M₁₆	—	94	8	no	—
213	M₁₆	—	94	4	no	—
214	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	47	8	no	—
221	C₂×C₈	—	47	8	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	8	no	—
224	C₂×C₈	—	94	8	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
227	C₄²	—	47	2	no	—
228	C₄²	—	47	4	no	—
229	C₄²	—	47	2	no	—
230	C₄²	—	47	4	no	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
235	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
236	C₂³	#13093989	1	4	yes	—
237	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
238	C₂×C₄	—	47	4	no	—
239	C₂×C₄	—	47	8	no	—
240	C₂×C₄	—	47	4	no	—
241	C₂×C₄	—	94	8	no	—
242	C₂×C₄	—	94	8	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	C₄²×D₄₇	1	2	yes	—
245	C₂×C₄	C₄²×D₄₇	1	2	yes	—
246	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	#13093980	1	2	yes	—
249	C₂×C₄	#13093980	1	2	yes	—
250	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
251	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
252	C₂×C₄	#13093989	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
255	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂×#10625352	1	2	yes	—
257	C₈	—	2	4	no	—
258	C₈	—	2	8	no	—
259	C₈	—	47	8	no	—
260	C₈	—	94	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	#15688974	1	2	yes	—
265	C₂²	#15688974	1	2	yes	—
266	C₂²	#108×D₄₇	1	4	yes	—
267	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
268	C₂²	C₂×#13093989	1	2	yes	—
269	C₄	—	47	8	no	—
270	C₄	—	94	4	no	—
271	C₄	#15688974	1	2	yes	—
272	C₄	#15688974	1	2	yes	—
273	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
274	C₄	C₂×#13093980	1	1	yes	characteristic
275	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
276	C₂	—	2	2	no	—
277	C₂	—	2	2	no	—
278	C₂	#20243289	1	4	yes	direct product factor
279	C₂	C₂×#15688974	1	1	yes	characteristic
280	C₂	C₂×#108×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
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×C₄×#108
LCS	G ▷ 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
4. #20243289	C₂	direct (N × H)
10. C₂×#13093959	C₄	semidirect
40. #13093959	C₂×C₄	semidirect
113. C₂×Dic₄₇	#108	semidirect
135. Dic₄₇	C₂×#108	semidirect
144. C₂×C₄₇	C₄×#108	semidirect
163. C₄₇	C₂×C₄×#108	semidirect
278. C₂	#20243289	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×#13093959	2
C₂×M₁₆×Dic₄₇	2
C₂²×C₈×Dic₄₇	2
#20243289	2
C₂×#15688888	2
C₂×C₄₇×#341	2
C₂×C₄×#108	47

Subgroups — C2×#20243289 #26214373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243289 #26214373