Subgroups — D₇×#926 #13059573

Order 1344.

660 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	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	#10621409	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#926	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×#217	C₂	1	2	yes	maximal
5	#10621090	C₂	1	2	yes	maximal, complemented
6	D₇×C₄⋊Dic₃	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₃×D₇	C₂	1	2	yes	maximal, complemented
8	#10620920	C₂	1	2	yes	maximal, complemented
9	#10620919	C₂	1	2	yes	maximal, complemented
10	D₇×#117	—	3	1	no	maximal, Hall
11	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₇	C₄	1	8	yes	complemented
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×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₃×D₇	C₂²	1	4	yes	—
20	#66636	C₂²	1	2	yes	—
21	C₄⋊Dic₂₁	C₂²	1	2	yes	complemented
22	C₄×Dic₂₁	C₂²	1	2	yes	complemented
23	C₇×#217	C₂²	1	2	yes	—
24	C₇×C₄⋊Dic₃	C₂²	1	2	yes	complemented
25	C₄×C₇×Dic₃	C₂²	1	2	yes	complemented
26	Dic₇⋊Dic₃	C₂²	1	4	yes	complemented
27	(C₂×D₇)⋊Dic₃	C₂²	1	4	yes	—
28	Dic₃×Dic₇	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₇	D₃	1	1	yes	characteristic
30	C₇×#117	—	3	1	no	—
31	#6489	—	3	1	no	—
32	#6455	—	3	2	no	—
33	D₇×C₄⋊C₄	—	3	2	no	—
34	D₇×C₂²⋊C₄	—	3	2	no	—
35	#6440	—	3	2	no	—
36	#6435	—	3	2	no	—
37	C₄²×D₇	—	3	2	no	—
38	C₂×#926	—	7	1	no	maximal, Hall
39	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₃×D₇	—	2	4	no	—
41	C₂²×C₃×D₇	C₂×C₄	1	4	yes	—
42	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
43	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
45	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₁	C₂³	1	4	yes	—
48	C₂×C₇×Dic₃	C₂³	1	4	yes	—
49	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	complemented
50	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
52	C₃×C₄×D₇	C₂×C₄	1	4	yes	—
53	C₃×C₄×D₇	C₂×C₄	1	4	yes	—
54	C₃×C₄×D₇	C₂×C₄	1	4	yes	complemented
55	C₃×C₄×D₇	C₂×C₄	1	4	yes	complemented
56	Dic₃×D₇	—	2	8	no	—
57	C₂³×D₇	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
59	C₂²×Dic₇	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×D₇	—	3	4	no	—
61	C₂×C₄×D₇	Dic₃	1	8	yes	complemented
62	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
64	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	3	2	no	—
67	C₇×C₂²⋊C₄	—	3	2	no	—
68	C₄²×C₇	—	3	2	no	—
69	C₂²⋊Dic₇	—	3	2	no	—
70	(C₂×D₇)⋊C₄	—	3	4	no	—
71	C₄⋊Dic₇	—	3	2	no	—
72	Dic₇⋊C₄	—	3	4	no	—
73	C₄×Dic₇	—	3	2	no	—
74	C₄×Dic₇	—	3	4	no	—
75	C₂³×C₃×C₄	—	7	1	no	—
76	C₂×#217	—	7	2	no	—
77	#926	—	7	1	no	—
78	#926	—	7	2	no	—
79	#926	—	7	2	no	—
80	#926	—	7	2	no	—
81	#926	D₇	1	1	yes	characteristic, direct product factor
82	C₂×C₄⋊Dic₃	—	7	2	no	—
83	C₂×C₄×Dic₃	—	7	2	no	—
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₃×D₇	—	2	2	no	—
88	C₂×C₃×D₇	—	2	8	no	—
89	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
90	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
91	C₂×C₃×D₇	#41	1	4	yes	—
92	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
93	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
94	Dic₂₁	—	2	4	no	—
95	C₃×Dic₇	C₂²×C₄	1	2	yes	—
96	C₃×Dic₇	C₂²×C₄	1	2	yes	—
97	C₇×Dic₃	—	2	4	no	—
98	C₂×#117	—	21	1	no	Sylow
99	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
100	C₂²×D₇	—	2	4	no	—
101	C₂²×D₇	C₂×Dic₃	1	4	yes	—
102	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
103	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₄×C₇	—	3	4	no	—
105	C₂×C₄×C₇	C₂×Dic₃	1	4	yes	complemented
106	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
108	C₂×Dic₇	—	3	4	no	—
109	C₂×Dic₇	C₂×Dic₃	1	4	yes	complemented
110	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
111	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
112	C₄×D₇	—	6	8	no	—
113	C₄×D₇	C₂×Dic₃	1	4	yes	—
114	C₄×D₇	C₂×Dic₃	1	4	yes	—
115	C₄×D₇	C₂×Dic₃	1	4	yes	complemented
116	C₄×D₇	C₂×Dic₃	1	4	yes	complemented
117	C₂⁴×C₃	—	7	1	no	—
118	C₂²×C₃×C₄	—	7	1	no	—
119	C₂²×C₃×C₄	—	7	8	no	—
120	C₂²×C₃×C₄	—	7	1	no	—
121	C₂²×C₃×C₄	—	7	1	no	—
122	C₂²×C₃×C₄	—	7	1	no	—
123	C₂²×C₃×C₄	—	7	1	no	—
124	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
125	C₂²×Dic₃	—	7	4	no	—
126	#217	—	7	2	no	—
127	#217	—	7	4	no	—
128	#217	C₂×D₇	1	2	yes	—
129	C₄⋊Dic₃	—	7	2	no	—
130	C₄⋊Dic₃	—	7	4	no	—
131	C₄⋊Dic₃	C₂×D₇	1	2	yes	complemented
132	C₄×Dic₃	—	7	2	no	—
133	C₄×Dic₃	—	7	4	no	—
134	C₄×Dic₃	C₂×D₇	1	2	yes	complemented
135	C₂×C₃×C₇	—	2	2	no	—
136	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₃×C₇	C₂×#41	1	2	yes	—
138	C₃×D₇	—	2	2	no	—
139	C₃×D₇	#117	1	4	yes	complemented
140	C₂³×C₄	—	7	1	no	—
141	#117	—	3	1	no	—
142	#117	—	21	2	no	—
143	#117	—	21	2	no	—
144	#117	—	21	2	no	—
145	#117	—	21	1	no	—
146	C₂×C₄⋊C₄	—	21	2	no	—
147	C₂×C₂²⋊C₄	—	21	2	no	—
148	C₂×C₄²	—	21	2	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	C₂²×Dic₃	1	2	yes	—
151	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
152	C₂×D₇	—	2	2	no	—
153	C₂×D₇	—	2	8	no	—
154	C₂×D₇	#235	1	4	yes	—
155	C₂×D₇	C₂²×Dic₃	1	2	yes	—
156	C₂×D₇	C₂²×Dic₃	1	2	yes	—
157	C₄×C₇	—	6	4	no	—
158	C₄×C₇	C₂²×Dic₃	1	2	yes	—
159	C₄×C₇	C₂²×Dic₃	1	2	yes	—
160	Dic₇	—	6	4	no	—
161	Dic₇	C₂²×Dic₃	1	2	yes	—
162	Dic₇	C₂²×Dic₃	1	2	yes	—
163	C₂³×C₃	—	7	1	no	—
164	C₂³×C₃	—	7	4	no	—
165	C₂³×C₃	—	7	1	no	—
166	C₂³×C₃	—	14	4	no	—
167	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
168	C₂×C₃×C₄	—	7	4	no	—
169	C₂×C₃×C₄	—	7	1	no	—
170	C₂×C₃×C₄	—	7	1	no	—
171	C₂×C₃×C₄	—	7	4	no	—
172	C₂×C₃×C₄	—	7	4	no	—
173	C₂×C₃×C₄	—	7	4	no	—
174	C₂×C₃×C₄	—	7	4	no	—
175	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
176	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
178	C₂×Dic₃	—	7	4	no	—
179	C₂×Dic₃	—	14	8	no	—
180	C₂×Dic₃	C₂²×D₇	1	4	yes	—
181	C₃×C₇	C₂×#117	1	1	yes	characteristic, complemented, Hall
182	C₂⁴	—	7	1	no	—
183	C₂²×C₄	—	7	1	no	—
184	C₂²×C₄	—	7	8	no	—
185	C₂²×C₄	—	7	1	no	—
186	C₂²×C₄	—	7	1	no	—
187	C₂²×C₄	—	7	1	no	—
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	21	4	no	—
190	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
191	C₄⋊C₄	—	3	2	no	—
192	C₄⋊C₄	—	21	4	no	—
193	C₄⋊C₄	—	21	2	no	—
194	C₂²⋊C₄	—	3	2	no	—
195	C₂²⋊C₄	—	21	4	no	—
196	C₂²⋊C₄	—	21	2	no	—
197	C₄²	—	3	2	no	—
198	C₄²	—	21	4	no	—
199	C₄²	—	21	2	no	—
200	C₂×C₇	—	2	2	no	—
201	C₂×C₇	C₂×#235	1	2	yes	—
202	C₂×C₇	C₂³×Dic₃	1	1	yes	characteristic
203	D₇	—	2	2	no	—
204	D₇	#926	1	4	yes	direct product factor
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	7	2	no	—
207	C₂²×C₃	—	7	4	no	—
208	C₂²×C₃	—	7	2	no	—
209	C₂²×C₃	—	14	8	no	—
210	C₂²×C₃	—	14	2	no	—
211	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
212	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
213	C₃×C₄	—	7	2	no	—
214	C₃×C₄	—	7	2	no	—
215	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
216	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
217	Dic₃	—	2	4	no	—
218	Dic₃	—	14	4	no	—
219	C₂³	—	7	1	no	—
220	C₂³	—	7	4	no	—
221	C₂³	—	7	1	no	—
222	C₂³	—	14	4	no	—
223	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
224	C₂×C₄	—	3	4	no	—
225	C₂×C₄	—	7	4	no	—
226	C₂×C₄	—	7	1	no	—
227	C₂×C₄	—	7	1	no	—
228	C₂×C₄	—	7	4	no	—
229	C₂×C₄	—	7	4	no	—
230	C₂×C₄	—	7	4	no	—
231	C₂×C₄	—	7	4	no	—
232	C₂×C₄	—	21	4	no	—
233	C₂×C₄	—	42	8	no	—
234	C₂×C₄	Dic₃×D₇	1	4	yes	complemented
235	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
237	C₇	C₂×#926	1	1	yes	characteristic, complemented, Sylow
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	7	4	no	—
240	C₂×C₃	—	14	2	no	—
241	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
242	C₂×C₃	D₇×#41	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	7	4	no	—
245	C₂²	—	7	2	no	—
246	C₂²	—	7	2	no	—
247	C₂²	—	14	8	no	—
248	C₂²	—	14	2	no	—
249	C₂²	C₂×Dic₃×D₇	1	2	yes	—
250	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
251	C₄	—	6	4	no	—
252	C₄	—	7	2	no	—
253	C₄	—	7	2	no	—
254	C₄	—	42	4	no	—
255	C₄	C₂×Dic₃×D₇	1	2	yes	—
256	C₄	C₂×Dic₃×D₇	1	2	yes	—
257	C₃	D₇×#117	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	2	2	no	—
259	C₂	—	7	4	no	—
260	C₂	—	14	2	no	—
261	C₂	D₇×#235	1	2	yes	—
262	C₂	C₂²×Dic₃×D₇	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	C₂²×C₃×C₄×C₇	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×D₇	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×#117
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #10621409	C₂	semidirect
3. C₇×#926	C₂	semidirect
5. #10621090	C₂	semidirect
6. D₇×C₄⋊Dic₃	C₂	semidirect
7. C₄×Dic₃×D₇	C₂	semidirect
8. #10620920	C₂	semidirect
9. #10620919	C₂	semidirect
14. C₂×C₃×C₄×D₇	C₄	semidirect
21. C₄⋊Dic₂₁	C₂²	semidirect
22. C₄×Dic₂₁	C₂²	semidirect
24. C₇×C₄⋊Dic₃	C₂²	semidirect
25. C₄×C₇×Dic₃	C₂²	semidirect
26. Dic₇⋊Dic₃	C₂²	semidirect
28. Dic₃×Dic₇	C₂²	semidirect
44. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
49. C₂×C₃×Dic₇	C₂×C₄	semidirect
54. C₃×C₄×D₇	C₂×C₄	semidirect
55. C₃×C₄×D₇	C₂×C₄	semidirect
61. C₂×C₄×D₇	Dic₃	semidirect
81. #926	D₇	direct (N × H)
105. C₂×C₄×C₇	C₂×Dic₃	semidirect
109. C₂×Dic₇	C₂×Dic₃	semidirect
115. C₄×D₇	C₂×Dic₃	semidirect
116. C₄×D₇	C₂×Dic₃	semidirect
131. C₄⋊Dic₃	C₂×D₇	semidirect
134. C₄×Dic₃	C₂×D₇	semidirect
139. C₃×D₇	#117	semidirect
175. C₂×C₃×C₄	C₄×D₇	semidirect
181. C₃×C₇	C₂×#117	semidirect
204. D₇	#926	direct (N × H)
234. C₂×C₄	Dic₃×D₇	semidirect
237. C₇	C₂×#926	semidirect
257. C₃	D₇×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×D₇	2
#10621409	2
C₇×#926	2
D₇×#217	2
#10621090	2
D₇×C₄⋊Dic₃	2
C₄×Dic₃×D₇	2
#10620920	2
#10620919	2
D₇×#117	3
C₂×#926	7

Subgroups — D7×#926 #13059573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#926 #13059573