Subgroups — C₂×#20241338 #26125462

Order 12032.

410 conjugacy classes of subgroups.

The subgroup lattice has 306 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	1	yes	maximal, characteristic
2	C₂×#15688905	C₂	1	1	yes	maximal, characteristic
3	#20241338	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#349	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
7	C₂×#13093964	—	2	1	no	—
8	#15689146	—	2	1	no	—
9	#15689146	—	2	1	no	—
10	#15689146	C₄	1	2	yes	—
11	#15689146	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₈	—	2	1	no	—
13	C₂×Dic₄₇⋊C₈	—	2	1	no	—
14	#15688905	—	2	1	no	—
15	#15688905	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₄₇	—	2	1	no	—
18	C₄₇×#349	—	2	1	no	—
19	C₄₇×#349	C₂²	1	2	yes	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂×#10625329	—	2	1	no	—
22	C₂×#10625329	—	4	1	no	—
23	C₂×#10625329	—	4	1	no	—
24	C₂×#10625329	C₂×C₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂×C₄×Dic₄₇	—	2	1	no	—
27	C₂×C₄×Dic₄₇	—	2	1	no	—
28	C₂×C₄×Dic₄₇	—	2	1	no	—
29	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	#13093964	—	2	2	no	—
34	#13093964	—	4	1	no	—
35	C₂×C₄₇⋊C₁₆	—	4	1	no	—
36	Dic₄₇⋊C₈	—	2	2	no	—
37	Dic₄₇⋊C₈	—	2	2	no	—
38	Dic₄₇⋊C₈	—	2	2	no	—
39	Dic₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	4	1	no	—
42	C₂×M₁₆×C₄₇	—	4	1	no	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
45	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	M₃₂×C₄₇	—	2	2	no	—
50	M₃₂×C₄₇	—	4	1	no	—
51	C₂×C₁₆×C₄₇	—	4	1	no	—
52	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	1	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	2	1	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	4	4	no	—
68	M₁₆×C₄₇	—	4	1	no	—
69	M₁₆×C₄₇	—	4	1	no	—
70	M₁₆×C₄₇	—	4	1	no	—
71	M₁₆×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₄₇	—	2	1	no	—
81	C₄×Dic₄₇	—	2	2	no	—
82	C₄×Dic₄₇	—	2	1	no	—
83	C₄×Dic₄₇	—	2	2	no	—
84	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
85	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
86	#10625329	—	4	2	no	—
87	#10625329	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	4	1	no	—
90	C₂×C₄₇⋊C₈	—	4	1	no	—
91	C₁₆×C₄₇	—	4	2	no	—
92	C₄₇⋊C₁₆	—	4	2	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₈×C₄₇	—	2	1	no	—
124	C₈×C₄₇	—	2	1	no	—
125	C₈×C₄₇	—	2	1	no	—
126	C₈×C₄₇	—	2	1	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	C₂×#1327	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	4	1	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	—	4	2	no	—
135	C₂²×C₄₇	—	4	2	no	—
136	C₂²×C₄₇	#324	1	1	yes	characteristic
137	C₂²×C₄₇	#324	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	#324	1	1	yes	characteristic
142	C₄×C₄₇	#324	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
145	Dic₄₇	—	2	4	no	—
146	Dic₄₇	—	4	2	no	—
147	C₂×#423	—	47	1	no	—
148	C₂×#349	—	47	1	no	—
149	C₂×#349	D₄₇	1	1	yes	characteristic
150	#1327	—	47	4	no	—
151	C₂×C₄₇	—	2	1	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	4	1	no	—
154	C₂×C₄₇	—	4	1	no	—
155	C₂×C₄₇	#1327	1	2	yes	complemented
156	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
157	C₂²×M₁₆	—	47	1	no	—
158	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
159	C₂²×C₄²	—	47	1	no	—
160	C₂×M₃₂	—	2	1	no	—
161	C₂×M₃₂	—	94	1	no	—
162	#423	—	47	2	no	—
163	#423	—	47	2	no	—
164	#423	—	94	1	no	—
165	#423	—	94	1	no	—
166	C₂×C₄⋊C₈	—	94	1	no	—
167	C₂×C₄⋊C₈	—	94	1	no	—
168	#349	—	2	1	no	—
169	#349	—	47	2	no	—
170	#349	—	94	1	no	—
171	#349	C₂×D₄₇	1	2	yes	—
172	C₄₇	C₂×#1327	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	94	1	no	—
174	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	4	1	no	—
176	C₂×M₁₆	—	4	1	no	—
177	C₂×M₁₆	—	47	2	no	—
178	C₂×M₁₆	—	94	1	no	—
179	C₂×M₁₆	—	188	1	no	—
180	C₂×M₁₆	—	188	1	no	—
181	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
183	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	—	94	1	no	—
186	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
187	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂×C₄²	—	47	1	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	47	1	no	—
191	C₂×C₄²	—	47	2	no	—
192	C₂×C₄²	—	94	1	no	—
193	C₂×C₄²	—	94	1	no	—
194	C₂×C₄²	—	94	1	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	4	1	no	—
197	M₃₂	—	94	2	no	—
198	M₃₂	—	188	1	no	—
199	C₂×C₁₆	—	4	1	no	—
200	C₂×C₁₆	—	188	1	no	—
201	C₄⋊C₈	—	94	2	no	—
202	C₄⋊C₈	—	94	2	no	—
203	C₄⋊C₈	—	94	2	no	—
204	C₄⋊C₈	—	94	2	no	—
205	C₂⁴	#10625324	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	94	1	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	—	94	1	no	—
213	C₂²×C₄	—	94	2	no	—
214	C₂²×C₄	—	188	4	no	—
215	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	#10625324	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	188	2	no	—
226	M₁₆	—	188	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	188	1	no	—
233	C₂×C₈	—	188	1	no	—
234	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₄²	—	47	2	no	—
239	C₄²	—	47	2	no	—
240	C₄²	—	94	1	no	—
241	C₄²	—	94	2	no	—
242	C₄²	—	94	1	no	—
243	C₄²	—	94	2	no	—
244	C₁₆	—	4	2	no	—
245	C₁₆	—	188	2	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
250	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
251	C₂³	C₂×#10625324	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	2	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
269	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
270	C₂×C₄	#13093938	1	1	yes	characteristic
271	C₂×C₄	#13093938	1	1	yes	characteristic
272	C₂×C₄	#13093942	1	1	yes	characteristic
273	C₂×C₄	#13093942	1	1	yes	characteristic
274	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	188	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	#15688848	1	1	yes	characteristic
289	C₂²	#15688848	1	1	yes	characteristic
290	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	94	4	no	—
294	C₄	—	188	2	no	—
295	C₄	#15688848	1	1	yes	characteristic
296	C₄	#15688848	1	1	yes	characteristic
297	C₄	C₂×#13093938	1	1	yes	characteristic
298	C₄	C₂×#13093942	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	4	1	no	—
303	C₂	#20241338	1	2	yes	direct product factor
304	C₂	C₂×#15688848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093938
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₄₇
Fitting	C₂×C₄₇×#349	C₂
socle	C₂²×C₄₇	#324
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂×M₁₆×C₄₇	C₂³
hypercenter	C₂×#349	D₄₇
hypocenter	C₄₇	C₂×#1327
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. #20241338	C₂	direct (N × H)
155. C₂×C₄₇	#1327	semidirect
172. C₄₇	C₂×#1327	semidirect
303. C₂	#20241338	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
C₂×#15688905	2
#20241338	2
C₂×C₄₇×#349	2
C₂×#1327	47

Subgroups — C2×#20241338 #26125462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241338 #26125462