Subgroups — C₂×#20241327 #26125408

Order 12032.

466 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241327	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
8	C₂×Dic₄₇⋊C₈	—	2	1	no	—
9	C₂×Dic₄₇⋊C₈	—	2	1	no	—
10	C₂×Dic₄₇⋊C₈	—	2	1	no	—
11	C₂×Dic₄₇⋊C₈	—	2	1	no	—
12	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
15	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	—	2	1	no	—
18	C₄₇×C₂²⋊C₁₆	—	2	1	no	—
19	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
20	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
21	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	4	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
30	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄₇⋊C₁₆	—	2	2	no	—
33	C₂×C₄₇⋊C₁₆	—	4	2	no	—
34	Dic₄₇⋊C₈	—	2	2	no	—
35	Dic₄₇⋊C₈	—	2	2	no	—
36	Dic₄₇⋊C₈	—	2	2	no	—
37	Dic₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	4	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₇	—	2	2	no	—
47	C₂×C₁₆×C₄₇	—	4	2	no	—
48	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	1	no	—
58	C₂²×Dic₄₇	—	2	1	no	—
59	C₂²×Dic₄₇	—	2	2	no	—
60	C₂²×Dic₄₇	—	2	1	no	—
61	C₂²×Dic₄₇	—	2	1	no	—
62	C₂²×Dic₄₇	—	4	2	no	—
63	C₂²×Dic₄₇	C₁₆	1	4	yes	complemented
64	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
65	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	4	1	no	—
69	C₂×C₈×C₄₇	—	4	1	no	—
70	C₂×C₈×C₄₇	—	4	1	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₄×Dic₄₇	—	2	2	no	—
77	C₄×Dic₄₇	—	2	2	no	—
78	C₄×Dic₄₇	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₁₆×C₄₇	—	4	2	no	—
85	C₄₇⋊C₁₆	—	4	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
89	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
91	C₂³×C₄₇	M₃₂	1	2	yes	—
92	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#99	1	2	yes	—
103	C₂×C₄×C₄₇	#101	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	2	4	no	—
108	C₂×Dic₄₇	—	2	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	C₂×C₁₆	1	4	yes	complemented
115	C₈×C₄₇	—	2	1	no	—
116	C₈×C₄₇	—	2	1	no	—
117	C₈×C₄₇	—	2	1	no	—
118	C₈×C₄₇	—	2	1	no	—
119	C₄₇⋊C₈	—	4	2	no	—
120	C₂×#1322	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	4	no	—
122	C₂²×C₄₇	—	2	4	no	—
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	1	no	—
126	C₂²×C₄₇	—	2	1	no	—
127	C₂²×C₄₇	#324	1	2	yes	—
128	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
129	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
130	C₂²×C₄₇	#349	1	2	yes	—
131	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	#324	1	2	yes	—
137	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
139	Dic₄₇	—	2	4	no	—
140	Dic₄₇	—	4	2	no	—
141	C₂²×C₄⋊C₈	—	47	1	no	—
142	C₂×C₂²⋊C₁₆	—	47	1	no	—
143	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
144	#1322	—	47	4	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	#1322	1	4	yes	complemented
149	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
150	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#349	1	1	yes	characteristic
152	C₂³×C₈	—	47	1	no	—
153	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂²×C₄²	—	47	1	no	—
155	C₂²×C₁₆	—	2	1	no	—
156	C₂²×C₁₆	—	94	1	no	—
157	C₂×C₄⋊C₈	—	47	2	no	—
158	C₂×C₄⋊C₈	—	47	2	no	—
159	C₂×C₄⋊C₈	—	94	1	no	—
160	C₂×C₄⋊C₈	—	94	1	no	—
161	C₂×C₄⋊C₈	—	94	1	no	—
162	C₂×C₄⋊C₈	—	94	1	no	—
163	C₂²⋊C₁₆	—	2	1	no	—
164	C₂²⋊C₁₆	—	47	2	no	—
165	C₂²⋊C₁₆	—	94	1	no	—
166	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
167	C₄₇	C₂×#1322	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	—	47	1	no	—
169	C₂³×C₄	—	47	1	no	—
170	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	—	4	2	no	—
172	C₂²×C₈	—	47	2	no	—
173	C₂²×C₈	—	94	1	no	—
174	C₂²×C₈	—	94	1	no	—
175	C₂²×C₈	—	188	2	no	—
176	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
178	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	2	no	—
184	C₂×C₄²	—	94	1	no	—
185	C₂×C₄²	—	94	1	no	—
186	C₂×C₄²	—	94	1	no	—
187	C₂×C₄²	—	94	1	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	4	2	no	—
190	C₂×C₁₆	—	94	2	no	—
191	C₂×C₁₆	—	188	2	no	—
192	C₄⋊C₈	—	94	2	no	—
193	C₄⋊C₈	—	94	2	no	—
194	C₄⋊C₈	—	94	2	no	—
195	C₄⋊C₈	—	94	2	no	—
196	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	—	47	2	no	—
202	C₂²×C₄	—	94	1	no	—
203	C₂²×C₄	—	94	1	no	—
204	C₂²×C₄	—	94	2	no	—
205	C₂²×C₄	—	94	1	no	—
206	C₂²×C₄	—	94	1	no	—
207	C₂²×C₄	—	188	2	no	—
208	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	#10625324	1	1	yes	characteristic
210	C₂²×C₄	#10625324	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	94	2	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₂×C₈	—	188	2	no	—
224	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
227	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₄²	—	94	2	no	—
229	C₄²	—	94	2	no	—
230	C₄²	—	94	2	no	—
231	C₄²	—	94	2	no	—
232	C₁₆	—	4	2	no	—
233	C₁₆	—	188	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
237	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
238	C₂³	C₁₆×D₄₇	1	2	yes	—
239	C₂³	#13093963	1	2	yes	—
240	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	47	4	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	94	4	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	188	2	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
258	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
259	C₂×C₄	#13093938	1	2	yes	—
260	C₂×C₄	#13093942	1	2	yes	—
261	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	188	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	#15688848	1	2	yes	—
275	C₂²	(C₂×D₄₇)⋊C₁₆	1	2	yes	—
276	C₂²	(C₂×D₄₇)⋊C₁₆	1	2	yes	—
277	C₂²	#15688904	1	2	yes	—
278	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
279	C₂²	C₂×C₁₆×D₄₇	1	1	yes	characteristic
280	C₂²	C₂×#13093963	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	94	4	no	—
284	C₄	—	188	2	no	—
285	C₄	#15688848	1	2	yes	—
286	C₄	C₂×#13093938	1	1	yes	characteristic
287	C₄	C₂×#13093942	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#20241327	1	4	yes	direct product factor
292	C₂	C₂×#15688848	1	1	yes	characteristic
293	C₂	C₂×(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
294	C₂	C₂×#15688904	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#1322
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20241327	C₂	direct (N × H)
63. C₂²×Dic₄₇	C₁₆	semidirect
114. C₂×Dic₄₇	C₂×C₁₆	semidirect
148. C₂×C₄₇	#1322	semidirect
167. C₄₇	C₂×#1322	semidirect
291. C₂	#20241327	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×Dic₄₇⋊C₈	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241327	2
C₂×C₄₇×C₂²⋊C₁₆	2
C₂×#1322	47

Subgroups — C2×#20241327 #26125408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241327 #26125408