Subgroups — C₂×#20245316 #26370019

Order 12032.

1382 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093967	C₂	1	2	yes	maximal, complemented
3	#20245316	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688923	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	—	2	1	no	—
8	C₂²×D₃₇₆	—	2	8	no	—
9	C₂²×D₃₇₆	—	2	2	no	—
10	C₂²×D₃₇₆	—	2	1	no	—
11	C₂²×D₃₇₆	C₂²	1	4	yes	—
12	C₂²×D₃₇₆	C₂²	1	2	yes	—
13	C₂×#13093967	—	2	8	no	—
14	C₂×#13093967	C₂²	1	8	yes	complemented
15	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
16	#15688923	C₂²	1	8	yes	complemented
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
18	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
21	C₄₇×#467	—	2	2	no	—
22	C₄₇×#467	C₂²	1	4	yes	complemented
23	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#103×C₄₇	—	2	1	no	—
25	C₂⁴×D₄₇	—	4	1	no	—
26	C₂²×D₁₈₈	—	2	8	no	—
27	C₂²×D₁₈₈	—	2	4	no	—
28	C₂²×D₁₈₈	—	2	2	no	—
29	C₂²×D₁₈₈	—	4	4	no	—
30	C₂²×D₁₈₈	—	4	1	no	—
31	C₂²×D₁₈₈	—	4	2	no	—
32	C₂×D₃₇₆	—	2	8	no	—
33	C₂×D₃₇₆	—	2	16	no	—
34	C₂×D₃₇₆	—	2	4	no	—
35	C₂×D₃₇₆	—	2	2	no	—
36	C₂×D₃₇₆	—	2	4	no	—
37	C₂×D₃₇₆	—	2	16	no	—
38	C₂×D₃₇₆	D₄	1	8	yes	—
39	C₂×D₃₇₆	C₂³	1	4	yes	—
40	#13093967	—	2	16	no	—
41	C₂×C₄₇⋊C₁₆	—	2	4	no	—
42	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	complemented
43	C₂²×Q₈×C₄₇	—	2	1	no	—
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂×Q₁₆×C₄₇	—	2	4	no	—
46	C₂×Q₁₆×C₄₇	—	2	1	no	—
47	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	complemented
48	C₂²×C₈×C₄₇	—	2	4	no	—
49	C₂²×C₈×C₄₇	D₄	1	2	yes	—
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²⋊Q₈×C₄₇	—	2	4	no	—
54	C₂²⋊Q₈×C₄₇	—	4	2	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
57	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
58	#107×C₄₇	—	2	1	no	—
59	#107×C₄₇	C₂³	1	2	yes	complemented
60	#103×C₄₇	—	2	4	no	—
61	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
62	C₂³×D₄₇	—	4	8	no	—
63	C₂³×D₄₇	—	4	4	no	—
64	C₂³×D₄₇	—	4	2	no	—
65	C₂³×D₄₇	—	8	4	no	—
66	C₂³×D₄₇	—	8	4	no	—
67	C₂×Q₈×C₄₇	—	2	2	no	—
68	C₂×Q₈×C₄₇	—	4	1	no	—
69	C₂×Q₈×C₄₇	—	4	2	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	4	1	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	D₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×D₁₈₈	—	2	16	no	—
80	C₂×D₁₈₈	—	2	8	no	—
81	C₂×D₁₈₈	—	2	4	no	—
82	C₂×D₁₈₈	—	4	8	no	—
83	C₂×D₁₈₈	—	4	8	no	—
84	C₂×D₁₈₈	—	4	16	no	—
85	C₂×D₁₈₈	—	4	4	no	—
86	C₂×D₁₈₈	—	4	2	no	—
87	C₂×D₁₈₈	—	4	4	no	—
88	Q₁₆×C₄₇	—	2	4	no	—
89	Q₁₆×C₄₇	—	4	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	4	no	—
92	C₂×C₈×C₄₇	—	2	4	no	—
93	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	4	1	no	—
99	C₄⋊C₄×C₄₇	—	4	4	no	—
100	C₂²⋊C₄×C₄₇	—	4	4	no	—
101	D₃₇₆	—	2	16	no	—
102	D₃₇₆	—	2	8	no	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	C₂×D₄	1	8	yes	—
105	C₄₇⋊C₁₆	—	2	4	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	SD₃₂	1	4	yes	complemented
109	C₂³×C₄₇	C₂×D₈	1	2	yes	—
110	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₄₇	—	4	16	no	—
112	C₂²×D₄₇	—	4	4	no	—
113	C₂²×D₄₇	—	4	8	no	—
114	C₂²×D₄₇	—	8	8	no	—
115	C₂²×D₄₇	—	8	32	no	—
116	C₂²×D₄₇	—	8	4	no	—
117	C₂²×D₄₇	—	8	8	no	—
118	C₂²×D₄₇	—	8	4	no	—
119	Q₈×C₄₇	—	4	2	no	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	8	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	#120	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	D₁₈₈	—	2	8	no	—
136	D₁₈₈	—	4	16	no	—
137	D₁₈₈	—	4	4	no	—
138	D₁₈₈	—	4	8	no	—
139	C₈×C₄₇	—	2	2	no	—
140	C₈×C₄₇	#120	1	2	yes	—
141	C₈×C₄₇	C₂²×D₄	1	2	yes	—
142	C₂×#2187	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	8	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	#447	1	2	yes	—
147	C₂²×C₄₇	C₂×SD₃₂	1	2	yes	—
148	C₂²×C₄₇	C₂×SD₃₂	1	4	yes	complemented
149	C₂²×C₄₇	#509	1	2	yes	—
150	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₄₇	—	4	8	no	—
152	C₂×D₄₇	—	8	32	no	—
153	C₂×D₄₇	—	8	4	no	—
154	C₂×D₄₇	—	8	4	no	—
155	C₂×D₄₇	—	8	16	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	—	8	2	no	—
159	C₄×C₄₇	#447	1	2	yes	—
160	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
161	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂³×D₈	—	47	1	no	—
163	C₂²×SD₃₂	—	47	2	no	—
164	C₂×#467	D₄₇	1	1	yes	characteristic, complemented
165	#2187	—	47	8	no	—
166	C₂×#357	—	47	2	no	—
167	C₂×C₂²⋊C₁₆	—	47	1	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	#2187	1	4	yes	complemented
170	C₂×C₄₇	C₂×#447	1	1	yes	characteristic
171	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
173	D₄₇	—	8	8	no	—
174	C₂³×D₄	—	94	1	no	—
175	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
176	C₂²×D₈	—	47	2	no	—
177	C₂²×D₈	—	47	4	no	—
178	C₂²×D₈	—	94	8	no	—
179	C₂²×D₈	—	94	1	no	—
180	C₂²×D₈	—	94	2	no	—
181	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₂²⋊Q₈	—	2	1	no	—
183	C₂×SD₃₂	—	47	8	no	—
184	C₂×SD₃₂	—	94	8	no	—
185	C₂²×C₁₆	—	47	2	no	—
186	#467	—	2	2	no	—
187	#467	C₂×D₄₇	1	4	yes	complemented
188	C₂×#107	C₂×D₄₇	1	1	yes	characteristic, complemented
189	C₂×#103	—	2	1	no	—
190	#357	—	47	8	no	—
191	C₂²⋊C₁₆	—	47	4	no	—
192	C₄₇	C₂×#2187	1	1	yes	characteristic, complemented, Sylow
193	C₂⁵	—	188	1	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×D₄	—	94	8	no	—
196	C₂²×D₄	—	94	4	no	—
197	C₂²×D₄	—	94	2	no	—
198	C₂²×D₄	—	188	4	no	—
199	C₂²×D₄	—	188	2	no	—
200	C₂²×D₄	—	188	1	no	—
201	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	4	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	complemented
205	C₂×D₈	—	47	8	no	—
206	C₂×D₈	—	47	4	no	—
207	C₂×D₈	—	94	16	no	—
208	C₂×D₈	—	94	8	no	—
209	C₂×D₈	—	94	16	no	—
210	C₂×D₈	—	94	4	no	—
211	C₂×D₈	—	94	4	no	—
212	C₂×D₈	—	94	2	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²⋊Q₈	—	2	4	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	4	1	no	—
222	C₂×C₂²⋊C₄	—	4	1	no	—
223	SD₃₂	—	94	16	no	—
224	C₂×C₁₆	—	47	4	no	—
225	C₂×C₁₆	—	94	4	no	—
226	#107	—	2	1	no	—
227	#107	C₂²×D₄₇	1	2	yes	complemented
228	#103	—	2	4	no	—
229	C₂⁴	—	376	4	no	—
230	C₂⁴	—	376	4	no	—
231	C₂⁴	—	188	8	no	—
232	C₂⁴	—	188	4	no	—
233	C₂⁴	—	188	2	no	—
234	C₂⁴	#10625334	1	1	yes	characteristic
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	94	16	no	—
240	C₂×D₄	—	94	8	no	—
241	C₂×D₄	—	94	4	no	—
242	C₂×D₄	—	188	8	no	—
243	C₂×D₄	—	188	16	no	—
244	C₂×D₄	—	188	8	no	—
245	C₂×D₄	—	188	4	no	—
246	C₂×D₄	—	188	2	no	—
247	C₂×D₄	—	188	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	#10625334	1	2	yes	—
253	C₂²×C₄	#10625334	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	Q₁₆	—	2	4	no	—
257	Q₁₆	—	4	2	no	—
258	D₈	—	47	8	no	—
259	D₈	—	94	16	no	—
260	D₈	—	94	8	no	—
261	D₈	—	94	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	D₄×D₄₇	1	4	yes	—
266	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₁₆	—	94	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	376	32	no	—
277	C₂³	—	376	8	no	—
278	C₂³	—	376	4	no	—
279	C₂³	—	376	8	no	—
280	C₂³	—	376	4	no	—
281	C₂³	—	188	16	no	—
282	C₂³	—	188	8	no	—
283	C₂³	—	188	4	no	—
284	C₂³	#13093967	1	4	yes	complemented
285	C₂³	C₂×#10625334	1	2	yes	—
286	C₂³	C₂×#10625334	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	94	8	no	—
291	D₄	—	188	16	no	—
292	D₄	—	188	8	no	—
293	D₄	—	188	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	#13093992	1	2	yes	—
303	C₂×C₄	C₂×#10625334	1	2	yes	—
304	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
305	C₂×C₄	#13094049	1	2	yes	—
306	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	#13093992	1	2	yes	—
309	C₈	C₂×D₄×D₄₇	1	2	yes	—
310	C₂²	—	2	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	376	32	no	—
314	C₂²	—	376	16	no	—
315	C₂²	—	376	4	no	—
316	C₂²	—	376	4	no	—
317	C₂²	—	188	8	no	—
318	C₂²	#15689256	1	2	yes	—
319	C₂²	C₂×#13093967	1	2	yes	—
320	C₂²	C₂×#13093967	1	4	yes	complemented
321	C₂²	#15689558	1	2	yes	—
322	C₂²	C₂²×#10625334	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#15689256	1	2	yes	—
327	C₄	C₂×#13093992	1	1	yes	characteristic
328	C₄	C₂×#13094049	1	1	yes	characteristic
329	C₂	—	2	4	no	—
330	C₂	—	376	8	no	—
331	C₂	#20245316	1	4	yes	direct product factor
332	C₂	C₂×#15689256	1	1	yes	characteristic
333	C₂	C₂²×#13093967	1	1	yes	characteristic
334	C₂	C₂×#15689558	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#467	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂³×D₃₇₆	C₂
hypercenter	C₂×#467	D₄₇
hypocenter	C₄₇	C₂×#2187
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂²×#13093967	C₂	semidirect
3. #20245316	C₂	direct (N × H)
4. C₂×#15688923	C₂	semidirect
5. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂×C₄₇×#467	C₂	semidirect
14. C₂×#13093967	C₂²	semidirect
15. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
16. #15688923	C₂²	semidirect
17. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
18. C₂²×Q₁₆×C₄₇	C₂²	semidirect
22. C₄₇×#467	C₂²	semidirect
23. C₂×#107×C₄₇	C₂²	semidirect
42. C₂×C₄₇⋊C₁₆	C₂³	semidirect
47. C₂×Q₁₆×C₄₇	C₂³	semidirect
59. #107×C₄₇	C₂³	semidirect
108. C₂³×C₄₇	SD₃₂	semidirect
148. C₂²×C₄₇	C₂×SD₃₂	semidirect
164. C₂×#467	D₄₇	semidirect
169. C₂×C₄₇	#2187	semidirect
175. C₂²×Q₁₆	C₂×D₄₇	semidirect
187. #467	C₂×D₄₇	semidirect
188. C₂×#107	C₂×D₄₇	semidirect
192. C₄₇	C₂×#2187	semidirect
204. C₂×Q₁₆	C₂²×D₄₇	semidirect
227. #107	C₂²×D₄₇	semidirect
284. C₂³	#13093967	semidirect
320. C₂²	C₂×#13093967	semidirect
331. C₂	#20245316	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×D₃₇₆	2
C₂²×#13093967	2
#20245316	2
C₂×#15688923	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×C₄₇×#467	2
C₂×#2187	47

Subgroups — C2×#20245316 #26370019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245316 #26370019