Subgroups — D₇×#890 #13058634

Order 1344.

452 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621373	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#890	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	D₇×C₄⋊Dic₃	C₂	1	2	yes	maximal
5	D₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	D₇×Dic₃⋊C₄	C₂	1	2	yes	maximal
7	C₄×Dic₃×D₇	C₂	1	1	yes	maximal, characteristic
8	#10620946	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620943	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620897	C₂	1	2	yes	maximal, complemented
11	#10620896	C₂	1	2	yes	maximal, complemented
12	D₇×#125	—	3	1	no	maximal, Hall
13	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃×D₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃×D₇	C₂²	1	2	yes	—
18	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₂₁	C₂²	1	2	yes	—
21	#66629	C₂²	1	2	yes	—
22	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₇×C₄⋊Dic₃	C₂²	1	2	yes	—
25	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×#125	—	3	1	no	—
37	D₇×C₄⋊C₄	—	3	2	no	—
38	D₇×C₄⋊C₄	—	3	2	no	—
39	D₇×C₄⋊C₄	—	3	1	no	—
40	D₇×C₄⋊C₄	D₃	1	1	yes	characteristic
41	#6453	—	3	1	no	—
42	#6453	—	3	1	no	—
43	#6452	—	3	2	no	—
44	#6452	—	3	2	no	—
45	C₄²×D₇	—	3	1	no	—
46	#6433	—	3	1	no	—
47	C₂×#890	—	7	1	no	maximal, Hall
48	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
55	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₇×Dic₃	C₂³	1	2	yes	—
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
59	C₃×C₄×D₇	—	2	4	no	—
60	C₃×C₄×D₇	Q₈	1	4	yes	—
61	Dic₃×D₇	—	2	2	no	—
62	Dic₃×D₇	—	2	2	no	—
63	Dic₃×D₇	—	2	4	no	—
64	C₂×C₄×D₇	—	3	1	no	—
65	C₂×C₄×D₇	—	3	1	no	—
66	C₂×C₄×D₇	—	3	2	no	—
67	C₂×C₄×D₇	C₂×D₃	1	2	yes	—
68	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	3	2	no	—
70	C₇×C₄⋊C₄	—	3	2	no	—
71	C₇×C₄⋊C₄	—	3	1	no	—
72	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₄²×C₇	—	3	1	no	—
74	C₄⋊Dic₇	—	3	1	no	—
75	C₄⋊Dic₇	—	3	2	no	—
76	C₄⋊Dic₇	—	3	2	no	—
77	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
78	Dic₇⋊C₄	—	3	2	no	—
79	Dic₇⋊C₄	—	3	2	no	—
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
84	C₄×Dic₇	—	3	1	no	—
85	C₄×Dic₇	—	3	1	no	—
86	C₄×Dic₇	—	3	1	no	—
87	C₂×C₃×C₄⋊C₄	—	7	1	no	—
88	C₂×C₄⋊Dic₃	—	7	2	no	—
89	C₂×C₄⋊Dic₃	—	7	1	no	—
90	C₂×Dic₃⋊C₄	—	7	2	no	—
91	C₂×C₄×Dic₃	—	7	1	no	—
92	#890	—	7	1	no	—
93	#890	—	7	1	no	—
94	#890	—	7	2	no	—
95	#890	—	7	2	no	—
96	#890	—	7	1	no	—
97	#890	D₇	1	1	yes	characteristic, direct product factor
98	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₇	C₂×Q₈	1	2	yes	—
100	C₂×C₃×D₇	#41	1	2	yes	—
101	C₂×C₃×D₇	#41	1	2	yes	—
102	C₃×C₄×C₇	—	2	2	no	—
103	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
104	Dic₂₁	—	2	1	no	—
105	Dic₂₁	—	2	1	no	—
106	Dic₂₁	—	2	2	no	—
107	C₃×Dic₇	—	2	2	no	—
108	C₃×Dic₇	C₂×Q₈	1	2	yes	—
109	C₇×Dic₃	—	2	1	no	—
110	C₇×Dic₃	—	2	1	no	—
111	C₇×Dic₃	—	2	2	no	—
112	C₂×#125	—	21	1	no	Sylow
113	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₄×C₇	—	3	1	no	—
115	C₂×C₄×C₇	—	3	1	no	—
116	C₂×C₄×C₇	—	3	2	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
118	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
119	C₂×Dic₇	—	3	1	no	—
120	C₂×Dic₇	—	3	1	no	—
121	C₂×Dic₇	—	3	2	no	—
122	C₂×Dic₇	C₂²×D₃	1	2	yes	—
123	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
124	C₄×D₇	—	2	4	no	—
125	C₄×D₇	—	6	2	no	—
126	C₄×D₇	—	6	2	no	—
127	C₄×D₇	—	6	4	no	—
128	C₄×D₇	Dic₆	1	4	yes	—
129	C₂²×C₃×C₄	—	7	1	no	—
130	C₂²×C₃×C₄	—	7	2	no	—
131	C₂²×Dic₃	—	7	1	no	—
132	C₂²×Dic₃	—	7	1	no	—
133	C₂²×Dic₃	—	7	2	no	—
134	C₃×C₄⋊C₄	—	7	2	no	—
135	C₃×C₄⋊C₄	—	7	1	no	—
136	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
137	C₄⋊Dic₃	—	7	2	no	—
138	C₄⋊Dic₃	—	7	1	no	—
139	C₄⋊Dic₃	—	7	2	no	—
140	C₄⋊Dic₃	—	7	2	no	—
141	C₄⋊Dic₃	—	7	2	no	—
142	C₄⋊Dic₃	C₂×D₇	1	2	yes	—
143	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
144	Dic₃⋊C₄	—	7	2	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
148	C₄×Dic₃	—	7	1	no	—
149	C₄×Dic₃	—	7	1	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
152	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
155	C₃×D₇	#125	1	4	yes	complemented
156	#125	—	3	1	no	—
157	#125	—	21	1	no	—
158	#125	—	21	2	no	—
159	#125	—	21	2	no	—
160	#125	—	21	1	no	—
161	#125	—	21	1	no	—
162	C₂×C₄⋊C₄	—	7	1	no	—
163	C₂×C₄⋊C₄	—	21	2	no	—
164	C₂×C₄⋊C₄	—	21	1	no	—
165	C₂×C₄⋊C₄	—	21	2	no	—
166	C₂×C₄²	—	21	1	no	—
167	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
168	C₂×D₇	C₂×Dic₆	1	2	yes	—
169	C₂×D₇	#237	1	2	yes	—
170	C₂×D₇	#239	1	2	yes	—
171	C₄×C₇	—	2	2	no	—
172	C₄×C₇	—	6	1	no	—
173	C₄×C₇	—	6	1	no	—
174	C₄×C₇	—	6	2	no	—
175	C₄×C₇	C₂×Dic₆	1	2	yes	—
176	Dic₇	—	2	2	no	—
177	Dic₇	—	6	1	no	—
178	Dic₇	—	6	1	no	—
179	Dic₇	—	6	2	no	—
180	Dic₇	C₂×Dic₆	1	2	yes	—
181	C₂³×C₃	—	7	1	no	—
182	C₂×C₃×C₄	—	7	1	no	—
183	C₂×C₃×C₄	—	7	4	no	—
184	C₂×C₃×C₄	—	7	2	no	—
185	C₂×C₃×C₄	—	14	4	no	—
186	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₂×Dic₃	—	7	1	no	—
189	C₂×Dic₃	—	7	1	no	—
190	C₂×Dic₃	—	7	2	no	—
191	C₂×Dic₃	—	14	2	no	—
192	C₂×Dic₃	—	14	4	no	—
193	C₂×Dic₃	—	14	2	no	—
194	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
195	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
196	C₂×Dic₃	C₂²×D₇	1	2	yes	—
197	C₃×C₇	C₂×#125	1	1	yes	characteristic, complemented, Hall
198	C₂²×C₄	—	7	1	no	—
199	C₂²×C₄	—	7	2	no	—
200	C₂²×C₄	—	21	1	no	—
201	C₂²×C₄	—	21	1	no	—
202	C₂²×C₄	—	21	2	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	—	3	2	no	—
205	C₄⋊C₄	—	3	1	no	—
206	C₄⋊C₄	—	7	2	no	—
207	C₄⋊C₄	—	7	1	no	—
208	C₄⋊C₄	—	21	2	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	2	no	—
211	C₄⋊C₄	—	21	2	no	—
212	C₄⋊C₄	—	21	2	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	—	21	2	no	—
215	C₄⋊C₄	—	21	1	no	—
216	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
217	C₄²	—	3	1	no	—
218	C₄²	—	21	1	no	—
219	C₄²	—	21	1	no	—
220	C₄²	—	21	1	no	—
221	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
222	C₂×C₇	C₂×#237	1	1	yes	characteristic
223	C₂×C₇	C₂×#239	1	1	yes	characteristic
224	D₇	#890	1	4	yes	direct product factor
225	C₂²×C₃	—	7	2	no	—
226	C₂²×C₃	—	7	2	no	—
227	C₂²×C₃	—	7	2	no	—
228	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	7	2	no	—
231	C₃×C₄	—	14	2	no	—
232	C₃×C₄	Q₈×D₇	1	2	yes	—
233	Dic₃	—	2	1	no	—
234	Dic₃	—	2	1	no	—
235	Dic₃	—	2	2	no	—
236	Dic₃	—	14	1	no	—
237	Dic₃	—	14	1	no	—
238	Dic₃	—	14	2	no	—
239	C₂³	—	7	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	3	2	no	—
243	C₂×C₄	—	7	1	no	—
244	C₂×C₄	—	7	4	no	—
245	C₂×C₄	—	7	2	no	—
246	C₂×C₄	—	14	4	no	—
247	C₂×C₄	—	21	1	no	—
248	C₂×C₄	—	21	1	no	—
249	C₂×C₄	—	21	2	no	—
250	C₂×C₄	—	42	2	no	—
251	C₂×C₄	—	42	2	no	—
252	C₂×C₄	—	42	4	no	—
253	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
255	C₇	C₂×#890	1	1	yes	characteristic, complemented, Sylow
256	C₂×C₃	—	7	4	no	—
257	C₂×C₃	C₂×Q₈×D₇	1	1	yes	characteristic
258	C₂×C₃	D₇×#41	1	1	yes	characteristic
259	C₂×C₃	D₇×#41	1	1	yes	characteristic
260	C₂²	—	7	2	no	—
261	C₂²	—	7	2	no	—
262	C₂²	—	7	2	no	—
263	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	6	1	no	—
266	C₄	—	6	1	no	—
267	C₄	—	6	2	no	—
268	C₄	—	7	2	no	—
269	C₄	—	14	2	no	—
270	C₄	—	42	1	no	—
271	C₄	—	42	1	no	—
272	C₄	—	42	2	no	—
273	C₄	D₇×Dic₆	1	2	yes	—
274	C₃	D₇×#125	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	7	4	no	—
276	C₂	C₂×D₇×Dic₆	1	1	yes	characteristic
277	C₂	D₇×#237	1	1	yes	characteristic
278	C₂	D₇×#239	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×C₄⋊C₄	C₂²
prime radical	C₂²×C₃×D₇	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	C₂×#125
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #10621373	C₂	semidirect
2. C₇×#890	C₂	semidirect
8. #10620946	C₂	semidirect
9. #10620943	C₂	semidirect
10. #10620897	C₂	semidirect
11. #10620896	C₂	semidirect
97. #890	D₇	direct (N × H)
155. C₃×D₇	#125	semidirect
197. C₃×C₇	C₂×#125	semidirect
224. D₇	#890	direct (N × H)
255. C₇	C₂×#890	semidirect
274. C₃	D₇×#125	semidirect

Maximal subgroups

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

subgroup	index
#10621373	2
C₇×#890	2
C₃×D₇×C₄⋊C₄	2
D₇×C₄⋊Dic₃	2
D₇×Dic₃⋊C₄	2
C₄×Dic₃×D₇	2
#10620946	2
#10620943	2
#10620897	2
#10620896	2
D₇×#125	3
C₂×#890	7

Subgroups — D7×#890 #13058634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#890 #13058634