Subgroups — C₂×#15689141 #20251931

Order 6016.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10625350	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689141	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13093952	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	—	2	1	no	—
7	C₂×#10625350	—	2	2	no	—
8	C₂×#10625350	—	2	1	no	—
9	C₂×#10625350	—	2	2	no	—
10	C₂×#10625350	C₄	1	2	yes	complemented
11	C₂×#10625350	C₄	1	4	yes	complemented
12	C₂×#10625350	C₂²	1	2	yes	—
13	C₂×#10625350	C₂²	1	4	yes	—
14	C₂²×Dic₉₄	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₈₈	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₄₇	—	2	1	no	—
17	C₂²×C₄×D₄₇	—	2	1	no	—
18	C₂×#10625329	—	2	1	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	C₂²	1	2	yes	complemented
21	C₂×#10625329	C₂²	1	4	yes	complemented
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	#13093952	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
25	#117×C₄₇	—	2	2	no	—
26	#117×C₄₇	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
29	C₂×C₄²×C₄₇	C₂²	1	2	yes	complemented
30	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×D₄₇	—	2	1	no	—
32	C₂²×C₄×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	D₄	1	2	yes	—
38	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	D₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₄₇	—	2	2	no	—
44	C₂×C₂²⋊D₄₇	—	2	4	no	—
45	C₂×C₂²⋊D₄₇	—	2	2	no	—
46	C₂×C₂²⋊D₄₇	—	2	4	no	—
47	C₂²×Dic₄₇	—	2	1	no	—
48	#10625350	—	2	4	no	—
49	#10625350	—	2	4	no	—
50	#10625350	—	2	4	no	—
51	#10625350	—	2	2	no	—
52	#10625350	—	2	2	no	—
53	#10625350	—	2	4	no	—
54	#10625350	—	2	4	no	—
55	#10625350	C₂×C₄	1	4	yes	—
56	#10625350	C₂×C₄	1	8	yes	complemented
57	#10625350	C₂³	1	4	yes	—
58	C₂×D₁₈₈	—	2	2	no	—
59	C₂×D₁₈₈	—	2	1	no	—
60	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
61	C₂×D₁₈₈	C₂×C₄	1	4	yes	complemented
62	C₂×C₄×D₄₇	—	2	4	no	—
63	C₂×C₄×D₄₇	—	2	2	no	—
64	C₂×C₄×D₄₇	—	2	2	no	—
65	C₂×C₄×D₄₇	—	2	4	no	—
66	C₂×Dic₉₄	—	2	2	no	—
67	C₂×Dic₉₄	—	2	1	no	—
68	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
69	C₂×Dic₉₄	C₂×C₄	1	4	yes	complemented
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₂²⋊C₄×C₄₇	—	2	4	no	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	C₂³	1	4	yes	complemented
74	#10625329	—	2	4	no	—
75	#10625329	—	2	2	no	—
76	#10625329	C₂³	1	4	yes	complemented
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂³×C₄₇	—	4	2	no	—
80	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×D₄₇	—	2	2	no	—
86	C₂²×D₄₇	—	2	4	no	—
87	C₂²×D₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊D₄₇	—	2	4	no	—
109	C₂²⋊D₄₇	—	2	8	no	—
110	C₂²⋊D₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	D₁₈₈	—	2	2	no	—
114	D₁₈₈	—	2	4	no	—
115	D₁₈₈	C₂²×C₄	1	4	yes	—
116	C₄×D₄₇	—	2	4	no	—
117	C₄×D₄₇	—	2	8	no	—
118	C₄×D₄₇	—	2	4	no	—
119	Dic₉₄	—	2	2	no	—
120	Dic₉₄	—	2	4	no	—
121	Dic₉₄	C₂²×C₄	1	4	yes	—
122	C₄₇⋊C₈	—	2	4	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	4	2	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	4	8	no	—
135	C₂×D₄₇	—	4	4	no	—
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
139	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
140	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	Dic₄₇	—	2	4	no	—
144	C₂×#421	—	47	1	no	maximal, Sylow
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	#421	1	2	yes	complemented
150	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	D₄₇	—	4	4	no	—
152	C₂²×#41	—	47	1	no	—
153	C₂²×M₁₆	—	47	1	no	—
154	C₂×#117	D₄₇	1	1	yes	characteristic, complemented
155	#421	—	47	8	no	—
156	C₂×#104	—	47	4	no	—
157	C₄₇	C₂×#421	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	47	2	no	—
159	C₂×#41	—	47	4	no	—
160	C₂×#41	—	47	2	no	—
161	C₂×#41	—	47	4	no	—
162	C₂×#41	—	94	2	no	—
163	C₂×#41	—	94	1	no	—
164	C₂×#41	—	94	1	no	—
165	C₂×#41	—	94	2	no	—
166	C₂²×Q₈	—	47	1	no	—
167	C₂²×D₄	—	47	1	no	—
168	C₂²×D₄	—	94	1	no	—
169	C₂³×C₄	—	94	1	no	—
170	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
171	C₂×M₁₆	—	47	2	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	1	no	—
175	C₂²×C₈	—	94	1	no	—
176	#117	—	2	2	no	—
177	#117	C₂×D₄₇	1	4	yes	complemented
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂×D₄₇	1	2	yes	complemented
181	#104	—	47	16	no	—
182	C₂⁴	—	94	1	no	—
183	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
184	#41	—	47	4	no	—
185	#41	—	47	8	no	—
186	#41	—	47	4	no	—
187	#41	—	94	4	no	—
188	#41	—	94	4	no	—
189	#41	—	94	4	no	—
190	#41	—	94	4	no	—
191	#41	—	94	2	no	—
192	#41	—	94	2	no	—
193	#41	—	94	4	no	—
194	C₂×Q₈	—	47	2	no	—
195	C₂×Q₈	—	47	4	no	—
196	C₂×Q₈	—	94	2	no	—
197	C₂×Q₈	—	94	1	no	—
198	C₂×D₄	—	47	2	no	—
199	C₂×D₄	—	47	4	no	—
200	C₂×D₄	—	94	2	no	—
201	C₂×D₄	—	94	4	no	—
202	C₂×D₄	—	94	2	no	—
203	C₂×D₄	—	94	4	no	—
204	C₂×D₄	—	94	2	no	—
205	C₂×D₄	—	94	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	94	1	no	—
208	C₂²×C₄	—	94	4	no	—
209	C₂²×C₄	—	94	2	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₄×D₄₇	1	2	yes	—
215	C₂²×C₄	D₁₈₈	1	2	yes	—
216	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
217	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
220	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
221	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
222	M₁₆	—	47	4	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	94	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂²×D₄₇	1	4	yes	complemented
231	C₂³	—	4	2	no	—
232	C₂³	—	94	4	no	—
233	C₂³	—	94	2	no	—
234	C₂³	—	188	4	no	—
235	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
237	C₂³	(C₂×D₄₇)⋊C₄	1	2	yes	—
238	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
240	Q₈	—	47	4	no	—
241	Q₈	—	94	2	no	—
242	Q₈	—	94	4	no	—
243	D₄	—	47	4	no	—
244	D₄	—	94	4	no	—
245	D₄	—	94	8	no	—
246	D₄	—	94	4	no	—
247	D₄	—	94	2	no	—
248	D₄	—	94	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	8	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
258	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
259	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
261	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
262	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
264	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
266	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
267	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
268	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
269	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
274	C₈	—	94	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	94	4	no	—
281	C₂²	—	188	8	no	—
282	C₂²	—	188	4	no	—
283	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
284	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
285	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
286	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
287	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₄	—	4	4	no	—
289	C₄	—	94	4	no	—
290	C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
291	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
292	C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
293	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
294	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	188	4	no	—
301	C₂	#15689141	1	2	yes	direct product factor
302	C₂	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#421
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂²×#10625329	C₂	semidirect
3. #15689141	C₂	direct (N × H)
4. C₂×#13093952	C₂	semidirect
5. C₂×#117×C₄₇	C₂	semidirect
10. C₂×#10625350	C₄	semidirect
11. C₂×#10625350	C₄	semidirect
14. C₂²×Dic₉₄	C₄	semidirect
15. C₂²×D₁₈₈	C₄	semidirect
20. C₂×#10625329	C₂²	semidirect
21. C₂×#10625329	C₂²	semidirect
23. #13093952	C₂²	semidirect
26. #117×C₄₇	C₂²	semidirect
29. C₂×C₄²×C₄₇	C₂²	semidirect
56. #10625350	C₂×C₄	semidirect
61. C₂×D₁₈₈	C₂×C₄	semidirect
69. C₂×Dic₉₄	C₂×C₄	semidirect
73. C₄²×C₄₇	C₂³	semidirect
76. #10625329	C₂³	semidirect
149. C₂×C₄₇	#421	semidirect
154. C₂×#117	D₄₇	semidirect
157. C₄₇	C₂×#421	semidirect
177. #117	C₂×D₄₇	semidirect
180. C₂×C₄²	C₂×D₄₇	semidirect
230. C₄²	C₂²×D₄₇	semidirect
301. C₂	#15689141	direct (N × H)

Subgroups — C2×#15689141 #20251931

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689141 #20251931