Subgroups — C₂×D₃×Q₈×D₇ #13062254

Order 1344.

1700 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²	C₂²×D₃×D₇
commutator, cosocle	C₂×C₃×C₇	C₂⁵
Frattini	C₂	C₂³×D₃×D₇
Fitting	C₂×C₃×C₇×Q₈	C₂²
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂²×D₃×D₇	C₂²
hypercenter	C₂×Q₈	D₃×D₇
hypocenter	C₃×C₇	C₂³×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₂₁	C₂	semidirect
2. C₂×D₃×C₇×Q₈	C₂	semidirect
3. C₂×C₃×Q₈×D₇	C₂	semidirect
4. D₃×Q₈×D₇	C₂	direct (N × H)
6. C₂×D₂₁⋊Q₈	C₂	semidirect
7. C₂×D₃×Dic₁₄	C₂	semidirect
8. C₂×D₇×Dic₆	C₂	semidirect
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
12. Q₈×D₂₁	C₂²	semidirect
14. C₂×Dic₄₂	C₂²	semidirect
15. D₃×C₇×Q₈	C₂²	semidirect
17. C₂×C₇×Dic₆	C₂²	semidirect
18. C₃×Q₈×D₇	C₂²	semidirect
20. C₂×C₃×Dic₁₄	C₂²	semidirect
21. C₂×#4248	C₂²	semidirect
26. D₂₁⋊Q₈	C₂²	semidirect
27. D₃×Dic₁₄	C₂²	semidirect
28. D₇×Dic₆	C₂²	semidirect
34. C₂×Q₈×D₇	D₃	direct (N × H)
41. C₂×D₃×D₇	Q₈	direct (N × H)
43. C₃×C₇×Q₈	C₂³	semidirect
47. Dic₄₂	C₂³	semidirect
50. C₇×Dic₆	C₂³	semidirect
53. C₃×Dic₁₄	C₂³	semidirect
54. #4248	C₂³	semidirect
61. C₂×C₇×Q₈	C₂×D₃	semidirect
67. Q₈×D₇	C₂×D₃	direct (N × H)
76. C₂×Dic₁₄	C₂×D₃	semidirect
82. C₂×D₃×Q₈	D₇	direct (N × H)
86. C₂×D₂₁	C₂×Q₈	semidirect
88. C₂×D₃×C₇	C₂×Q₈	semidirect
90. C₂×C₃×D₇	C₂×Q₈	semidirect
92. D₃×D₇	C₂×Q₈	direct (N × H)
104. C₇×Q₈	C₂²×D₃	semidirect
117. Dic₁₄	C₂²×D₃	semidirect
121. C₂×C₃×Q₈	C₂×D₇	semidirect
127. D₃×Q₈	C₂×D₇	direct (N × H)
136. C₂×Dic₆	C₂×D₇	semidirect
137. C₂×C₃×C₇	C₂²×Q₈	semidirect
139. D₂₁	C₂²×Q₈	semidirect
140. C₃×D₇	C₂²×Q₈	semidirect
141. D₃×C₇	C₂²×Q₈	semidirect
156. C₂×D₇	D₃×Q₈	direct (N × H)
168. C₃×Q₈	C₂²×D₇	semidirect
181. Dic₆	C₂²×D₇	semidirect
182. C₃×C₇	C₂³×Q₈	semidirect
194. C₂×Q₈	D₃×D₇	direct (N × H)
203. C₂×C₇	C₂×D₃×Q₈	semidirect
206. D₇	C₂×D₃×Q₈	direct (N × H)
213. C₂×D₃	Q₈×D₇	direct (N × H)
228. Q₈	C₂×D₃×D₇	direct (N × H)
239. C₇	C₂²×D₃×Q₈	semidirect
241. C₂×C₃	C₂×Q₈×D₇	semidirect
244. D₃	C₂×Q₈×D₇	direct (N × H)
257. C₃	C₂²×Q₈×D₇	semidirect
261. C₂	D₃×Q₈×D₇	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×Q₈×D₂₁	2
C₂×D₃×C₇×Q₈	2
C₂×C₃×Q₈×D₇	2
D₃×Q₈×D₇	2
C₂×C₄×D₃×D₇	2
C₂×D₂₁⋊Q₈	2
C₂×D₃×Dic₁₄	2
C₂×D₇×Dic₆	2
C₂²×Q₈×D₇	3
C₂²×D₃×Q₈	7

Subgroups — C2×D3×Q8×D7 #13062254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×Q₈×D₇ #13062254