Subgroups — C₄×D₇×Dic₆ #13057713

Order 1344.

596 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	C₂×D₇×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄²×D₇	C₂	1	1	yes	maximal, characteristic
5	C₄×#4248	C₂	1	2	yes	maximal, complemented
6	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
7	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal
8	C₄×Dic₃×D₇	C₂	1	2	yes	maximal
9	#10620895	C₂	1	1	yes	maximal, characteristic, complemented
10	Dic₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620885	C₂	1	2	yes	maximal, complemented
12	C₄×Q₈×D₇	—	3	1	no	maximal, Hall
13	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
14	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
18	C₂×#4248	C₂²	1	2	yes	—
19	C₂×Dic₃×D₇	C₂²	1	2	yes	—
20	C₂×Dic₃×D₇	C₂²	1	2	yes	—
21	D₇×Dic₆	C₄	1	8	yes	direct product factor
22	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
24	#66629	C₂²	1	2	yes	—
25	C₄×Dic₂₁	C₂²	1	2	yes	—
26	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₄×C₇×Dic₃	C₂²	1	2	yes	—
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₇	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	Dic₃×Dic₇	C₂²	1	2	yes	—
36	Dic₃×Dic₇	C₂²	1	2	yes	—
37	C₂×Q₈×D₇	—	3	1	no	—
38	C₄×C₇×Q₈	—	3	1	no	—
39	Q₈×Dic₇	—	3	1	no	—
40	D₇×C₄⋊C₄	—	3	2	no	—
41	D₇×C₄⋊C₄	—	3	1	no	—
42	#6450	—	3	2	no	—
43	#6450	—	3	1	no	—
44	C₄²×D₇	—	3	2	no	—
45	C₄²×D₇	D₃	1	1	yes	characteristic
46	C₄×Dic₁₄	—	3	1	no	—
47	C₄×Dic₁₄	—	3	2	no	—
48	C₂×C₄×Dic₆	—	7	1	no	maximal, Hall
49	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	Dic₄₂	C₂×C₄	1	4	yes	complemented
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₂×C₇×Dic₃	C₂³	1	2	yes	—
58	C₇×Dic₆	C₂×C₄	1	4	yes	complemented
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₇	—	2	2	no	—
63	C₃×C₄×D₇	C₂×C₄	1	4	yes	—
64	C₃×C₄×D₇	Q₈	1	4	yes	complemented
65	#4248	C₂×C₄	1	8	yes	complemented
66	Dic₃×D₇	—	2	4	no	—
67	Dic₃×D₇	C₂×C₄	1	8	yes	—
68	C₂×C₇×Q₈	—	3	1	no	—
69	Q₈×D₇	—	3	8	no	—
70	C₂×C₄×D₇	—	3	2	no	—
71	C₂×C₄×D₇	—	3	2	no	—
72	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
74	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
75	C₂×Dic₁₄	—	3	1	no	—
76	C₂×Dic₁₄	—	3	2	no	—
77	C₇×C₄⋊C₄	—	3	2	no	—
78	C₇×C₄⋊C₄	—	3	1	no	—
79	C₄²×C₇	—	3	2	no	—
80	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
81	C₄⋊Dic₇	—	3	1	no	—
82	C₄⋊Dic₇	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	—	3	2	no	—
85	Dic₇⋊C₄	—	3	2	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₄×Dic₇	—	3	2	no	—
88	C₄×Dic₇	—	3	2	no	—
89	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
90	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
91	C₄×Dic₇	C₂×D₃	1	1	yes	characteristic
92	C₂²×Dic₆	—	7	1	no	—
93	C₂×C₃×C₄²	—	7	1	no	—
94	C₂×C₄⋊Dic₃	—	7	1	no	—
95	C₂×Dic₃⋊C₄	—	7	2	no	—
96	C₂×C₄×Dic₃	—	7	2	no	—
97	C₄×Dic₆	—	7	1	no	—
98	C₄×Dic₆	—	7	1	no	—
99	C₄×Dic₆	—	7	1	no	—
100	C₄×Dic₆	—	7	2	no	—
101	C₄×Dic₆	—	7	2	no	—
102	C₄×Dic₆	D₇	1	1	yes	characteristic, direct product factor
103	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
104	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
105	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
106	C₂×C₃×D₇	#41	1	2	yes	—
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
109	C₃×C₄×C₇	C₂×Q₈	1	2	yes	complemented
110	Dic₂₁	—	2	2	no	—
111	Dic₂₁	C₂²×C₄	1	4	yes	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	C₂²×C₄	1	2	yes	—
114	C₃×Dic₇	C₂×Q₈	1	2	yes	complemented
115	C₇×Dic₃	—	2	2	no	—
116	C₇×Dic₃	C₂²×C₄	1	4	yes	—
117	C₂×C₄×Q₈	—	21	1	no	Sylow
118	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
119	C₇×Q₈	—	3	4	no	—
120	C₂×C₄×C₇	—	3	2	no	—
121	C₂×C₄×C₇	—	3	2	no	—
122	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
125	C₂×Dic₇	—	3	2	no	—
126	C₂×Dic₇	—	3	2	no	—
127	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
128	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
129	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
130	C₄×D₇	—	2	2	no	—
131	C₄×D₇	—	3	8	no	—
132	C₄×D₇	—	6	4	no	—
133	C₄×D₇	Dic₆	1	4	yes	direct product factor
134	C₄×D₇	C₄×D₃	1	4	yes	—
135	Dic₁₄	—	3	4	no	—
136	Dic₁₄	—	3	8	no	—
137	C₂²×C₃×C₄	—	7	1	no	—
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₂²×C₃×C₄	—	7	1	no	—
140	C₂²×Dic₃	—	7	2	no	—
141	C₂²×Dic₃	—	7	2	no	—
142	C₂×Dic₆	—	7	1	no	—
143	C₂×Dic₆	—	7	2	no	—
144	C₂×Dic₆	—	7	8	no	—
145	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
146	C₃×C₄²	—	7	1	no	—
147	C₃×C₄²	—	7	1	no	—
148	C₃×C₄²	—	7	1	no	—
149	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
150	C₄⋊Dic₃	—	7	1	no	—
151	C₄⋊Dic₃	—	7	2	no	—
152	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
153	Dic₃⋊C₄	—	7	2	no	—
154	Dic₃⋊C₄	—	7	2	no	—
155	Dic₃⋊C₄	—	7	2	no	—
156	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
157	C₄×Dic₃	—	7	2	no	—
158	C₄×Dic₃	—	7	2	no	—
159	C₄×Dic₃	—	7	2	no	—
160	C₄×Dic₃	C₂×D₇	1	2	yes	—
161	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
164	C₃×D₇	C₄×Q₈	1	4	yes	complemented
165	C₂²×Q₈	—	21	1	no	—
166	C₄×Q₈	—	3	1	no	—
167	C₄×Q₈	—	21	2	no	—
168	C₄×Q₈	—	21	2	no	—
169	C₄×Q₈	—	21	1	no	—
170	C₄×Q₈	—	21	1	no	—
171	C₄×Q₈	—	21	1	no	—
172	C₂×C₄⋊C₄	—	21	2	no	—
173	C₂×C₄⋊C₄	—	21	1	no	—
174	C₂×C₄²	—	7	1	no	—
175	C₂×C₄²	—	21	2	no	—
176	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
177	C₂×D₇	C₂×Dic₆	1	2	yes	—
178	C₂×D₇	C₂×C₄×D₃	1	2	yes	—
179	C₂×D₇	#235	1	2	yes	—
180	C₄×C₇	—	2	1	no	—
181	C₄×C₇	—	3	4	no	—
182	C₄×C₇	—	6	2	no	—
183	C₄×C₇	C₂×Dic₆	1	2	yes	complemented
184	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	3	4	no	—
187	Dic₇	—	6	2	no	—
188	Dic₇	C₂×Dic₆	1	2	yes	complemented
189	Dic₇	C₂×C₄×D₃	1	2	yes	—
190	C₂³×C₃	—	7	1	no	—
191	C₂×C₃×C₄	—	7	1	no	—
192	C₂×C₃×C₄	—	7	1	no	—
193	C₂×C₃×C₄	—	7	4	no	—
194	C₂×C₃×C₄	—	7	4	no	—
195	C₂×C₃×C₄	—	7	1	no	—
196	C₂×C₃×C₄	—	14	2	no	—
197	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
200	C₂×Dic₃	—	7	2	no	—
201	C₂×Dic₃	—	7	8	no	—
202	C₂×Dic₃	—	7	2	no	—
203	C₂×Dic₃	—	14	4	no	—
204	C₂×Dic₃	C₂²×D₇	1	2	yes	—
205	C₂×Dic₃	C₂²×D₇	1	2	yes	—
206	Dic₆	—	7	4	no	—
207	Dic₆	—	7	8	no	—
208	Dic₆	C₄×D₇	1	4	yes	direct product factor
209	C₃×C₇	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Hall
210	C₂×Q₈	—	3	1	no	—
211	C₂×Q₈	—	21	2	no	—
212	C₂×Q₈	—	21	8	no	—
213	C₂×Q₈	—	21	1	no	—
214	C₂²×C₄	—	7	1	no	—
215	C₂²×C₄	—	7	1	no	—
216	C₂²×C₄	—	7	1	no	—
217	C₂²×C₄	—	21	2	no	—
218	C₂²×C₄	—	21	2	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	21	2	no	—
222	C₄⋊C₄	—	21	2	no	—
223	C₄⋊C₄	—	21	2	no	—
224	C₄⋊C₄	—	21	2	no	—
225	C₄⋊C₄	—	21	1	no	—
226	C₄²	—	3	2	no	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	21	2	no	—
231	C₄²	—	21	2	no	—
232	C₄²	—	21	2	no	—
233	C₄²	D₃×D₇	1	1	yes	characteristic
234	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
235	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
236	C₂×C₇	C₂×#235	1	1	yes	characteristic
237	D₇	C₄×Dic₆	1	4	yes	direct product factor
238	C₂²×C₃	—	7	2	no	—
239	C₂²×C₃	—	7	2	no	—
240	C₂²×C₃	—	7	2	no	—
241	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	7	2	no	—
244	C₃×C₄	—	7	2	no	—
245	C₃×C₄	—	14	1	no	—
246	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
247	C₃×C₄	Q₈×D₇	1	2	yes	complemented
248	Dic₃	—	2	2	no	—
249	Dic₃	—	7	4	no	—
250	Dic₃	—	14	2	no	—
251	Dic₃	C₂×C₄×D₇	1	4	yes	—
252	C₂³	—	7	1	no	—
253	Q₈	—	3	4	no	—
254	Q₈	—	21	8	no	—
255	Q₈	—	21	4	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	7	1	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	4	no	—
261	C₂×C₄	—	7	4	no	—
262	C₂×C₄	—	7	1	no	—
263	C₂×C₄	—	14	2	no	—
264	C₂×C₄	—	21	8	no	—
265	C₂×C₄	—	21	2	no	—
266	C₂×C₄	—	21	2	no	—
267	C₂×C₄	—	42	4	no	—
268	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
269	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
271	C₇	C₂×C₄×Dic₆	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	7	4	no	—
273	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
274	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
275	C₂×C₃	D₇×#41	1	1	yes	characteristic
276	C₂²	—	7	2	no	—
277	C₂²	—	7	2	no	—
278	C₂²	—	7	2	no	—
279	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	3	4	no	—
282	C₄	—	6	2	no	—
283	C₄	—	7	2	no	—
284	C₄	—	7	2	no	—
285	C₄	—	14	1	no	—
286	C₄	—	21	4	no	—
287	C₄	—	42	2	no	—
288	C₄	D₇×Dic₆	1	2	yes	direct product factor
289	C₄	C₄×D₃×D₇	1	2	yes	—
290	C₃	C₄×Q₈×D₇	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	7	4	no	—
292	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
293	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
294	C₂	D₇×#235	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Frattini	C₂²	C₂²×D₃×D₇
Fitting	C₃×C₄²×C₇	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂²×C₃×D₇	C₂³
hypercenter	C₄²	D₃×D₇
hypocenter	C₃×C₇	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₄₂	C₂	semidirect
3. C₄×C₇×Dic₆	C₂	semidirect
5. C₄×#4248	C₂	semidirect
9. #10620895	C₂	semidirect
10. Dic₆×Dic₇	C₂	semidirect
11. #10620885	C₂	semidirect
21. D₇×Dic₆	C₄	direct (N × H)
55. Dic₄₂	C₂×C₄	semidirect
58. C₇×Dic₆	C₂×C₄	semidirect
64. C₃×C₄×D₇	Q₈	semidirect
65. #4248	C₂×C₄	semidirect
102. C₄×Dic₆	D₇	direct (N × H)
109. C₃×C₄×C₇	C₂×Q₈	semidirect
114. C₃×Dic₇	C₂×Q₈	semidirect
133. C₄×D₇	Dic₆	direct (N × H)
164. C₃×D₇	C₄×Q₈	semidirect
183. C₄×C₇	C₂×Dic₆	semidirect
188. Dic₇	C₂×Dic₆	semidirect
208. Dic₆	C₄×D₇	direct (N × H)
209. C₃×C₇	C₂×C₄×Q₈	semidirect
237. D₇	C₄×Dic₆	direct (N × H)
247. C₃×C₄	Q₈×D₇	semidirect
271. C₇	C₂×C₄×Dic₆	semidirect
288. C₄	D₇×Dic₆	direct (N × H)
290. C₃	C₄×Q₈×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₇×Dic₆	2
C₄×Dic₄₂	2
C₄×C₇×Dic₆	2
C₃×C₄²×D₇	2
C₄×#4248	2
D₇×C₄⋊Dic₃	2
D₇×Dic₃⋊C₄	2
C₄×Dic₃×D₇	2
#10620895	2
Dic₆×Dic₇	2
#10620885	2
C₄×Q₈×D₇	3
C₂×C₄×Dic₆	7

Subgroups — C4×D7×Dic6 #13057713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₇×Dic₆ #13057713