Subgroups — Dic₆×C₂²⋊D₇ #13059514

Order 1344.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621407	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
5	#10621137	C₂	1	2	yes	maximal, complemented
6	Dic₃×C₂²⋊D₇	C₂	1	2	yes	maximal
7	#10620983	C₂	1	2	yes	maximal, complemented
8	#10620938	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620921	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620891	C₂	1	2	yes	maximal, complemented
12	Q₈×C₂²⋊D₇	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
15	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
16	C₂×C₇×Dic₆	—	2	4	no	—
17	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
18	C₂×C₇×Dic₆	C₂²	1	2	yes	—
19	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊D₇	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
22	C₂×#4248	C₂²	1	2	yes	—
23	C₂×Dic₃×D₇	C₂²	1	2	yes	—
24	D₇×Dic₆	—	2	4	no	—
25	#66636	C₂²	1	2	yes	—
26	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
27	#66629	C₂²	1	2	yes	—
28	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	Dic₃⋊Dic₇	C₂²	1	2	yes	—
33	#66577	C₂²	1	2	yes	—
34	Dic₇⋊Dic₃	C₂²	1	2	yes	—
35	Dic₇⋊Dic₃	C₂²	1	2	yes	—
36	(C₂×D₇)⋊Dic₃	C₂²	1	2	yes	—
37	Dic₃×Dic₇	C₂²	1	2	yes	—
38	C₂²×C₇×Q₈	—	3	1	no	—
39	C₂×Q₈×D₇	—	3	1	no	—
40	#6509	—	3	1	no	—
41	#6509	—	3	2	no	—
42	Q₈×Dic₇	—	3	1	no	—
43	#6507	—	3	2	no	—
44	#6507	—	3	1	no	—
45	C₄×C₂²⋊D₇	—	3	2	no	—
46	C₄×C₂²⋊D₇	D₃	1	1	yes	characteristic
47	#6487	—	3	1	no	—
48	#6487	—	3	2	no	—
49	D₄×Dic₆	—	7	1	no	maximal, Hall
50	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	—	2	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₂×Dic₂₁	C₂³	1	2	yes	—
57	Dic₄₂	—	2	2	no	—
58	C₂×C₇×Dic₃	—	2	4	no	—
59	C₂×C₇×Dic₃	C₂³	1	2	yes	—
60	C₂×C₇×Dic₃	C₂³	1	2	yes	—
61	C₇×Dic₆	—	2	2	no	—
62	C₇×Dic₆	—	2	4	no	—
63	C₇×Dic₆	D₄	1	4	yes	complemented
64	C₃×C₂²⋊D₇	Q₈	1	4	yes	complemented
65	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
66	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
68	C₃×C₄×D₇	—	2	2	no	—
69	#4248	—	2	4	no	—
70	Dic₃×D₇	—	2	4	no	—
71	C₂×C₇×Q₈	—	3	1	no	—
72	C₂×C₇×Q₈	—	3	2	no	—
73	C₂×C₇×Q₈	—	3	1	no	—
74	C₂×C₇×Q₈	—	6	4	no	—
75	C₂²×C₄×C₇	—	3	2	no	—
76	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
77	C₂×C₂²⋊D₇	C₂×D₃	1	1	yes	characteristic
78	Q₈×D₇	—	6	4	no	—
79	C₂×C₄×D₇	—	3	2	no	—
80	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
81	C₂×Dic₁₄	—	3	1	no	—
82	C₂×Dic₁₄	—	3	2	no	—
83	C₂²⋊Dic₇	—	3	2	no	—
84	C₂²⋊Dic₇	C₂×D₃	1	1	yes	characteristic
85	(C₂×D₇)⋊C₄	—	3	2	no	—
86	(C₂×D₇)⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₄⋊Dic₇	—	3	1	no	—
88	C₄⋊Dic₇	—	3	2	no	—
89	Dic₇⋊C₄	—	3	2	no	—
90	Dic₇⋊C₄	—	3	2	no	—
91	Dic₇⋊C₄	—	3	2	no	—
92	Dic₇⋊C₄	—	3	2	no	—
93	Dic₇⋊C₄	C₂×D₃	1	1	yes	characteristic
94	C₄×Dic₇	—	3	2	no	—
95	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
96	C₂²×Dic₆	—	7	1	no	—
97	C₂²×Dic₆	D₇	1	1	yes	characteristic, complemented
98	C₃×C₄×D₄	—	7	1	no	—
99	D₄×Dic₃	—	7	2	no	—
100	#924	—	7	1	no	—
101	#924	—	7	1	no	—
102	#888	—	7	2	no	—
103	#878	—	7	2	no	—
104	#878	—	7	2	no	—
105	#869	—	7	1	no	—
106	C₄×Dic₆	—	7	1	no	—
107	C₂²×C₃×C₇	—	2	1	no	—
108	C₂²×C₃×C₇	—	2	1	no	—
109	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	complemented
110	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
111	C₂×C₃×D₇	—	2	1	no	—
112	C₂×C₃×D₇	—	2	1	no	—
113	C₂×C₃×D₇	C₂×Q₈	1	2	yes	complemented
114	C₃×C₄×C₇	—	2	1	no	—
115	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
116	Dic₂₁	—	2	2	no	—
117	Dic₂₁	—	2	2	no	—
118	C₃×Dic₇	—	2	1	no	—
119	C₃×Dic₇	—	2	1	no	—
120	C₃×Dic₇	C₂×Q₈	1	2	yes	complemented
121	C₇×Dic₃	—	2	2	no	—
122	C₇×Dic₃	C₂×D₄	1	4	yes	—
123	D₄×Q₈	—	21	1	no	Sylow
124	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
126	C₇×Q₈	—	3	4	no	—
127	C₇×Q₈	—	6	4	no	—
128	C₇×Q₈	—	6	2	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	3	2	no	—
131	C₂×C₄×C₇	—	3	2	no	—
132	C₂×C₄×C₇	—	6	4	no	—
133	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
135	C₂²⋊D₇	Dic₆	1	4	yes	direct product factor
136	C₂×Dic₇	—	3	2	no	—
137	C₂×Dic₇	—	3	2	no	—
138	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
139	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
140	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
141	C₄×D₇	—	2	2	no	—
142	C₄×D₇	—	6	4	no	—
143	Dic₁₄	—	6	4	no	—
144	Dic₁₄	—	6	2	no	—
145	C₂×C₃×D₄	—	7	1	no	—
146	C₂²×C₃×C₄	—	7	1	no	—
147	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
148	C₂²×Dic₃	—	7	2	no	—
149	C₂²×Dic₃	C₂×D₇	1	2	yes	—
150	C₂×Dic₆	—	2	4	no	—
151	C₂×Dic₆	—	7	1	no	—
152	C₂×Dic₆	—	7	2	no	—
153	C₂×Dic₆	—	14	4	no	—
154	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
155	C₂×Dic₆	C₂×D₇	1	2	yes	—
156	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	7	1	no	—
158	C₃×C₂²⋊C₄	—	7	1	no	—
159	C₃×C₂²⋊C₄	—	7	1	no	—
160	C₃×C₄²	—	7	1	no	—
161	#217	—	7	2	no	—
162	#217	—	7	2	no	—
163	C₄⋊Dic₃	—	7	1	no	—
164	C₄⋊Dic₃	—	7	2	no	—
165	C₄⋊Dic₃	—	7	2	no	—
166	Dic₃⋊C₄	—	7	2	no	—
167	Dic₃⋊C₄	—	7	2	no	—
168	Dic₃⋊C₄	—	7	2	no	—
169	C₄×Dic₃	—	7	2	no	—
170	C₂×C₃×C₇	—	2	2	no	—
171	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
174	C₃×D₇	—	2	2	no	—
175	C₂²×Q₈	—	3	1	no	—
176	C₂²×Q₈	—	21	1	no	—
177	C₄⋊Q₈	—	21	1	no	—
178	C₄⋊Q₈	—	21	2	no	—
179	C₂²⋊Q₈	—	21	2	no	—
180	C₂²⋊Q₈	—	21	2	no	—
181	C₂²⋊Q₈	—	21	1	no	—
182	C₂²⋊Q₈	—	21	1	no	—
183	C₄×Q₈	—	21	1	no	—
184	C₄×D₄	—	7	1	no	—
185	C₄×D₄	—	21	2	no	—
186	C₂²×C₇	—	2	1	no	—
187	C₂²×C₇	—	2	1	no	—
188	C₂²×C₇	C₂×Dic₆	1	2	yes	complemented
189	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
190	C₂×D₇	—	2	1	no	—
191	C₂×D₇	—	2	1	no	—
192	C₂×D₇	C₂×Dic₆	1	2	yes	complemented
193	C₄×C₇	—	2	1	no	—
194	C₄×C₇	—	3	4	no	—
195	C₄×C₇	—	6	2	no	—
196	C₄×C₇	D₃×D₄	1	2	yes	—
197	Dic₇	—	2	1	no	—
198	Dic₇	—	2	1	no	—
199	Dic₇	—	6	2	no	—
200	Dic₇	—	6	2	no	—
201	Dic₇	C₂×Dic₆	1	2	yes	complemented
202	C₂³×C₃	—	7	1	no	—
203	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
204	C₃×D₄	—	7	4	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	7	1	no	—
207	C₂×C₃×C₄	—	7	1	no	—
208	C₂×C₃×C₄	—	7	1	no	—
209	C₂×C₃×C₄	—	14	2	no	—
210	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	7	2	no	—
214	C₂×Dic₃	—	7	2	no	—
215	C₂×Dic₃	—	14	4	no	—
216	C₂×Dic₃	C₂²×D₇	1	2	yes	—
217	C₂×Dic₃	C₂²×D₇	1	2	yes	—
218	Dic₆	—	2	2	no	—
219	Dic₆	—	2	4	no	—
220	Dic₆	—	14	2	no	—
221	Dic₆	—	14	4	no	—
222	Dic₆	C₂²⋊D₇	1	4	yes	direct product factor
223	C₃×C₇	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
224	C₂×Q₈	—	3	1	no	—
225	C₂×Q₈	—	3	2	no	—
226	C₂×Q₈	—	3	1	no	—
227	C₂×Q₈	—	6	4	no	—
228	C₂×Q₈	—	21	2	no	—
229	C₂×Q₈	—	21	1	no	—
230	C₂×Q₈	—	42	4	no	—
231	C₂×D₄	—	7	1	no	—
232	C₂²×C₄	—	3	2	no	—
233	C₂²×C₄	—	7	1	no	—
234	C₂²×C₄	—	21	2	no	—
235	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
236	C₄⋊C₄	—	7	1	no	—
237	C₄⋊C₄	—	21	2	no	—
238	C₄⋊C₄	—	21	2	no	—
239	C₄⋊C₄	—	21	2	no	—
240	C₄⋊C₄	—	21	2	no	—
241	C₄⋊C₄	—	21	2	no	—
242	C₄⋊C₄	—	21	1	no	—
243	C₂²⋊C₄	—	7	1	no	—
244	C₂²⋊C₄	—	7	1	no	—
245	C₂²⋊C₄	—	21	2	no	—
246	C₂²⋊C₄	—	21	2	no	—
247	C₄²	—	7	1	no	—
248	C₄²	—	21	2	no	—
249	C₂×C₇	—	2	2	no	—
250	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
251	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
252	C₂×C₇	#1010	1	1	yes	characteristic
253	D₇	—	2	2	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	7	2	no	—
257	C₂²×C₃	—	14	1	no	—
258	C₂²×C₃	—	14	1	no	—
259	C₂²×C₃	Q₈×D₇	1	2	yes	complemented
260	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	7	2	no	—
263	C₃×C₄	—	14	1	no	—
264	C₃×C₄	—	14	1	no	—
265	C₃×C₄	C₂×C₂²⋊D₇	1	2	yes	—
266	Dic₃	—	2	2	no	—
267	Dic₃	—	14	2	no	—
268	Dic₃	—	14	2	no	—
269	Dic₃	C₂×C₂²⋊D₇	1	4	yes	—
270	C₂³	—	7	1	no	—
271	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
272	Q₈	—	3	4	no	—
273	Q₈	—	6	4	no	—
274	Q₈	—	6	2	no	—
275	Q₈	—	42	4	no	—
276	Q₈	—	42	2	no	—
277	D₄	—	7	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	3	2	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	7	1	no	—
283	C₂×C₄	—	7	1	no	—
284	C₂×C₄	—	7	1	no	—
285	C₂×C₄	—	14	2	no	—
286	C₂×C₄	—	21	2	no	—
287	C₂×C₄	—	21	2	no	—
288	C₂×C₄	—	42	4	no	—
289	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
290	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
291	C₇	D₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	—	14	2	no	—
294	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
295	C₂×C₃	#6551	1	1	yes	characteristic
296	C₂×C₃	C₂²×C₂²⋊D₇	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	7	2	no	—
300	C₂²	—	14	1	no	—
301	C₂²	—	14	1	no	—
302	C₂²	D₇×Dic₆	1	2	yes	complemented
303	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	3	4	no	—
306	C₄	—	6	2	no	—
307	C₄	—	7	2	no	—
308	C₄	—	14	1	no	—
309	C₄	—	14	1	no	—
310	C₄	—	42	2	no	—
311	C₄	—	42	2	no	—
312	C₄	D₃×C₂²⋊D₇	1	2	yes	—
313	C₃	Q₈×C₂²⋊D₇	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	14	2	no	—
316	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
317	C₂	#10621547	1	1	yes	characteristic
318	C₂	C₂×D₃×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₂²×C₃×C₄×C₇	C₂²
prime radical	C₂×C₃×C₂²⋊D₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	D₄×Q₈
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×Dic₆	C₂	semidirect
2. C₂×D₇×Dic₆	C₂	semidirect
3. #10621407	C₂	semidirect
5. #10621137	C₂	semidirect
7. #10620983	C₂	semidirect
8. #10620938	C₂	semidirect
9. #10620921	C₂	semidirect
10. Dic₆×Dic₇	C₂	semidirect
11. #10620891	C₂	semidirect
63. C₇×Dic₆	D₄	semidirect
64. C₃×C₂²⋊D₇	Q₈	semidirect
97. C₂²×Dic₆	D₇	semidirect
109. C₂²×C₃×C₇	C₂×Q₈	semidirect
113. C₂×C₃×D₇	C₂×Q₈	semidirect
120. C₃×Dic₇	C₂×Q₈	semidirect
135. C₂²⋊D₇	Dic₆	direct (N × H)
188. C₂²×C₇	C₂×Dic₆	semidirect
192. C₂×D₇	C₂×Dic₆	semidirect
201. Dic₇	C₂×Dic₆	semidirect
222. Dic₆	C₂²⋊D₇	direct (N × H)
223. C₃×C₇	D₄×Q₈	semidirect
259. C₂²×C₃	Q₈×D₇	semidirect
291. C₇	D₄×Dic₆	semidirect
302. C₂²	D₇×Dic₆	semidirect
313. C₃	Q₈×C₂²⋊D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×Dic₆	2
C₂×D₇×Dic₆	2
#10621407	2
C₃×C₄×C₂²⋊D₇	2
#10621137	2
Dic₃×C₂²⋊D₇	2
#10620983	2
#10620938	2
#10620921	2
Dic₆×Dic₇	2
#10620891	2
Q₈×C₂²⋊D₇	3
D₄×Dic₆	7

Subgroups — Dic6×C22⋊D7 #13059514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆×C₂²⋊D₇ #13059514