Subgroups — C₂×#20242284 #26186019

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689250	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20242284	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688853	C₂	1	2	yes	maximal
5	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₈₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
9	#15689250	—	2	4	no	—
10	#15689250	C₂²	1	8	yes	—
11	C₄×Dic₁₈₈	C₄	1	4	yes	—
12	C₄×Dic₁₈₈	C₂²	1	4	yes	—
13	C₂×#13093958	C₄	1	1	yes	characteristic
14	C₂×Dic₄₇⋊C₈	—	2	2	no	—
15	C₂×#13093949	C₄	1	2	yes	—
16	C₂×#13093934	—	2	2	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#15688853	C₂²	1	8	yes	—
20	C₈×C₄₇⋊C₈	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#335	C₂²	1	4	yes	—
24	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₈₈	—	2	4	no	—
26	C₂×Dic₁₈₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄×Dic₉₄	—	2	2	no	—
31	C₄×Dic₉₄	—	2	2	no	—
32	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
33	C₄×Dic₉₄	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	#13093958	C₂×C₄	1	4	yes	—
38	Dic₄₇⋊C₈	—	2	8	no	—
39	#13093949	C₂×C₄	1	8	yes	—
40	#13093934	—	2	8	no	—
41	C₄×C₄₇⋊C₈	—	2	1	no	—
42	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
43	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	2	no	—
50	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	2	no	—
57	C₂²×Dic₄₇	—	2	2	no	—
58	C₂×Dic₉₄	—	2	4	no	—
59	C₂×Dic₉₄	—	2	4	no	—
60	C₂×Dic₉₄	—	2	2	no	—
61	C₂×Dic₉₄	C₂²×C₄	1	4	yes	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	4	4	no	—
65	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	#41	1	1	yes	characteristic
71	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊Dic₄₇	—	2	2	no	—
73	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
74	Dic₄₇⋊C₄	—	2	4	no	—
75	Dic₄₇⋊C₄	—	2	4	no	—
76	C₄×Dic₄₇	—	2	4	no	—
77	C₄×Dic₄₇	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	complemented
82	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
83	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
84	Dic₁₈₈	—	2	8	no	—
85	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	#131	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₄₇	#135	1	2	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
97	C₂×Dic₄₇	—	2	4	no	—
98	C₂×Dic₄₇	—	2	4	no	—
99	C₂×Dic₄₇	—	4	4	no	—
100	C₂×Dic₄₇	—	4	4	no	—
101	Dic₉₄	—	2	4	no	—
102	Dic₉₄	—	2	4	no	—
103	Dic₉₄	—	4	4	no	—
104	C₈×C₄₇	—	2	2	no	—
105	C₈×C₄₇	—	4	4	no	—
106	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
107	C₄₇⋊C₈	—	2	4	no	—
108	C₄₇⋊C₈	—	2	2	no	—
109	C₄₇⋊C₈	C₂×Q₁₆	1	4	yes	complemented
110	C₂×#1591	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	#436	1	2	yes	—
112	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
113	C₂²×C₄₇	#443	1	2	yes	—
114	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	#436	1	2	yes	—
118	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
119	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
120	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
121	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
123	Dic₄₇	—	4	4	no	—
124	Dic₄₇	—	4	4	no	—
125	C₂×#446	—	47	2	no	—
126	C₂×C₄×Q₁₆	—	47	1	no	—
127	#1591	—	47	8	no	—
128	C₂×#335	D₄₇	1	1	yes	characteristic
129	C₂×Q₈⋊C₈	—	47	2	no	—
130	C₂×C₈²	—	47	1	no	—
131	C₂×C₄₇	#1591	1	4	yes	complemented
132	C₂×C₄₇	C₂×#436	1	1	yes	characteristic
133	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
135	C₂²×Q₁₆	—	47	1	no	—
136	C₂×C₄×Q₈	—	47	2	no	—
137	#446	—	47	8	no	—
138	#446	—	94	4	no	—
139	C₄×Q₁₆	—	47	4	no	—
140	C₄×Q₁₆	—	47	4	no	—
141	C₂×#107	—	47	1	no	—
142	C₂×C₄⋊C₈	—	94	2	no	—
143	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
144	C₂×#103	—	47	2	no	—
145	C₂×#97	—	94	2	no	—
146	C₂×C₄×C₈	—	47	1	no	—
147	C₂×C₄×C₈	—	47	1	no	—
148	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
149	#335	C₂×D₄₇	1	4	yes	—
150	Q₈⋊C₈	—	47	8	no	—
151	C₈²	—	47	4	no	—
152	C₄₇	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	47	2	no	—
154	C₂×Q₁₆	—	47	4	no	—
155	C₂×Q₁₆	—	94	4	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	47	1	no	—
158	C₂²×C₈	—	47	1	no	—
159	C₂²×C₈	—	94	1	no	—
160	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
162	C₄×Q₈	—	47	4	no	—
163	C₄×Q₈	—	47	4	no	—
164	C₄×Q₈	—	94	2	no	—
165	C₄×Q₈	—	94	2	no	—
166	C₂×C₄⋊C₄	—	47	2	no	—
167	C₂×C₄⋊C₄	—	94	2	no	—
168	C₂×C₄²	—	94	2	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#107	—	47	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	94	8	no	—
173	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
174	#103	—	47	8	no	—
175	#97	—	94	8	no	—
176	C₄×C₈	—	47	2	no	—
177	C₄×C₈	—	47	2	no	—
178	C₄×C₈	—	47	2	no	—
179	C₄×C₈	—	94	1	no	—
180	C₄×C₈	C₄×D₄₇	1	2	yes	—
181	C₄×C₈	C₂²×D₄₇	1	2	yes	—
182	C₂×Q₈	—	47	4	no	—
183	C₂×Q₈	—	94	2	no	—
184	C₂×Q₈	—	94	4	no	—
185	C₂×Q₈	—	94	4	no	—
186	C₂²×C₄	—	94	2	no	—
187	C₂²×C₄	—	94	2	no	—
188	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	#10625354	1	1	yes	characteristic
191	Q₁₆	—	94	8	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	47	4	no	—
196	C₂×C₈	—	47	2	no	—
197	C₂×C₈	—	47	2	no	—
198	C₂×C₈	—	94	4	no	—
199	C₂×C₈	—	94	2	no	—
200	C₂×C₈	—	94	2	no	—
201	C₂×C₈	#10625324	1	4	yes	—
202	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
204	C₄⋊C₄	—	47	4	no	—
205	C₄⋊C₄	—	94	4	no	—
206	C₄⋊C₄	—	94	2	no	—
207	C₄⋊C₄	—	94	4	no	—
208	C₄²	—	94	4	no	—
209	C₄²	—	94	4	no	—
210	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₄²	D₄×D₄₇	1	1	yes	characteristic
212	C₄²	#10625354	1	1	yes	characteristic
213	C₄²	C₂³×D₄₇	1	1	yes	characteristic
214	C₂³	#13093986	1	1	yes	characteristic
215	Q₈	—	94	4	no	—
216	Q₈	—	94	4	no	—
217	Q₈	—	188	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	94	4	no	—
220	C₂×C₄	—	94	4	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	4	no	—
223	C₂×C₄	#13093986	1	1	yes	characteristic
224	C₂×C₄	#13093986	1	1	yes	characteristic
225	C₂×C₄	#13093986	1	1	yes	characteristic
226	C₂×C₄	C₂×#10625324	1	2	yes	—
227	C₂×C₄	#13094032	1	2	yes	—
228	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
229	C₂×C₄	#13094043	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	—	47	4	no	—
236	C₈	—	94	4	no	—
237	C₈	—	94	2	no	—
238	C₈	C₂×#10625324	1	4	yes	—
239	C₂²	#15689200	1	2	yes	—
240	C₂²	#15689217	1	2	yes	—
241	C₂²	#15689239	1	2	yes	—
242	C₂²	C₂×#13093986	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	188	4	no	—
246	C₄	—	188	4	no	—
247	C₄	#15689200	1	2	yes	—
248	C₄	#15689217	1	2	yes	—
249	C₄	C₂²×#10625324	1	1	yes	characteristic
250	C₄	C₂×#13094032	1	1	yes	characteristic
251	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
252	C₄	C₂×#13094043	1	1	yes	characteristic
253	C₂	#20242284	1	4	yes	direct product factor
254	C₂	C₂×#15689200	1	1	yes	characteristic
255	C₂	C₂×#15689217	1	1	yes	characteristic
256	C₂	C₂×#15689239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	C₂×#1591
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20242284	C₂	direct (N × H)
81. C₂×C₄₇⋊C₈	Q₁₆	semidirect
109. C₄₇⋊C₈	C₂×Q₁₆	semidirect
131. C₂×C₄₇	#1591	semidirect
152. C₄₇	C₂×#1591	semidirect
253. C₂	#20242284	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×#15689250	2
C₂×C₄×Dic₁₈₈	2
#20242284	2
C₂×#15688853	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×#335	2
C₂×#1591	47

Subgroups — C2×#20242284 #26186019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242284 #26186019