Subgroups — C₂×#20241479 #26131154

Order 12032.

694 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	complemented
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×#13093958	—	2	1	no	—
11	C₂×#13093958	—	2	2	no	—
12	C₂×#13093958	C₄	1	4	yes	complemented
13	C₂×#13093958	C₂²	1	2	yes	—
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂×#107×C₄₇	—	2	1	no	—
17	C₂×#107×C₄₇	—	2	2	no	—
18	C₂×#107×C₄₇	C₄	1	4	yes	complemented
19	C₂×#107×C₄₇	C₂²	1	2	yes	—
20	C₂³×Dic₄₇	—	4	1	no	—
21	C₂×C₄⋊Dic₄₇	—	2	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	4	1	no	—
24	C₂×C₄⋊Dic₄₇	—	4	2	no	—
25	C₂×C₄₇⋊C₁₆	—	2	8	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	C₂×C₄₇⋊C₁₆	—	2	2	no	—
28	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
29	#13093958	—	2	4	no	—
30	#13093958	—	2	2	no	—
31	#13093958	C₂×C₄	1	4	yes	complemented
32	C₂³×C₄×C₄₇	—	4	1	no	—
33	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	D₄	1	2	yes	—
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
46	#107×C₄₇	—	2	4	no	—
47	#107×C₄₇	—	2	2	no	—
48	#107×C₄₇	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	8	2	no	—
53	C₂²×C₄×C₄₇	—	8	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	2	no	—
64	C₂²×Dic₄₇	—	8	2	no	—
65	C₂²×Dic₄₇	—	8	2	no	—
66	C₂×C₈×C₄₇	C₄²	1	4	yes	—
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	2	no	—
85	C₄₇⋊C₁₆	—	2	4	no	—
86	C₂³×C₄₇	#102	1	2	yes	—
87	C₂³×C₄₇	#102	1	2	yes	—
88	C₂³×C₄₇	#107	1	2	yes	—
89	C₂³×C₄₇	D₁₆	1	2	yes	—
90	C₂³×C₄₇	SD₃₂	1	2	yes	—
91	C₂³×C₄₇	SD₃₂	1	2	yes	—
92	C₂³×C₄₇	Q₃₂	1	2	yes	—
93	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	8	8	no	—
96	C₂×C₄×C₄₇	—	8	2	no	—
97	C₂×C₄×C₄₇	—	8	2	no	—
98	C₂×C₄×C₄₇	#95	1	4	yes	—
99	C₂×C₄×C₄₇	#102	1	2	yes	—
100	C₂×C₄×C₄₇	#102	1	2	yes	—
101	C₂×C₄×C₄₇	#103	1	4	yes	—
102	C₂×C₄×C₄₇	#103	1	4	yes	—
103	C₂×C₄×C₄₇	#106	1	4	yes	—
104	C₂×C₄×C₄₇	#107	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	8	8	no	—
113	C₂×Dic₄₇	—	8	2	no	—
114	C₂×Dic₄₇	—	8	2	no	—
115	C₈×C₄₇	#95	1	4	yes	—
116	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
117	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×#1380	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	#340	1	4	yes	—
122	C₂²×C₄₇	#357	1	4	yes	—
123	C₂²×C₄₇	#357	1	4	yes	—
124	C₂²×C₄₇	#358	1	4	yes	—
125	C₂²×C₄₇	#358	1	4	yes	—
126	C₂²×C₄₇	#366	1	4	yes	—
127	C₂²×C₄₇	#367	1	4	yes	—
128	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
135	C₄×C₄₇	—	8	4	no	—
136	C₄×C₄₇	#340	1	4	yes	—
137	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
141	Dic₄₇	—	8	4	no	—
142	C₂³×C₁₆	—	47	1	no	—
143	C₂²×#107	—	47	1	no	—
144	C₂²×#107	D₄₇	1	1	yes	characteristic
145	#1380	—	47	4	no	—
146	C₂×C₄₇	#1380	1	8	yes	complemented
147	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#367	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₄⋊C₄	—	94	1	no	—
157	C₂²×C₁₆	—	47	4	no	—
158	C₂²×C₁₆	—	47	2	no	—
159	C₂²×C₁₆	—	94	2	no	—
160	C₂²×C₁₆	—	94	2	no	—
161	C₂×#107	—	2	1	no	—
162	C₂×#107	—	2	2	no	—
163	C₂×#107	—	47	4	no	—
164	C₂×#107	—	47	2	no	—
165	C₂×#107	—	94	2	no	—
166	C₂×#107	—	94	1	no	—
167	C₂×#107	Dic₄₇	1	4	yes	complemented
168	C₂×#107	C₂×D₄₇	1	2	yes	—
169	C₄₇	C₂×#1380	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	4	1	no	—
171	C₂³×C₄	—	188	1	no	—
172	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂²×C₈	Dic₉₄	1	2	yes	—
174	C₂²×C₈	C₄×D₄₇	1	2	yes	—
175	C₂²×C₈	C₄×D₄₇	1	2	yes	—
176	C₂²×C₈	D₁₈₈	1	2	yes	—
177	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
178	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	4	1	no	—
184	C₂×C₄⋊C₄	—	4	2	no	—
185	C₂×C₄⋊C₄	—	94	4	no	—
186	C₂×C₄⋊C₄	—	94	2	no	—
187	C₂×C₄⋊C₄	—	188	1	no	—
188	C₂×C₄⋊C₄	—	188	2	no	—
189	C₂×C₁₆	—	47	4	no	—
190	C₂×C₁₆	—	94	8	no	—
191	C₂×C₁₆	—	94	2	no	—
192	C₂×C₁₆	—	94	2	no	—
193	#107	—	2	4	no	—
194	#107	—	2	2	no	—
195	#107	—	47	4	no	—
196	#107	—	94	4	no	—
197	#107	—	94	2	no	—
198	#107	C₂×Dic₄₇	1	4	yes	complemented
199	C₂⁴	#10625334	1	1	yes	characteristic
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	2	no	—
203	C₂²×C₄	—	8	2	no	—
204	C₂²×C₄	—	376	2	no	—
205	C₂²×C₄	—	376	2	no	—
206	C₂²×C₄	—	188	4	no	—
207	C₂²×C₄	—	188	2	no	—
208	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
210	C₂²×C₄	#10625334	1	1	yes	characteristic
211	C₂²×C₄	#10625335	1	2	yes	—
212	C₂²×C₄	#10625336	1	2	yes	—
213	C₂²×C₄	#10625337	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
217	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
218	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
219	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
220	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
221	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
222	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
223	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₄⋊C₄	—	188	4	no	—
234	C₄⋊C₄	—	188	2	no	—
235	C₁₆	—	94	4	no	—
236	C₂³	#13093944	1	2	yes	—
237	C₂³	#13093945	1	2	yes	—
238	C₂³	#13093957	1	2	yes	—
239	C₂³	#13093965	1	2	yes	—
240	C₂³	#13093967	1	2	yes	—
241	C₂³	#13093968	1	2	yes	—
242	C₂³	#13093970	1	2	yes	—
243	C₂³	C₂×#10625334	1	1	yes	characteristic
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	8	no	—
248	C₂×C₄	—	376	2	no	—
249	C₂×C₄	—	376	2	no	—
250	C₂×C₄	—	376	8	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	C₄₇⋊#95	1	4	yes	—
253	C₂×C₄	#13093944	1	2	yes	—
254	C₂×C₄	#13093945	1	2	yes	—
255	C₂×C₄	#13093947	1	4	yes	—
256	C₂×C₄	#13093948	1	4	yes	—
257	C₂×C₄	#13093955	1	4	yes	—
258	C₂×C₄	#13093957	1	2	yes	—
259	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
263	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
264	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
265	C₈	C₄₇⋊#95	1	4	yes	—
266	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
267	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
269	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂²	#15688885	1	4	yes	—
271	C₂²	#15688923	1	4	yes	—
272	C₂²	#15688924	1	4	yes	—
273	C₂²	#15688926	1	4	yes	—
274	C₂²	#15688927	1	4	yes	—
275	C₂²	#15688948	1	4	yes	—
276	C₂²	#15688950	1	4	yes	—
277	C₂²	C₂×#13093944	1	1	yes	characteristic
278	C₂²	C₂×#13093945	1	1	yes	characteristic
279	C₂²	C₂×#13093957	1	1	yes	characteristic
280	C₂²	C₂×#13093965	1	1	yes	characteristic
281	C₂²	C₂×#13093967	1	1	yes	characteristic
282	C₂²	C₂×#13093968	1	1	yes	characteristic
283	C₂²	C₂×#13093970	1	1	yes	characteristic
284	C₄	—	8	4	no	—
285	C₄	—	376	4	no	—
286	C₄	#15688885	1	4	yes	—
287	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
288	C₄	C₂×#13093947	1	1	yes	characteristic
289	C₄	C₂×#13093948	1	1	yes	characteristic
290	C₄	C₂×#13093955	1	1	yes	characteristic
291	C₂	#20241479	1	8	yes	direct product factor
292	C₂	C₂×#15688885	1	1	yes	characteristic
293	C₂	C₂×#15688923	1	1	yes	characteristic
294	C₂	C₂×#15688924	1	1	yes	characteristic
295	C₂	C₂×#15688926	1	1	yes	characteristic
296	C₂	C₂×#15688927	1	1	yes	characteristic
297	C₂	C₂×#15688948	1	1	yes	characteristic
298	C₂	C₂×#15688950	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×#107×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#107	D₄₇
hypocenter	C₄₇	C₂×#1380
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #20241479	C₂	direct (N × H)
8. C₂²×C₄₇⋊C₁₆	C₄	semidirect
12. C₂×#13093958	C₄	semidirect
18. C₂×#107×C₄₇	C₄	semidirect
28. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
31. #13093958	C₂×C₄	semidirect
48. #107×C₄₇	C₂×C₄	semidirect
146. C₂×C₄₇	#1380	semidirect
167. C₂×#107	Dic₄₇	semidirect
169. C₄₇	C₂×#1380	semidirect
198. #107	C₂×Dic₄₇	semidirect
291. C₂	#20241479	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂²×#13093958	2
#20241479	2
C₂²×#107×C₄₇	2
C₂×#1380	47

Subgroups — C2×#20241479 #26131154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241479 #26131154