Subgroups — D₃×#6551 #13062260

Order 1344.

1588 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621660	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6551	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×#1161	C₂	1	4	yes	maximal, complemented
5	D₃×Q₈×D₇	C₂	1	4	yes	maximal, complemented
6	#10621576	C₂	1	4	yes	maximal, complemented
7	#10621575	C₂	1	4	yes	maximal, complemented
8	D₃×#1157	C₂	1	6	yes	maximal, complemented
9	#10621547	C₂	1	6	yes	maximal, complemented
10	C₂×#6551	—	3	1	no	maximal, Hall
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
12	#66732	C₂²	1	4	yes	complemented
13	Q₈×D₂₁	C₂²	1	4	yes	complemented
14	#66728	C₂²	1	6	yes	complemented
15	D₃×C₇×Q₈	C₂²	1	4	yes	—
16	D₃×C₇×Q₈	C₂²	1	4	yes	complemented
17	C₂×C₄×D₃×C₇	C₂²	1	3	yes	—
18	C₂×C₇×Dic₆	C₂²	1	3	yes	complemented
19	C₃×#1161	C₂²	1	4	yes	complemented
20	C₃×Q₈×D₇	C₂²	1	4	yes	complemented
21	C₃×#1157	C₂²	1	6	yes	complemented
22	D₃×C₂²⋊D₇	C₂²	1	4	yes	—
23	#66683	C₂²	1	12	yes	complemented
24	D₃×D₂₈	C₂²	1	6	yes	—
25	C₄×D₃×D₇	C₂²	1	12	yes	—
26	#66677	C₂²	1	12	yes	complemented
27	#66675	C₂²	1	12	yes	complemented
28	D₂₁⋊Q₈	C₂²	1	12	yes	complemented
29	#66673	C₂²	1	6	yes	complemented
30	D₃×Dic₁₄	C₂²	1	6	yes	—
31	#66670	C₂²	1	12	yes	complemented
32	#66669	C₂²	1	6	yes	complemented
33	D₇×Dic₆	C₂²	1	12	yes	complemented
34	C₂²×C₇×Q₈	—	3	1	no	—
35	#6551	—	3	1	no	—
36	#6551	—	3	4	no	—
37	#6551	—	3	4	no	—
38	#6551	—	3	6	no	—
39	#6551	D₃	1	1	yes	characteristic, direct product factor
40	C₂×#1161	—	3	4	no	—
41	C₂×Q₈×D₇	—	3	4	no	—
42	C₂×#1157	—	3	6	no	—
43	D₃×2₋¹⁺⁴	—	7	1	no	maximal, Hall
44	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
45	C₂×D₃×D₇	C₂³	1	4	yes	—
46	C₃×C₇×Q₈	C₂³	1	4	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
48	#4268	C₂³	1	4	yes	—
49	D₈₄	C₂³	1	6	yes	—
50	C₄×D₂₁	C₂³	1	12	yes	—
51	Dic₄₂	C₂³	1	6	yes	—
52	C₂×C₇×Dic₃	C₂³	1	3	yes	—
53	C₄×D₃×C₇	C₂³	1	6	yes	—
54	C₄×D₃×C₇	C₂³	1	6	yes	—
55	C₇×Dic₆	C₂³	1	12	yes	—
56	C₃×C₂²⋊D₇	C₂³	1	4	yes	—
57	C₃×D₂₈	C₂³	1	6	yes	—
58	C₃×C₄×D₇	C₂³	1	12	yes	—
59	C₃×Dic₁₄	C₂³	1	6	yes	—
60	#4248	C₂³	1	12	yes	—
61	#4247	C₂³	1	4	yes	—
62	#4246	C₂³	1	12	yes	—
63	#4245	C₂³	1	4	yes	—
64	D₂₁⋊C₄	C₂³	1	12	yes	—
65	D₃×Dic₇	C₂³	1	4	yes	—
66	Dic₃×D₇	C₂³	1	12	yes	—
67	C₂×C₇×Q₈	—	3	4	no	—
68	C₂×C₇×Q₈	—	3	4	no	—
69	C₂×C₇×Q₈	—	3	3	no	—
70	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
71	C₂²×C₄×C₇	—	3	3	no	—
72	C₂×C₂²⋊D₇	—	3	4	no	—
73	#1161	—	3	4	no	—
74	#1161	—	3	12	no	—
75	#1161	—	3	12	no	—
76	#1161	C₂×D₃	1	4	yes	complemented
77	Q₈×D₇	—	3	4	no	—
78	Q₈×D₇	—	3	12	no	—
79	Q₈×D₇	—	3	12	no	—
80	Q₈×D₇	C₂×D₃	1	4	yes	complemented
81	#1157	—	3	6	no	—
82	#1157	—	3	12	no	—
83	#1157	—	3	6	no	—
84	#1157	—	3	12	no	—
85	#1157	—	3	6	no	—
86	#1157	C₂×D₃	1	6	yes	complemented
87	C₂×D₂₈	—	3	6	no	—
88	C₂×C₄×D₇	—	3	12	no	—
89	C₂×Dic₁₄	—	3	6	no	—
90	C₃×2₋¹⁺⁴	—	7	1	no	—
91	#1010	—	7	4	no	—
92	#1010	—	7	6	no	—
93	D₃×#41	—	7	4	no	—
94	D₃×#41	—	7	6	no	—
95	#1007	—	7	1	no	—
96	#1007	—	7	4	no	—
97	C₂×D₃×Q₈	—	7	4	no	—
98	C₂×D₃×Q₈	D₇	1	1	yes	characteristic, complemented
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×D₂₁	C₂⁴	1	4	yes	—
101	C₂×D₃×C₇	—	2	2	no	—
102	C₂×D₃×C₇	C₂⁴	1	1	yes	characteristic
103	C₂×D₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₃×D₇	C₂⁴	1	4	yes	—
105	D₃×D₇	—	2	8	no	—
106	C₃×C₄×C₇	C₂⁴	1	6	yes	—
107	Dic₂₁	C₂⁴	1	4	yes	—
108	C₃×Dic₇	C₂⁴	1	4	yes	—
109	C₇×Dic₃	C₂⁴	1	6	yes	—
110	C₂×2₋¹⁺⁴	—	21	1	no	Sylow
111	C₂³×C₇	—	3	1	no	—
112	C₂²×D₇	—	3	4	no	—
113	C₇×Q₈	—	3	12	no	—
114	C₇×Q₈	C₂²×D₃	1	4	yes	—
115	C₂×C₄×C₇	—	3	3	no	—
116	C₂×C₄×C₇	—	3	6	no	—
117	C₂×C₄×C₇	—	3	6	no	—
118	C₂×C₄×C₇	C₂²×D₃	1	3	yes	—
119	C₂²⋊D₇	—	3	4	no	—
120	C₂²⋊D₇	—	3	4	no	—
121	C₂²⋊D₇	—	3	4	no	—
122	C₂²⋊D₇	C₂²×D₃	1	4	yes	—
123	C₂×Dic₇	—	3	4	no	—
124	D₂₈	—	3	12	no	—
125	D₂₈	—	3	6	no	—
126	D₂₈	C₂²×D₃	1	6	yes	—
127	C₄×D₇	—	3	12	no	—
128	C₄×D₇	—	3	12	no	—
129	C₄×D₇	—	3	12	no	—
130	C₄×D₇	C₂²×D₃	1	12	yes	—
131	Dic₁₄	—	3	6	no	—
132	Dic₁₄	—	3	12	no	—
133	Dic₁₄	C₂²×D₃	1	6	yes	—
134	C₃×#41	—	7	4	no	—
135	C₃×#41	—	7	6	no	—
136	C₂×C₃×Q₈	—	7	4	no	—
137	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
138	#239	—	7	4	no	—
139	#239	—	7	6	no	—
140	D₃×Q₈	—	7	4	no	—
141	D₃×Q₈	—	7	6	no	—
142	D₃×Q₈	—	7	12	no	—
143	D₃×Q₈	C₂×D₇	1	4	yes	—
144	D₃×Q₈	C₂×D₇	1	4	yes	complemented
145	#237	—	7	12	no	—
146	#237	—	7	6	no	—
147	#237	—	7	12	no	—
148	D₃×D₄	—	7	4	no	—
149	D₃×D₄	—	7	6	no	—
150	#235	—	7	6	no	—
151	#235	—	7	12	no	—
152	#235	—	7	12	no	—
153	C₂×C₄×D₃	—	7	12	no	—
154	C₂×C₄×D₃	C₂×D₇	1	3	yes	—
155	C₂×Dic₆	—	7	12	no	—
156	C₂×Dic₆	C₂×D₇	1	3	yes	complemented
157	C₂×C₃×C₇	—	2	1	no	—
158	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
159	D₂₁	—	2	4	no	—
160	C₃×D₇	—	2	4	no	—
161	D₃×C₇	—	2	1	no	—
162	D₃×C₇	2₋¹⁺⁴	1	2	yes	complemented
163	2₋¹⁺⁴	—	7	1	no	—
164	2₋¹⁺⁴	—	21	4	no	—
165	2₋¹⁺⁴	—	21	4	no	—
166	2₋¹⁺⁴	—	21	6	no	—
167	2₋¹⁺⁴	—	21	1	no	—
168	C₂×#41	—	21	6	no	—
169	C₂×#41	—	21	4	no	—
170	C₂²×Q₈	—	3	1	no	—
171	C₂²×Q₈	—	21	4	no	—
172	C₂²×C₇	—	3	1	no	—
173	C₂²×C₇	—	3	1	no	—
174	C₂²×C₇	—	6	2	no	—
175	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
176	C₂×D₇	—	3	4	no	—
177	C₂×D₇	—	6	8	no	—
178	C₂×D₇	C₂³×D₃	1	4	yes	—
179	C₄×C₇	—	3	6	no	—
180	C₄×C₇	C₂³×D₃	1	6	yes	—
181	Dic₇	—	3	4	no	—
182	Dic₇	C₂³×D₃	1	4	yes	—
183	C₂²×D₃	—	7	4	no	—
184	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
185	C₃×Q₈	—	7	6	no	—
186	C₃×Q₈	C₂²×D₇	1	4	yes	—
187	C₃×D₄	—	7	4	no	—
188	C₃×D₄	—	7	6	no	—
189	C₂×C₃×C₄	—	7	12	no	—
190	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
191	#67	—	7	4	no	—
192	#67	—	7	4	no	—
193	#67	—	7	12	no	—
194	C₂×Dic₃	—	7	12	no	—
195	C₂×Dic₃	C₂²×D₇	1	3	yes	—
196	D₁₂	—	7	4	no	—
197	D₁₂	—	7	6	no	—
198	C₄×D₃	—	7	4	no	—
199	C₄×D₃	—	7	12	no	—
200	C₄×D₃	—	7	12	no	—
201	C₄×D₃	C₂²×D₇	1	6	yes	—
202	C₄×D₃	C₂²×D₇	1	6	yes	—
203	Dic₆	—	7	6	no	—
204	Dic₆	—	7	12	no	—
205	Dic₆	C₂²×D₇	1	12	yes	—
206	C₃×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
207	#41	—	7	4	no	—
208	#41	—	7	6	no	—
209	#41	—	21	12	no	—
210	#41	—	21	12	no	—
211	#41	—	21	6	no	—
212	#41	—	21	6	no	—
213	#41	—	21	12	no	—
214	#41	—	21	12	no	—
215	#41	—	21	4	no	—
216	#41	—	21	6	no	—
217	C₂×Q₈	—	3	4	no	—
218	C₂×Q₈	—	3	4	no	—
219	C₂×Q₈	—	3	3	no	—
220	C₂×Q₈	—	7	4	no	—
221	C₂×Q₈	—	21	6	no	—
222	C₂×Q₈	—	21	12	no	—
223	C₂×Q₈	—	21	12	no	—
224	C₂×Q₈	—	21	4	no	—
225	C₂×Q₈	D₃×D₇	1	1	yes	characteristic, complemented
226	C₂×D₄	—	21	6	no	—
227	C₂×D₄	—	21	4	no	—
228	C₂²×C₄	—	3	3	no	—
229	C₂²×C₄	—	21	12	no	—
230	C₂×C₇	—	2	1	no	—
231	C₂×C₇	—	3	2	no	—
232	C₂×C₇	—	6	1	no	—
233	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
234	D₇	—	2	4	no	—
235	D₇	—	6	4	no	—
236	C₂²×C₃	—	7	4	no	—
237	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
238	C₂×D₃	—	2	2	no	—
239	C₂×D₃	—	7	4	no	—
240	C₂×D₃	—	14	8	no	—
241	C₂×D₃	C₂³×D₇	1	1	yes	characteristic
242	C₂×D₃	C₂³×D₇	1	1	yes	characteristic
243	C₃×C₄	—	7	4	no	—
244	C₃×C₄	C₂³×D₇	1	6	yes	—
245	Dic₃	—	7	4	no	—
246	Dic₃	C₂³×D₇	1	6	yes	—
247	C₂³	—	3	1	no	—
248	C₂³	—	21	4	no	—
249	Q₈	—	3	12	no	—
250	Q₈	—	7	6	no	—
251	Q₈	—	21	12	no	—
252	Q₈	—	21	6	no	—
253	Q₈	C₂×D₃×D₇	1	4	yes	—
254	D₄	—	7	4	no	—
255	D₄	—	7	6	no	—
256	D₄	—	21	12	no	—
257	D₄	—	21	4	no	—
258	D₄	—	21	4	no	—
259	D₄	—	21	4	no	—
260	D₄	—	21	6	no	—
261	C₂×C₄	—	3	3	no	—
262	C₂×C₄	—	3	6	no	—
263	C₂×C₄	—	3	6	no	—
264	C₂×C₄	—	7	12	no	—
265	C₂×C₄	—	21	12	no	—
266	C₂×C₄	—	21	12	no	—
267	C₂×C₄	—	21	4	no	—
268	C₂×C₄	—	21	12	no	—
269	C₂×C₄	C₂×D₃×D₇	1	3	yes	—
270	C₇	D₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	14	4	no	—
273	C₂×C₃	C₂⁴×D₇	1	1	yes	characteristic
274	D₃	—	2	1	no	—
275	D₃	—	14	4	no	—
276	D₃	#6551	1	2	yes	direct product factor
277	C₂²	—	3	1	no	—
278	C₂²	—	3	1	no	—
279	C₂²	—	6	2	no	—
280	C₂²	—	7	4	no	—
281	C₂²	—	21	4	no	—
282	C₂²	—	42	8	no	—
283	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
284	C₄	—	3	6	no	—
285	C₄	—	7	4	no	—
286	C₄	—	21	4	no	—
287	C₄	C₂²×D₃×D₇	1	6	yes	—
288	C₃	C₂×#6551	1	1	yes	characteristic, complemented, Sylow
289	C₂	—	2	1	no	—
290	C₂	—	3	2	no	—
291	C₂	—	6	1	no	—
292	C₂	—	14	4	no	—
293	C₂	—	42	4	no	—
294	C₂	C₂³×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂⁵
Fitting	C₂×C₃×C₇×Q₈	C₂²
hypercenter	C₂×Q₈	D₃×D₇
hypocenter	C₃×C₇	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #10621660	C₂	semidirect
2. C₂×D₃×C₇×Q₈	C₂	semidirect
3. C₃×#6551	C₂	semidirect
4. D₃×#1161	C₂	semidirect
5. D₃×Q₈×D₇	C₂	semidirect
6. #10621576	C₂	semidirect
7. #10621575	C₂	semidirect
8. D₃×#1157	C₂	semidirect
9. #10621547	C₂	semidirect
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
12. #66732	C₂²	semidirect
13. Q₈×D₂₁	C₂²	semidirect
14. #66728	C₂²	semidirect
16. D₃×C₇×Q₈	C₂²	semidirect
18. C₂×C₇×Dic₆	C₂²	semidirect
19. C₃×#1161	C₂²	semidirect
20. C₃×Q₈×D₇	C₂²	semidirect
21. C₃×#1157	C₂²	semidirect
23. #66683	C₂²	semidirect
26. #66677	C₂²	semidirect
27. #66675	C₂²	semidirect
28. D₂₁⋊Q₈	C₂²	semidirect
29. #66673	C₂²	semidirect
31. #66670	C₂²	semidirect
32. #66669	C₂²	semidirect
33. D₇×Dic₆	C₂²	semidirect
39. #6551	D₃	direct (N × H)
70. C₂×C₇×Q₈	C₂×D₃	semidirect
76. #1161	C₂×D₃	semidirect
80. Q₈×D₇	C₂×D₃	semidirect
86. #1157	C₂×D₃	semidirect
98. C₂×D₃×Q₈	D₇	semidirect
137. C₂×C₃×Q₈	C₂×D₇	semidirect
144. D₃×Q₈	C₂×D₇	semidirect
156. C₂×Dic₆	C₂×D₇	semidirect
162. D₃×C₇	2₋¹⁺⁴	semidirect
206. C₃×C₇	C₂×2₋¹⁺⁴	semidirect
225. C₂×Q₈	D₃×D₇	semidirect
270. C₇	D₃×2₋¹⁺⁴	semidirect
276. D₃	#6551	direct (N × H)
288. C₃	C₂×#6551	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10621660	2
C₂×D₃×C₇×Q₈	2
C₃×#6551	2
D₃×#1161	2
D₃×Q₈×D₇	2
#10621576	2
#10621575	2
D₃×#1157	2
#10621547	2
C₂×#6551	3
D₃×2₋¹⁺⁴	7

Subgroups — D3×#6551 #13062260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#6551 #13062260