Subgroups — C₂×#20242874 #26198782

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 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×#15689202	C₂	1	1	yes	maximal, characteristic
3	#20242874	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688874	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×#10625324	—	2	1	no	—
8	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	#15689400	—	2	4	no	—
10	#15689400	C₂²	1	8	yes	—
11	#15689202	—	2	2	no	—
12	#15689202	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	#15688874	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	C₂×#10625324	—	2	4	no	—
29	C₂×#10625324	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	—	2	2	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	4	no	—
72	C₂×D₁₈₈	—	2	1	no	—
73	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	2	2	no	—
76	C₂×C₄×D₄₇	—	4	2	no	—
77	C₂×C₄×D₄₇	—	4	2	no	—
78	Q₁₆×C₄₇	—	4	8	no	—
79	Q₁₆×C₄₇	—	4	8	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	D₈	1	4	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	4	2	no	—
87	C₄⋊C₄×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	#41	1	1	yes	characteristic
91	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
92	(C₂×D₄₇)⋊C₄	—	2	2	no	—
93	(C₂×D₄₇)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	1	no	—
95	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	4	4	no	—
100	#10625324	—	4	8	no	—
101	C₂³×C₄₇	#121	1	1	yes	characteristic
102	C₂²×D₄₇	—	2	2	no	—
103	C₂²×D₄₇	—	4	2	no	—
104	C₂²×D₄₇	—	4	2	no	—
105	C₂²×D₄₇	—	4	2	no	—
106	Q₈×C₄₇	—	4	4	no	—
107	Q₈×C₄₇	—	4	4	no	—
108	Q₈×C₄₇	—	8	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	4	no	—
112	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₄₇	#137	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	D₁₈₈	—	2	4	no	—
124	D₁₈₈	—	4	2	no	—
125	C₄×D₄₇	—	4	2	no	—
126	C₄×D₄₇	—	4	2	no	—
127	C₄×D₄₇	—	4	4	no	—
128	C₈×C₄₇	—	2	2	no	—
129	C₈×C₄₇	C₂×D₈	1	4	yes	—
130	C₄₇⋊C₈	—	4	2	no	—
131	C₄₇⋊C₈	—	4	4	no	—
132	C₂×#1689	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	#459	1	2	yes	—
134	C₂²×C₄₇	#470	1	2	yes	—
135	C₂²×C₄₇	#473	1	2	yes	—
136	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
137	C₂×D₄₇	—	4	2	no	—
138	C₂×D₄₇	—	4	2	no	—
139	C₂×D₄₇	—	4	2	no	—
140	C₂×D₄₇	—	8	4	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	8	4	no	—
144	C₄×C₄₇	#459	1	2	yes	—
145	C₄×C₄₇	#470	1	2	yes	—
146	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₄₇	C₂×#137	1	2	yes	—
149	Dic₄₇	—	4	2	no	—
150	C₂×#494	D₄₇	1	1	yes	characteristic, complemented
151	C₂×#474	—	47	2	no	—
152	C₂×#436	—	47	1	no	—
153	#1689	—	47	8	no	—
154	C₂×#334	—	47	1	no	—
155	C₂×#332	—	47	2	no	—
156	C₂×C₄₇	#1689	1	4	yes	complemented
157	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#470	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
160	D₄₇	—	8	2	no	—
161	C₂²×Q₁₆	—	2	1	no	—
162	C₂²×Q₁₆	—	2	1	no	—
163	C₂²×M₁₆	—	94	1	no	—
164	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
165	C₂×C₄×D₄	—	47	1	no	—
166	#494	—	2	2	no	—
167	#494	C₂×D₄₇	1	4	yes	complemented
168	#474	—	47	8	no	—
169	#474	—	94	4	no	—
170	#436	—	47	4	no	—
171	#436	—	94	2	no	—
172	C₂×#107	—	94	2	no	—
173	C₂×C₄⋊C₈	—	47	1	no	—
174	C₂×C₄⋊C₈	—	47	2	no	—
175	C₂×#102	—	94	2	no	—
176	C₂×C₂²⋊C₈	—	94	1	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	#334	—	47	4	no	—
179	#332	—	47	8	no	—
180	C₄₇	C₂×#1689	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×D₄	—	47	1	no	—
183	C₂³×C₄	—	94	1	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	C₂×Q₁₆	—	4	4	no	—
188	C₂×M₁₆	—	94	4	no	—
189	C₂×M₁₆	—	188	4	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₂²×C₈	—	94	1	no	—
192	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
198	C₄×D₄	—	47	2	no	—
199	C₄×D₄	—	47	2	no	—
200	C₄×D₄	—	94	1	no	—
201	C₄×D₄	—	94	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	47	1	no	—
205	C₂×C₂²⋊C₄	—	94	1	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	94	8	no	—
208	C₄⋊C₈	—	47	2	no	—
209	C₄⋊C₈	—	47	4	no	—
210	C₄⋊C₈	—	94	2	no	—
211	C₄⋊C₈	—	94	1	no	—
212	#102	—	94	8	no	—
213	C₂²⋊C₈	—	94	4	no	—
214	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
215	C₄×C₈	C₂²×D₄₇	1	2	yes	—
216	C₂⁴	—	94	1	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×D₄	—	47	2	no	—
222	C₂×D₄	—	94	1	no	—
223	C₂×D₄	—	94	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	94	1	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	188	2	no	—
229	C₂²×C₄	—	188	2	no	—
230	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	#10625352	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	M₁₆	—	188	8	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	94	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	4	no	—
241	C₂×C₈	#10625334	1	4	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	47	2	no	—
249	C₄⋊C₄	—	94	1	no	—
250	C₂²⋊C₄	—	94	2	no	—
251	C₂²⋊C₄	—	94	2	no	—
252	C₄²	D₄×D₄₇	1	1	yes	characteristic
253	C₄²	#10625352	1	1	yes	characteristic
254	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₄²	C₂³×D₄₇	1	1	yes	characteristic
256	C₂³	—	94	2	no	—
257	C₂³	—	188	2	no	—
258	C₂³	—	188	2	no	—
259	C₂³	—	188	2	no	—
260	C₂³	#13093995	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	94	4	no	—
265	D₄	—	188	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	#13093995	1	1	yes	characteristic
275	C₂×C₄	#13093995	1	1	yes	characteristic
276	C₂×C₄	#13093995	1	1	yes	characteristic
277	C₂×C₄	C₂×#10625334	1	2	yes	—
278	C₂×C₄	#13094050	1	2	yes	—
279	C₂×C₄	#13094056	1	4	yes	—
280	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	188	2	no	—
285	C₈	—	188	4	no	—
286	C₈	C₂×#10625334	1	4	yes	—
287	C₂²	—	376	4	no	—
288	C₂²	—	188	2	no	—
289	C₂²	—	188	2	no	—
290	C₂²	—	188	2	no	—
291	C₂²	#15689315	1	2	yes	—
292	C₂²	#15689378	1	2	yes	—
293	C₂²	#15689395	1	2	yes	—
294	C₂²	C₂×#13093995	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	4	no	—
298	C₄	—	188	2	no	—
299	C₄	#15689315	1	2	yes	—
300	C₄	#15689378	1	2	yes	—
301	C₄	C₂²×#10625334	1	1	yes	characteristic
302	C₄	C₂×#13094050	1	1	yes	characteristic
303	C₄	C₂×#13094056	1	2	yes	—
304	C₂	—	376	2	no	—
305	C₂	#20242874	1	4	yes	direct product factor
306	C₂	C₂×#15689315	1	1	yes	characteristic
307	C₂	C₂×#15689378	1	1	yes	characteristic
308	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×#1689
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. #20242874	C₂	direct (N × H)
4. C₂×#15688874	C₂	semidirect
5. C₂×#15688869	C₂	semidirect
6. C₂×C₄₇×#494	C₂	semidirect
18. #15688874	C₂²	semidirect
19. #15688869	C₂²	semidirect
24. C₄₇×#494	C₂²	semidirect
150. C₂×#494	D₄₇	semidirect
156. C₂×C₄₇	#1689	semidirect
167. #494	C₂×D₄₇	semidirect
180. C₄₇	C₂×#1689	semidirect
305. C₂	#20242874	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×#15689202	2
#20242874	2
C₂×#15688874	2
C₂×#15688869	2
C₂×C₄₇×#494	2
C₂×#1689	47

Subgroups — C2×#20242874 #26198782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242874 #26198782