Subgroups — C₂×#20242750 #26198510

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689400	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20242750	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688869	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₄₇	—	2	1	no	—
8	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	#15689400	—	2	4	no	—
10	#15689400	C₂²	1	8	yes	—
11	C₈×D₁₈₈	—	2	2	no	—
12	C₈×D₁₈₈	C₂²	1	4	yes	—
13	C₂×#13093957	—	2	2	no	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093944	—	2	2	no	—
17	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	#15688876	C₂²	1	4	yes	complemented
19	#15688869	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₄₇	—	2	1	no	—
21	C₂²×Q₁₆×C₄₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#494	—	2	2	no	—
24	C₄₇×#494	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂×C₈×D₄₇	—	2	4	no	—
31	C₂×C₈×D₄₇	—	4	4	no	—
32	C₄×D₁₈₈	—	2	1	no	—
33	C₄×D₁₈₈	—	2	1	no	—
34	C₄×D₁₈₈	D₄	1	2	yes	—
35	C₄×D₁₈₈	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
37	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
38	#13093957	—	2	8	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
41	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
42	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
43	#13093944	—	2	8	no	—
44	(C₂×D₄₇)⋊C₈	—	2	4	no	—
45	C₂²×Q₈×C₄₇	—	2	2	no	—
46	C₂×Q₁₆×C₄₇	—	2	4	no	—
47	C₂×Q₁₆×C₄₇	—	2	4	no	—
48	C₂×Q₁₆×C₄₇	—	4	4	no	—
49	C₂×Q₁₆×C₄₇	—	4	4	no	—
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₇	—	2	2	no	—
53	C₄⋊Q₈×C₄₇	—	2	2	no	—
54	C₄⋊Q₈×C₄₇	—	2	2	no	—
55	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₇	D₄	1	2	yes	—
60	C₄×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂³×D₄₇	—	2	1	no	—
62	C₂×Q₈×C₄₇	—	2	4	no	—
63	C₂×Q₈×C₄₇	—	4	2	no	—
64	C₂×Q₈×C₄₇	—	4	4	no	—
65	C₂×Q₈×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	1	no	—
71	C₂×D₁₈₈	—	2	1	no	—
72	C₂×D₁₈₈	—	2	2	no	—
73	C₂×D₁₈₈	Q₁₆	1	4	yes	complemented
74	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₄₇	—	2	2	no	—
76	C₂×C₄×D₄₇	—	2	2	no	—
77	C₂×C₄×D₄₇	—	4	2	no	—
78	C₂×C₄×D₄₇	—	4	2	no	—
79	Q₁₆×C₄₇	—	4	8	no	—
80	Q₁₆×C₄₇	—	4	8	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	D₈	1	4	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	4	2	no	—
88	C₄⋊C₄×C₄₇	—	4	2	no	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
93	(C₂×D₄₇)⋊C₄	—	2	2	no	—
94	(C₂×D₄₇)⋊C₄	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	1	no	—
96	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	4	no	—
101	C₈×D₄₇	—	4	8	no	—
102	C₂³×C₄₇	#121	1	1	yes	characteristic
103	C₂²×D₄₇	—	2	2	no	—
104	C₂²×D₄₇	—	2	4	no	—
105	C₂²×D₄₇	—	4	2	no	—
106	C₂²×D₄₇	—	4	2	no	—
107	Q₈×C₄₇	—	4	4	no	—
108	Q₈×C₄₇	—	4	4	no	—
109	Q₈×C₄₇	—	8	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	8	4	no	—
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₄₇	#135	1	2	yes	—
119	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	D₁₈₈	—	2	2	no	—
126	D₁₈₈	—	2	4	no	—
127	D₁₈₈	C₂×Q₁₆	1	4	yes	complemented
128	C₄×D₄₇	—	4	2	no	—
129	C₄×D₄₇	—	4	2	no	—
130	C₄×D₄₇	—	4	4	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	C₂×D₈	1	4	yes	—
133	C₄₇⋊C₈	—	4	2	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	C₂×#1669	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	#459	1	2	yes	—
137	C₂²×C₄₇	#467	1	2	yes	—
138	C₂²×C₄₇	#473	1	2	yes	—
139	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
140	C₂×D₄₇	—	2	4	no	—
141	C₂×D₄₇	—	4	8	no	—
142	C₂×D₄₇	—	4	2	no	—
143	C₂×D₄₇	—	4	2	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	8	4	no	—
147	C₄×C₄₇	#459	1	2	yes	—
148	C₄×C₄₇	#467	1	2	yes	—
149	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
151	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
152	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	Dic₄₇	—	4	2	no	—
154	C₂×#494	D₄₇	1	1	yes	characteristic, complemented
155	C₂×#474	—	47	2	no	—
156	C₂×C₈×D₄	—	47	1	no	—
157	#1669	—	47	8	no	—
158	C₂×#335	—	47	1	no	—
159	C₂×#332	—	47	2	no	—
160	C₂×C₄₇	#1669	1	4	yes	complemented
161	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
164	D₄₇	—	4	4	no	—
165	C₂²×Q₁₆	—	2	1	no	—
166	C₂²×Q₁₆	—	2	1	no	—
167	C₂³×C₈	—	94	1	no	—
168	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
169	C₂×C₄×D₄	—	47	1	no	—
170	#494	—	2	2	no	—
171	#494	C₂×D₄₇	1	4	yes	complemented
172	#474	—	47	8	no	—
173	#474	—	94	4	no	—
174	C₈×D₄	—	47	4	no	—
175	C₈×D₄	—	94	2	no	—
176	C₂×#107	—	94	2	no	—
177	C₂×C₄⋊C₈	—	47	1	no	—
178	C₂×C₄⋊C₈	—	47	2	no	—
179	C₂×#102	—	94	2	no	—
180	C₂×C₂²⋊C₈	—	94	1	no	—
181	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
182	#335	—	47	4	no	—
183	#332	—	47	8	no	—
184	C₄₇	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×D₄	—	47	1	no	—
187	C₂³×C₄	—	94	1	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	4	4	no	—
191	C₂×Q₁₆	—	4	4	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	—	94	1	no	—
194	C₂²×C₈	—	94	4	no	—
195	C₂²×C₈	—	188	4	no	—
196	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
202	C₄×D₄	—	47	2	no	—
203	C₄×D₄	—	47	2	no	—
204	C₄×D₄	—	94	1	no	—
205	C₄×D₄	—	94	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	47	1	no	—
209	C₂×C₂²⋊C₄	—	94	1	no	—
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	#107	—	94	8	no	—
212	C₄⋊C₈	—	47	2	no	—
213	C₄⋊C₈	—	47	4	no	—
214	C₄⋊C₈	—	94	2	no	—
215	C₄⋊C₈	—	94	1	no	—
216	#102	—	94	8	no	—
217	C₂²⋊C₈	—	94	4	no	—
218	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
219	C₄×C₈	C₂²×D₄₇	1	2	yes	—
220	C₂⁴	—	94	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×D₄	—	47	4	no	—
226	C₂×D₄	—	47	2	no	—
227	C₂×D₄	—	94	1	no	—
228	C₂×D₄	—	94	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	#10625352	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	188	2	no	—
244	C₂×C₈	—	188	4	no	—
245	C₂×C₈	—	188	8	no	—
246	C₂×C₈	#10625334	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	47	2	no	—
254	C₄⋊C₄	—	94	1	no	—
255	C₂²⋊C₄	—	94	2	no	—
256	C₂²⋊C₄	—	94	2	no	—
257	C₄²	D₄×D₄₇	1	1	yes	characteristic
258	C₄²	#10625352	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	C₄²	C₂³×D₄₇	1	1	yes	characteristic
261	C₂³	—	94	2	no	—
262	C₂³	—	94	4	no	—
263	C₂³	—	188	2	no	—
264	C₂³	—	188	2	no	—
265	C₂³	#13093995	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	47	4	no	—
270	D₄	—	94	4	no	—
271	D₄	—	94	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	—	94	2	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	#13093995	1	1	yes	characteristic
281	C₂×C₄	#13093995	1	1	yes	characteristic
282	C₂×C₄	#13093995	1	1	yes	characteristic
283	C₂×C₄	C₂×#10625334	1	2	yes	—
284	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
285	C₂×C₄	#13094047	1	2	yes	—
286	C₂×C₄	#13094050	1	2	yes	—
287	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	188	2	no	—
292	C₈	—	188	4	no	—
293	C₈	C₂×#10625334	1	4	yes	—
294	C₂²	—	94	4	no	—
295	C₂²	—	188	8	no	—
296	C₂²	—	188	2	no	—
297	C₂²	—	188	2	no	—
298	C₂²	#15689315	1	2	yes	—
299	C₂²	#15689361	1	2	yes	—
300	C₂²	#15689395	1	2	yes	—
301	C₂²	C₂×#13093995	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	4	no	—
305	C₄	—	188	2	no	—
306	C₄	#15689315	1	2	yes	—
307	C₄	#15689361	1	2	yes	—
308	C₄	C₂²×#10625334	1	1	yes	characteristic
309	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
310	C₄	C₂×#13094047	1	1	yes	characteristic
311	C₄	C₂×#13094050	1	1	yes	characteristic
312	C₂	—	188	4	no	—
313	C₂	#20242750	1	4	yes	direct product factor
314	C₂	C₂×#15689315	1	1	yes	characteristic
315	C₂	C₂×#15689361	1	1	yes	characteristic
316	C₂	C₂×#15689395	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	C₂×#1669
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20242750	C₂	direct (N × H)
4. C₂×#15688876	C₂	semidirect
5. C₂×#15688869	C₂	semidirect
6. C₂×C₄₇×#494	C₂	semidirect
18. #15688876	C₂²	semidirect
19. #15688869	C₂²	semidirect
24. C₄₇×#494	C₂²	semidirect
73. C₂×D₁₈₈	Q₁₆	semidirect
127. D₁₈₈	C₂×Q₁₆	semidirect
154. C₂×#494	D₄₇	semidirect
160. C₂×C₄₇	#1669	semidirect
171. #494	C₂×D₄₇	semidirect
184. C₄₇	C₂×#1669	semidirect
313. C₂	#20242750	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×#15689400	2
C₂×C₈×D₁₈₈	2
#20242750	2
C₂×#15688876	2
C₂×#15688869	2
C₂×C₄₇×#494	2
C₂×#1669	47

Subgroups — C2×#20242750 #26198510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242750 #26198510