Subgroups — C₂×#20243527 #26216506

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 260 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×#13093934	C₂	1	1	yes	maximal, characteristic
3	#20243527	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×#13093934	C₄	1	4	yes	—
14	C₂×#13093934	C₄	1	4	yes	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	#15688888	C₂²	1	4	yes	—
17	#15688887	C₂²	1	8	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#341	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	Q₈	1	2	yes	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	C₂³	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	Dic₄₇⋊C₈	—	2	16	no	—
38	#13093934	C₂×C₄	1	16	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	—	2	4	no	—
43	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
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	—	4	4	no	—
59	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
60	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
62	M₁₆×C₄₇	—	2	4	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	4	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	4	4	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
71	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
72	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
74	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
75	#10625329	—	2	4	no	—
76	#10625329	—	2	4	no	—
77	#10625329	—	4	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	4	4	no	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₇	#125	1	4	yes	—
88	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#117	1	2	yes	—
92	C₂×C₄×C₄₇	#117	1	2	yes	—
93	C₂×C₄×C₄₇	#117	1	2	yes	—
94	C₂×C₄×C₄₇	#117	1	2	yes	—
95	C₂×C₄×C₄₇	#124	1	4	yes	—
96	C₂×C₄×C₄₇	#124	1	4	yes	—
97	C₂×C₄×C₄₇	#125	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
102	C₂×Dic₄₇	—	2	8	no	—
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
106	C₈×C₄₇	—	2	4	no	—
107	C₈×C₄₇	—	4	4	no	—
108	C₄₇⋊C₈	—	2	4	no	—
109	C₄₇⋊C₈	—	4	4	no	—
110	C₂×#1838	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	#382	1	2	yes	—
115	C₂²×C₄₇	#382	1	2	yes	—
116	C₂²×C₄₇	#430	1	4	yes	—
117	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#125	1	2	yes	—
119	C₄×C₄₇	#382	1	2	yes	—
120	C₄×C₄₇	#382	1	2	yes	—
121	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
123	C₄×C₄₇	C₂×#124	1	2	yes	—
124	Dic₄₇	—	2	8	no	—
125	C₂×#423	—	47	2	no	—
126	C₂²×#97	—	47	1	no	—
127	#1838	—	47	8	no	—
128	C₂×#341	—	47	2	no	—
129	C₂×#341	—	47	1	no	—
130	C₂×#341	D₄₇	1	1	yes	characteristic
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	#1838	1	4	yes	complemented
134	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
135	C₂×C₄₇	C₂×#430	1	2	yes	—
136	C₂²×M₁₆	—	47	2	no	—
137	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
138	C₂³×C₈	—	47	1	no	—
139	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂²×C₄²	—	47	1	no	—
141	#423	—	47	8	no	—
142	#423	—	94	4	no	—
143	C₂×C₄⋊C₈	—	94	4	no	—
144	C₂×#97	—	47	4	no	—
145	C₂×#97	—	47	4	no	—
146	C₂×#97	—	47	4	no	—
147	#341	—	47	8	no	—
148	#341	—	47	4	no	—
149	#341	C₂×D₄₇	1	4	yes	—
150	C₄₇	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	—	47	2	no	—
152	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	47	4	no	—
157	C₂×M₁₆	—	94	2	no	—
158	C₂×M₁₆	—	94	4	no	—
159	C₂×M₁₆	—	94	4	no	—
160	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	47	2	no	—
164	C₂²×C₈	—	47	2	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	—	94	4	no	—
167	C₂²×C₈	—	94	2	no	—
168	C₂²×C₈	C₄×D₄₇	1	2	yes	—
169	C₂²×C₈	C₄×D₄₇	1	2	yes	—
170	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
171	C₂×C₄²	—	47	2	no	—
172	C₂×C₄²	—	47	2	no	—
173	C₂×C₄²	—	47	2	no	—
174	C₂×C₄²	—	47	2	no	—
175	C₂×C₄²	—	47	2	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₄⋊C₈	—	94	16	no	—
178	#97	—	47	16	no	—
179	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	—	47	4	no	—
181	C₂²×C₄	—	47	4	no	—
182	C₂²×C₄	—	47	4	no	—
183	C₂²×C₄	—	188	4	no	—
184	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₄	#10625350	1	4	yes	—
187	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	#10625352	1	4	yes	—
190	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	4	4	no	—
195	M₁₆	—	94	4	no	—
196	M₁₆	—	94	4	no	—
197	M₁₆	—	188	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	94	4	no	—
205	C₂×C₈	—	94	4	no	—
206	C₂×C₈	—	94	4	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	188	4	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
210	C₄²	—	47	4	no	—
211	C₄²	—	47	4	no	—
212	C₄²	—	47	4	no	—
213	C₄²	—	47	4	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
217	C₂³	#13094014	1	4	yes	—
218	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
219	C₂×C₄	—	47	8	no	—
220	C₂×C₄	—	94	8	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	4	no	—
223	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₄	#13093980	1	2	yes	—
226	C₂×C₄	#13093980	1	2	yes	—
227	C₂×C₄	#13093981	1	2	yes	—
228	C₂×C₄	#13093981	1	2	yes	—
229	C₂×C₄	#13094010	1	4	yes	—
230	C₂×C₄	#13094010	1	4	yes	—
231	C₂×C₄	#13094014	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
234	C₂×C₄	C₂×#10625352	1	2	yes	—
235	C₂×C₄	C₂×#10625350	1	2	yes	—
236	C₈	—	2	4	no	—
237	C₈	—	4	4	no	—
238	C₈	—	94	4	no	—
239	C₈	—	188	4	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	#15688987	1	2	yes	—
244	C₂²	#15688987	1	2	yes	—
245	C₂²	#15689170	1	4	yes	—
246	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
247	C₂²	C₂×#13094014	1	2	yes	—
248	C₄	—	94	8	no	—
249	C₄	#15688987	1	2	yes	—
250	C₄	#15688987	1	2	yes	—
251	C₄	C₂×#13093980	1	1	yes	characteristic
252	C₄	C₂×#13093981	1	1	yes	characteristic
253	C₄	C₂×#13094010	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	#20243527	1	4	yes	direct product factor
257	C₂	C₂×#15688987	1	1	yes	characteristic
258	C₂	C₂×#15689170	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×#1838
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. #20243527	C₂	direct (N × H)
133. C₂×C₄₇	#1838	semidirect
150. C₄₇	C₂×#1838	semidirect
256. C₂	#20243527	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×#13093934	2
#20243527	2
C₂×#15688888	2
C₂×#15688887	2
C₂×C₄₇×#341	2
C₂×#1838	47

Subgroups — C2×#20243527 #26216506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243527 #26216506