Subgroups — C₂×#20243293 #26214378

Order 12032.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#13093934	C₂	1	1	yes	maximal, characteristic
4	#20243293	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×#13093934	C₄	1	4	yes	—
16	C₂×#13093934	C₄	1	4	yes	—
17	C₂×#13093934	C₂²	1	4	yes	—
18	C₂×C₈×Dic₄₇	—	2	2	no	—
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	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	#13093959	C₂×C₄	1	32	yes	complemented
40	#13093934	—	2	8	no	—
41	#13093934	C₂×C₄	1	16	yes	—
42	C₈×Dic₄₇	—	2	8	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	4	no	—
46	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₄₇	—	2	2	no	—
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
65	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
66	M₁₆×C₄₇	—	2	8	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
76	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
77	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
78	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
79	#10625329	—	2	8	no	—
80	#10625329	—	2	4	no	—
81	#10625329	C₂²×C₄	1	8	yes	—
82	C₂×C₄₇⋊C₈	—	2	8	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
85	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
92	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
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	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×Dic₄₇	—	2	8	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
110	C₈×C₄₇	—	2	4	no	—
111	C₈×C₄₇	—	2	8	no	—
112	C₄₇⋊C₈	—	2	8	no	—
113	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
114	C₂×#1778	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	8	no	—
118	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
119	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
120	C₂²×C₄₇	#430	1	4	yes	—
121	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
123	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
124	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
125	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
126	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
127	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
128	Dic₄₇	—	2	8	no	—
129	C₂²×#108	—	47	2	no	—
130	C₂×C₄×M₁₆	—	47	2	no	—
131	C₂²×#97	—	47	1	no	—
132	#1778	—	47	8	no	—
133	C₂×#341	—	47	1	no	—
134	C₂×#341	D₄₇	1	1	yes	characteristic
135	C₂×C₄₇	—	2	2	no	—
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	#1778	1	4	yes	complemented
138	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#430	1	2	yes	—
140	C₂²×M₁₆	—	47	2	no	—
141	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
142	C₂³×C₈	—	47	1	no	—
143	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂²×C₄²	—	47	1	no	—
145	C₂×#108	—	47	16	no	—
146	C₂×#108	—	47	8	no	—
147	C₄×M₁₆	—	47	8	no	—
148	C₄×M₁₆	—	94	4	no	—
149	C₂×#97	—	47	4	no	—
150	C₂×#97	—	47	4	no	—
151	C₂×#97	—	47	4	no	—
152	C₂×#97	—	94	2	no	—
153	C₂×C₄×C₈	—	94	2	no	—
154	#341	—	47	4	no	—
155	#341	C₂×D₄₇	1	4	yes	—
156	C₄₇	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	47	2	no	—
158	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	47	8	no	—
162	C₂×M₁₆	—	47	4	no	—
163	C₂×M₁₆	—	94	2	no	—
164	C₂×M₁₆	—	94	4	no	—
165	C₂×M₁₆	C₄×D₄₇	1	8	yes	—
166	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	—	47	8	no	—
172	C₂²×C₈	—	47	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₂²×C₈	C₄×D₄₇	1	2	yes	—
175	C₂²×C₈	C₄×D₄₇	1	2	yes	—
176	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	2	no	—
180	C₂×C₄²	—	47	2	no	—
181	C₂×C₄²	—	47	2	no	—
182	C₂×C₄²	—	47	2	no	—
183	#108	—	47	32	no	—
184	#97	—	47	16	no	—
185	#97	—	94	8	no	—
186	C₄×C₈	—	94	8	no	—
187	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	—	47	4	no	—
189	C₂²×C₄	—	47	4	no	—
190	C₂²×C₄	—	47	4	no	—
191	C₂²×C₄	—	188	4	no	—
192	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
195	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	#10625352	1	4	yes	—
198	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	47	8	no	—
203	M₁₆	—	94	8	no	—
204	M₁₆	—	94	4	no	—
205	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	47	8	no	—
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	47	4	no	—
214	C₂×C₈	—	47	8	no	—
215	C₂×C₈	—	94	8	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
218	C₄²	—	47	4	no	—
219	C₄²	—	47	4	no	—
220	C₄²	—	47	4	no	—
221	C₄²	—	47	4	no	—
222	C₂³	—	2	4	no	—
223	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
224	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
225	C₂³	#13093989	1	4	yes	—
226	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
227	C₂×C₄	—	47	8	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	—	188	4	no	—
230	C₂×C₄	—	188	4	no	—
231	C₂×C₄	C₄²×D₄₇	1	2	yes	—
232	C₂×C₄	C₄²×D₄₇	1	2	yes	—
233	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	#13093980	1	2	yes	—
236	C₂×C₄	#13093980	1	2	yes	—
237	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
238	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	4	yes	—
239	C₂×C₄	#13093989	1	4	yes	—
240	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
242	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄	C₂×#10625352	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	2	8	no	—
246	C₈	—	47	8	no	—
247	C₈	—	94	8	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	8	no	—
251	C₂²	#15688974	1	2	yes	—
252	C₂²	#15688974	1	2	yes	—
253	C₂²	#15689170	1	4	yes	—
254	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
255	C₂²	C₂×#13093989	1	2	yes	—
256	C₄	—	94	8	no	—
257	C₄	#15688974	1	2	yes	—
258	C₄	#15688974	1	2	yes	—
259	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
260	C₄	C₂×#13093980	1	1	yes	characteristic
261	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	#20243293	1	4	yes	direct product factor
265	C₂	C₂×#15688974	1	1	yes	characteristic
266	C₂	C₂×#15689170	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×#1778
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. #20243293	C₂	direct (N × H)
10. C₂×#13093959	C₄	semidirect
39. #13093959	C₂×C₄	semidirect
137. C₂×C₄₇	#1778	semidirect
156. C₄₇	C₂×#1778	semidirect
264. C₂	#20243293	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×#13093934	2
#20243293	2
C₂×#15688888	2
C₂×C₄₇×#341	2
C₂×#1778	47

Subgroups — C2×#20243293 #26214378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243293 #26214378