Subgroups — C₂×#20245305 #26370026

Order 12032.

1178 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093967	C₂	1	2	yes	maximal, complemented
2	C₂×#15689359	C₂	1	1	yes	maximal, characteristic
3	#20245305	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688927	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
8	C₂×#13093998	—	2	1	no	—
9	C₂×#13093967	—	2	8	no	—
10	C₂×#13093967	C₂²	1	8	yes	complemented
11	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	#15689359	—	2	2	no	—
13	#15689359	C₂²	1	4	yes	—
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093946	—	2	1	no	—
16	#15688927	C₂²	1	8	yes	complemented
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
18	C₂³×Q₈×C₄₇	—	2	1	no	—
19	C₂²×Q₁₆×C₄₇	—	2	8	no	—
20	C₂²×Q₁₆×C₄₇	—	2	2	no	—
21	C₂²×Q₁₆×C₄₇	—	2	1	no	—
22	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	complemented
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₈₈	—	2	1	no	—
26	C₂²×C₂²⋊D₄₇	—	4	1	no	—
27	C₂×D₃₇₆	—	2	4	no	—
28	C₂×D₃₇₆	—	2	1	no	—
29	C₂×D₃₇₆	C₂³	1	2	yes	—
30	#13093998	—	2	4	no	—
31	#13093998	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
34	#13093967	—	2	16	no	—
35	C₂×C₄₇⋊C₁₆	—	2	4	no	—
36	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
37	#13093958	—	2	1	no	—
38	#13093958	C₂³	1	2	yes	—
39	#13093946	—	2	4	no	—
40	C₂²×Q₈×C₄₇	—	2	8	no	—
41	C₂²×Q₈×C₄₇	—	2	2	no	—
42	C₂²×Q₈×C₄₇	—	2	4	no	—
43	C₂²×Q₈×C₄₇	—	4	4	no	—
44	C₂²×Q₈×C₄₇	—	4	1	no	—
45	C₂²×Q₈×C₄₇	—	4	2	no	—
46	C₂³×C₄×C₄₇	—	4	1	no	—
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₄₇	—	2	8	no	—
49	C₂×Q₁₆×C₄₇	—	2	16	no	—
50	C₂×Q₁₆×C₄₇	—	2	16	no	—
51	C₂×Q₁₆×C₄₇	—	2	2	no	—
52	C₂×Q₁₆×C₄₇	—	2	4	no	—
53	C₂×Q₁₆×C₄₇	—	2	4	no	—
54	C₂×Q₁₆×C₄₇	D₄	1	8	yes	complemented
55	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₇	—	2	4	no	—
57	C₂²×C₈×C₄₇	D₄	1	2	yes	—
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
62	C₂³×D₄₇	—	4	1	no	—
63	C₂×Q₈×C₄₇	—	2	16	no	—
64	C₂×Q₈×C₄₇	—	2	8	no	—
65	C₂×Q₈×C₄₇	—	2	4	no	—
66	C₂×Q₈×C₄₇	—	4	8	no	—
67	C₂×Q₈×C₄₇	—	4	8	no	—
68	C₂×Q₈×C₄₇	—	4	16	no	—
69	C₂×Q₈×C₄₇	—	4	4	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂×Q₈×C₄₇	—	4	4	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	4	8	no	—
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	D₈	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₄₇	—	4	4	no	—
82	C₂×C₂²⋊D₄₇	—	8	4	no	—
83	C₂²×Dic₄₇	—	4	1	no	—
84	C₂×D₁₈₈	—	2	2	no	—
85	C₂×D₁₈₈	—	4	2	no	—
86	C₂×D₁₈₈	—	4	2	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	Q₁₆×C₄₇	—	2	16	no	—
89	Q₁₆×C₄₇	—	2	8	no	—
90	Q₁₆×C₄₇	—	2	4	no	—
91	Q₁₆×C₄₇	C₂×D₄	1	8	yes	complemented
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	4	no	—
94	C₂×C₈×C₄₇	—	2	4	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
99	(C₂×D₄₇)⋊C₄	—	4	4	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	4	1	no	—
102	D₃₇₆	—	2	4	no	—
103	D₃₇₆	—	4	2	no	—
104	C₄₇⋊C₁₆	—	2	4	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	2	no	—
107	C₂³×C₄₇	SD₃₂	1	4	yes	complemented
108	C₂³×C₄₇	C₂×D₈	1	2	yes	—
109	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₄₇	—	4	2	no	—
111	C₂²×D₄₇	—	8	2	no	—
112	C₂²×D₄₇	—	8	2	no	—
113	C₂²×D₄₇	—	8	2	no	—
114	Q₈×C₄₇	—	2	8	no	—
115	Q₈×C₄₇	—	4	16	no	—
116	Q₈×C₄₇	—	4	4	no	—
117	Q₈×C₄₇	—	4	8	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	4	16	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	8	no	—
125	C₂×C₄×C₄₇	#120	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₄₇	—	8	8	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	8	2	no	—
133	D₁₈₈	—	4	2	no	—
134	D₁₈₈	—	4	2	no	—
135	D₁₈₈	—	8	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	#120	1	2	yes	—
138	C₈×C₄₇	C₂²×D₄	1	2	yes	—
139	C₂×#2185	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	8	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	#447	1	2	yes	—
144	C₂²×C₄₇	C₂×SD₃₂	1	2	yes	—
145	C₂²×C₄₇	C₂×SD₃₂	1	4	yes	complemented
146	C₂²×C₄₇	#510	1	2	yes	—
147	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₄₇	—	8	2	no	—
149	C₂×D₄₇	—	8	2	no	—
150	C₂×D₄₇	—	8	2	no	—
151	C₂×D₄₇	—	16	4	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	4	8	no	—
154	C₄×C₄₇	#447	1	2	yes	—
155	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
156	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	Dic₄₇	—	8	2	no	—
158	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
159	C₂²×SD₃₂	—	47	2	no	—
160	C₂×#466	—	47	1	no	—
161	#2185	—	47	8	no	—
162	C₂×#358	—	47	2	no	—
163	C₂×C₂²⋊C₁₆	—	47	1	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	#2185	1	4	yes	complemented
166	C₂×C₄₇	C₂×#447	1	1	yes	characteristic
167	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
169	D₄₇	—	16	2	no	—
170	C₂³×Q₈	—	2	1	no	—
171	C₂²×Q₁₆	—	2	8	no	—
172	C₂²×Q₁₆	—	2	1	no	—
173	C₂²×Q₁₆	—	2	2	no	—
174	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
175	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	complemented
176	C₂²×D₈	—	47	1	no	—
177	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#121	—	94	1	no	—
179	C₂×SD₃₂	—	47	8	no	—
180	C₂×SD₃₂	—	94	8	no	—
181	C₂²×C₁₆	—	47	2	no	—
182	#466	—	47	4	no	—
183	#466	—	94	2	no	—
184	C₂×#107	—	47	1	no	—
185	C₂×#102	—	94	1	no	—
186	#358	—	47	8	no	—
187	C₂²⋊C₁₆	—	47	4	no	—
188	C₄₇	C₂×#2185	1	1	yes	characteristic, complemented, Sylow
189	C₂²×Q₈	—	2	8	no	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	4	4	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂²×Q₈	—	4	1	no	—
195	C₂²×D₄	—	94	1	no	—
196	C₂²×D₄	—	188	1	no	—
197	C₂³×C₄	—	4	1	no	—
198	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂×Q₁₆	—	2	8	no	—
200	C₂×Q₁₆	—	2	16	no	—
201	C₂×Q₁₆	—	2	16	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	C₂²⋊D₄₇	1	8	yes	complemented
206	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
207	C₂×D₈	—	47	2	no	—
208	C₂×D₈	—	94	4	no	—
209	C₂×D₈	—	94	1	no	—
210	C₂²×C₈	—	2	4	no	—
211	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
213	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
214	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	#121	—	94	4	no	—
216	#121	—	188	2	no	—
217	C₂×C₄⋊C₄	—	94	1	no	—
218	C₂×C₂²⋊C₄	—	188	1	no	—
219	SD₃₂	—	94	16	no	—
220	C₂×C₁₆	—	47	4	no	—
221	C₂×C₁₆	—	94	4	no	—
222	#107	—	47	2	no	—
223	#107	—	94	1	no	—
224	#102	—	94	4	no	—
225	C₂⁴	—	188	1	no	—
226	C₂⁴	#10625334	1	1	yes	characteristic
227	C₂×Q₈	—	2	16	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	—	4	16	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂×Q₈	—	4	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×D₄	—	94	2	no	—
237	C₂×D₄	—	376	4	no	—
238	C₂×D₄	—	188	4	no	—
239	C₂×D₄	—	188	2	no	—
240	C₂×D₄	—	188	1	no	—
241	C₂×D₄	—	188	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	188	1	no	—
248	C₂²×C₄	#10625334	1	2	yes	—
249	C₂²×C₄	#10625334	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	Q₁₆	—	2	16	no	—
253	Q₁₆	—	2	8	no	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	C₂×C₂²⋊D₄₇	1	8	yes	complemented
256	D₈	—	94	4	no	—
257	D₈	—	188	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
262	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
264	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
265	C₄⋊C₄	—	94	2	no	—
266	C₄⋊C₄	—	188	1	no	—
267	C₂²⋊C₄	—	188	4	no	—
268	C₁₆	—	94	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	376	2	no	—
272	C₂³	—	376	2	no	—
273	C₂³	—	376	2	no	—
274	C₂³	—	188	2	no	—
275	C₂³	#13093967	1	4	yes	complemented
276	C₂³	C₂×#10625334	1	2	yes	—
277	C₂³	C₂×#10625334	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	4	4	no	—
282	D₄	—	376	8	no	—
283	D₄	—	376	2	no	—
284	D₄	—	188	2	no	—
285	D₄	—	188	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	376	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	#13093993	1	2	yes	—
296	C₂×C₄	C₂×#10625334	1	2	yes	—
297	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
298	C₂×C₄	#13094050	1	2	yes	—
299	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	#13093993	1	2	yes	—
302	C₈	C₂²×C₂²⋊D₄₇	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	376	2	no	—
307	C₂²	—	376	2	no	—
308	C₂²	—	376	2	no	—
309	C₂²	—	752	4	no	—
310	C₂²	#15689257	1	2	yes	—
311	C₂²	C₂×#13093967	1	2	yes	—
312	C₂²	C₂×#13093967	1	4	yes	complemented
313	C₂²	#15689564	1	2	yes	—
314	C₂²	C₂²×#10625334	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	376	2	no	—
318	C₄	#15689257	1	2	yes	—
319	C₄	C₂×#13093993	1	1	yes	characteristic
320	C₄	C₂×#13094050	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	—	752	2	no	—
323	C₂	#20245305	1	4	yes	direct product factor
324	C₂	C₂×#15689257	1	1	yes	characteristic
325	C₂	C₂²×#13093967	1	1	yes	characteristic
326	C₂	C₂×#15689564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂×#15689359	C₂
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2185
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093967	C₂	semidirect
3. #20245305	C₂	direct (N × H)
4. C₂×#15688927	C₂	semidirect
5. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂³×Q₁₆×C₄₇	C₂	semidirect
10. C₂×#13093967	C₂²	semidirect
16. #15688927	C₂²	semidirect
17. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
23. C₂²×Q₁₆×C₄₇	C₂²	semidirect
54. C₂×Q₁₆×C₄₇	D₄	semidirect
91. Q₁₆×C₄₇	C₂×D₄	semidirect
107. C₂³×C₄₇	SD₃₂	semidirect
145. C₂²×C₄₇	C₂×SD₃₂	semidirect
158. C₂³×Q₁₆	D₄₇	semidirect
165. C₂×C₄₇	#2185	semidirect
175. C₂²×Q₁₆	C₂×D₄₇	semidirect
188. C₄₇	C₂×#2185	semidirect
205. C₂×Q₁₆	C₂²⋊D₄₇	semidirect
255. Q₁₆	C₂×C₂²⋊D₄₇	semidirect
275. C₂³	#13093967	semidirect
312. C₂²	C₂×#13093967	semidirect
323. C₂	#20245305	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×#13093967	2
C₂×#15689359	2
#20245305	2
C₂×#15688927	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂³×Q₁₆×C₄₇	2
C₂×#2185	47

Subgroups — C2×#20245305 #26370026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245305 #26370026