Subgroups — #13058542

Order 1344.

532 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#10621370	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6455	C₂	1	1	yes	maximal, characteristic
5	C₄×#4248	C₂	1	2	yes	maximal, complemented
6	#10620956	C₂	1	2	yes	maximal
7	#10620955	C₂	1	2	yes	maximal
8	#10620919	C₂	1	1	yes	maximal, characteristic
9	#10620906	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620902	C₂	1	1	yes	maximal, characteristic, complemented
11	#10620885	C₂	1	2	yes	maximal, complemented
12	#90263	—	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×#4248	C₂²	1	2	yes	—
17	C₂×Dic₃×D₇	C₂²	1	2	yes	—
18	D₇×Dic₆	C₄	1	8	yes	complemented
19	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#66629	C₂²	1	2	yes	—
21	C₄×Dic₂₁	C₂²	1	2	yes	—
22	C₄×Dic₂₁	C₂²	1	1	yes	characteristic
23	C₇×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₄×C₇×Dic₃	C₂²	1	2	yes	—
25	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×C₄×Dic₇	C₂²	1	2	yes	—
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
32	Dic₇⋊Dic₃	C₂²	1	1	yes	characteristic
33	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
34	(C₂×D₇)⋊Dic₃	C₂²	1	1	yes	characteristic
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	#6455	—	3	2	no	—
41	#6455	D₃	1	1	yes	characteristic
42	#6450	—	3	1	no	—
43	#6450	—	3	2	no	—
44	#6435	—	3	1	no	—
45	#6435	—	3	2	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	C₄×Dic₁₄	—	3	2	no	—
48	#5516	—	7	1	no	maximal, Hall
49	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	Dic₄₂	C₂×C₄	1	4	yes	—
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₇×Dic₃	C₂³	1	2	yes	—
59	C₇×Dic₆	C₂×C₄	1	4	yes	—
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₇	C₂×C₄	1	4	yes	—
63	#4248	C₂×C₄	1	8	yes	—
64	Dic₃×D₇	C₂×C₄	1	8	yes	—
65	C₂×C₇×Q₈	—	3	1	no	—
66	Q₈×D₇	—	3	8	no	—
67	C₂×C₄×D₇	—	3	2	no	—
68	C₂×C₄×D₇	C₂×D₃	1	1	yes	characteristic
69	C₂×Dic₁₄	—	3	1	no	—
70	C₂×Dic₁₄	—	3	2	no	—
71	C₇×C₄⋊C₄	—	3	2	no	—
72	C₇×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₄²×C₇	—	3	2	no	—
74	C₄²×C₇	—	3	1	no	—
75	(C₂×D₇)⋊C₄	—	3	1	no	—
76	(C₂×D₇)⋊C₄	—	3	1	no	—
77	(C₂×D₇)⋊C₄	C₂×D₃	1	2	yes	—
78	C₄⋊Dic₇	—	3	2	no	—
79	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
80	Dic₇⋊C₄	—	3	2	no	—
81	Dic₇⋊C₄	—	3	1	no	—
82	Dic₇⋊C₄	—	3	1	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	C₄×Dic₇	—	3	2	no	—
85	C₄×Dic₇	—	3	1	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₄×Dic₇	—	3	2	no	—
88	C₄×Dic₇	C₂×D₃	1	2	yes	—
89	C₂²×Dic₆	—	7	1	no	—
90	C₃×#117	—	7	1	no	—
91	#926	—	7	1	no	—
92	#887	—	7	1	no	—
93	#887	—	7	1	no	—
94	#887	—	7	1	no	—
95	#887	D₇	1	1	yes	characteristic, complemented
96	#877	—	7	2	no	—
97	#877	—	7	2	no	—
98	C₄×Dic₆	—	7	2	no	—
99	C₄×Dic₆	—	7	2	no	—
100	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×D₇	—	2	1	no	—
102	C₂×C₃×D₇	—	2	1	no	—
103	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
104	C₃×C₄×C₇	—	2	2	no	—
105	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
106	Dic₂₁	—	2	1	no	—
107	Dic₂₁	—	2	1	no	—
108	Dic₂₁	C₂²×C₄	1	4	yes	—
109	C₃×Dic₇	—	2	2	no	—
110	C₃×Dic₇	C₂²×C₄	1	2	yes	—
111	C₇×Dic₃	—	2	1	no	—
112	C₇×Dic₃	—	2	1	no	—
113	C₇×Dic₃	C₂²×C₄	1	4	yes	—
114	#519	—	21	1	no	Sylow
115	C₂²×D₇	C₂²×D₃	1	1	yes	characteristic
116	C₇×Q₈	—	3	4	no	—
117	C₂×C₄×C₇	—	3	2	no	—
118	C₂×C₄×C₇	—	3	1	no	—
119	C₂×C₄×C₇	—	3	1	no	—
120	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
121	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
122	C₂×Dic₇	—	3	2	no	—
123	C₂×Dic₇	—	3	1	no	—
124	C₂×Dic₇	—	3	1	no	—
125	C₂×Dic₇	C₂²×D₃	1	2	yes	—
126	C₂×Dic₇	C₂²×D₃	1	1	yes	characteristic
127	C₄×D₇	—	3	8	no	—
128	C₄×D₇	C₄×D₃	1	4	yes	—
129	Dic₁₄	—	3	4	no	—
130	Dic₁₄	—	3	8	no	—
131	C₂²×C₃×C₄	—	7	1	no	—
132	C₂²×Dic₃	—	7	2	no	—
133	C₂×Dic₆	—	7	1	no	—
134	C₂×Dic₆	—	7	2	no	—
135	C₂×Dic₆	—	7	8	no	—
136	C₂×Dic₆	C₂×D₇	1	1	yes	characteristic
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
139	C₃×C₂²⋊C₄	—	7	2	no	—
140	C₃×C₄²	—	7	2	no	—
141	#217	—	7	1	no	—
142	#217	—	7	1	no	—
143	C₄⋊Dic₃	—	7	1	no	—
144	C₄⋊Dic₃	—	7	1	no	—
145	Dic₃⋊C₄	—	7	2	no	—
146	Dic₃⋊C₄	—	7	2	no	—
147	Dic₃⋊C₄	—	7	2	no	—
148	Dic₃⋊C₄	C₂×D₇	1	2	yes	—
149	C₄×Dic₃	—	7	2	no	—
150	C₄×Dic₃	—	7	1	no	—
151	C₄×Dic₃	—	7	2	no	—
152	C₄×Dic₃	—	7	2	no	—
153	C₄×Dic₃	C₂×D₇	1	2	yes	—
154	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
155	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
158	C₃×D₇	—	2	2	no	—
159	C₂²×Q₈	—	21	1	no	—
160	C₄×Q₈	—	3	1	no	—
161	C₄×Q₈	—	21	2	no	—
162	C₄×Q₈	—	21	1	no	—
163	C₄×Q₈	—	21	2	no	—
164	C₄×Q₈	—	21	1	no	—
165	C₄×Q₈	—	21	1	no	—
166	#117	—	7	1	no	—
167	#117	—	21	2	no	—
168	#117	—	21	1	no	—
169	#117	—	21	2	no	—
170	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
171	C₂×D₇	—	2	1	no	—
172	C₂×D₇	—	2	1	no	—
173	C₂×D₇	C₂×C₄×D₃	1	2	yes	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	3	4	no	—
176	C₄×C₇	—	6	1	no	—
177	C₄×C₇	—	6	1	no	—
178	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
179	Dic₇	—	2	2	no	—
180	Dic₇	—	3	4	no	—
181	Dic₇	—	6	1	no	—
182	Dic₇	—	6	1	no	—
183	Dic₇	C₂×C₄×D₃	1	2	yes	—
184	C₂³×C₃	—	7	1	no	—
185	C₂×C₃×C₄	—	7	1	no	—
186	C₂×C₃×C₄	—	7	4	no	—
187	C₂×C₃×C₄	—	7	2	no	—
188	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
190	C₂×Dic₃	—	7	2	no	—
191	C₂×Dic₃	—	7	8	no	—
192	C₂×Dic₃	—	7	1	no	—
193	C₂×Dic₃	—	7	1	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	Dic₆	—	7	4	no	—
198	Dic₆	—	7	8	no	—
199	Dic₆	C₄×D₇	1	4	yes	—
200	C₃×C₇	#519	1	1	yes	characteristic, complemented, Hall
201	C₂×Q₈	—	3	1	no	—
202	C₂×Q₈	—	21	2	no	—
203	C₂×Q₈	—	21	8	no	—
204	C₂×Q₈	—	21	1	no	—
205	C₂²×C₄	—	7	1	no	—
206	C₂²×C₄	—	21	2	no	—
207	C₄⋊C₄	—	3	2	no	—
208	C₄⋊C₄	—	7	1	no	—
209	C₄⋊C₄	—	21	2	no	—
210	C₄⋊C₄	—	21	1	no	—
211	C₄⋊C₄	—	21	1	no	—
212	C₄⋊C₄	—	21	2	no	—
213	C₄⋊C₄	—	21	2	no	—
214	C₄⋊C₄	D₃×D₇	1	1	yes	characteristic
215	C₂²⋊C₄	—	7	2	no	—
216	C₂²⋊C₄	—	21	1	no	—
217	C₂²⋊C₄	—	21	1	no	—
218	C₄²	—	3	2	no	—
219	C₄²	—	3	1	no	—
220	C₄²	—	7	2	no	—
221	C₄²	—	21	2	no	—
222	C₄²	—	21	2	no	—
223	C₄²	—	21	2	no	—
224	C₄²	—	21	1	no	—
225	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
226	C₂×C₇	#1010	1	1	yes	characteristic
227	C₂×C₇	#1010	1	1	yes	characteristic
228	D₇	—	2	2	no	—
229	C₂²×C₃	—	7	2	no	—
230	C₂²×C₃	—	14	1	no	—
231	C₂²×C₃	—	14	1	no	—
232	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	7	2	no	—
235	C₃×C₄	—	14	2	no	—
236	C₃×C₄	C₂×C₄×D₇	1	2	yes	—
237	Dic₃	—	2	1	no	—
238	Dic₃	—	2	1	no	—
239	Dic₃	—	7	4	no	—
240	Dic₃	—	14	1	no	—
241	Dic₃	—	14	1	no	—
242	Dic₃	C₂×C₄×D₇	1	4	yes	—
243	C₂³	—	7	1	no	—
244	Q₈	—	3	4	no	—
245	Q₈	—	21	8	no	—
246	Q₈	—	21	4	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	4	no	—
252	C₂×C₄	—	7	2	no	—
253	C₂×C₄	—	21	8	no	—
254	C₂×C₄	—	21	2	no	—
255	C₂×C₄	—	21	1	no	—
256	C₂×C₄	—	21	1	no	—
257	C₂×C₄	C₂×D₃×D₇	1	2	yes	—
258	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
259	C₇	#5516	1	1	yes	characteristic, complemented, Sylow
260	C₂×C₃	—	14	2	no	—
261	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
262	C₂×C₃	#6551	1	1	yes	characteristic
263	C₂×C₃	#6554	1	1	yes	characteristic
264	C₂²	—	7	2	no	—
265	C₂²	—	14	1	no	—
266	C₂²	—	14	1	no	—
267	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	3	4	no	—
270	C₄	—	6	1	no	—
271	C₄	—	6	1	no	—
272	C₄	—	7	2	no	—
273	C₄	—	14	2	no	—
274	C₄	—	21	4	no	—
275	C₄	—	42	1	no	—
276	C₄	—	42	1	no	—
277	C₄	C₄×D₃×D₇	1	2	yes	—
278	C₃	#90263	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	14	2	no	—
280	C₂	C₂×C₄×D₃×D₇	1	1	yes	characteristic
281	C₂	#10621567	1	1	yes	characteristic
282	C₂	#10621575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₇
commutator	C₂×C₃×C₇	C₂³×C₄
Fitting	C₃×C₇×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂²×C₃×D₇	C₂³
hypercenter	C₄⋊C₄	D₃×D₇
hypocenter	C₃×C₇	#519
LCS	G ▷ C₂×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
2. #10621370	C₂	semidirect
3. C₇×#887	C₂	semidirect
5. C₄×#4248	C₂	semidirect
9. #10620906	C₂	semidirect
10. #10620902	C₂	semidirect
11. #10620885	C₂	semidirect
18. D₇×Dic₆	C₄	semidirect
95. #887	D₇	semidirect
200. C₃×C₇	#519	semidirect
259. C₇	#5516	semidirect
278. C₃	#90263	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
#10621370	2
C₇×#887	2
C₃×#6455	2
C₄×#4248	2
#10620956	2
#10620955	2
#10620919	2
#10620906	2
#10620902	2
#10620885	2
#90263	3
#5516	7