Subgroups — #13059362

Order 1344.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇×#67	C₂	1	1	yes	maximal, characteristic
3	C₃×#6487	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621105	C₂	1	2	yes	maximal
5	#10621094	C₂	1	2	yes	maximal, complemented
6	#10620943	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620931	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620904	C₂	1	2	yes	maximal, complemented
9	#10620900	C₂	1	1	yes	maximal, characteristic, complemented
10	#10620896	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90232	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₁	C₂²	1	2	yes	—
15	C₂×C₇×#67	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₂₁	C₂²	1	2	yes	—
21	C₇×#217	C₂²	1	1	yes	characteristic
22	C₇×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
23	C₇×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
24	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
25	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	#66577	C₂²	1	2	yes	—
29	#66577	C₂²	1	2	yes	—
30	Dic₇⋊Dic₃	C₂²	1	2	yes	—
31	Dic₇⋊Dic₃	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₇	C₂²	1	2	yes	—
34	Dic₃×Dic₇	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₄×C₇×D₄	—	3	1	no	—
37	#6498	—	3	2	no	—
38	#6489	—	3	1	no	—
39	C₂×C₄⋊Dic₇	—	3	1	no	—
40	#6487	—	3	1	no	—
41	#6487	D₃	1	1	yes	characteristic, complemented
42	#6452	—	3	2	no	—
43	#6442	—	3	2	no	—
44	#6440	—	3	2	no	—
45	#6433	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5642	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂²×D₃×C₇	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₇	—	2	2	no	—
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₁	—	2	4	no	—
54	C₂×Dic₂₁	C₂³	1	2	yes	—
55	C₂×Dic₂₁	C₂³	1	2	yes	—
56	C₇×#67	—	2	2	no	—
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
60	C₄×D₃×C₇	—	2	2	no	—
61	C₂×C₃×Dic₇	C₂³	1	2	yes	—
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₃×Dic₁₄	—	2	2	no	—
64	C₂×C₇×D₄	—	3	1	no	—
65	C₂²×C₄×C₇	—	3	1	no	—
66	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
67	C₂²×Dic₇	—	3	2	no	—
68	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	3	1	no	—
70	C₇×C₂²⋊C₄	—	3	1	no	—
71	C₇×C₂²⋊C₄	—	3	1	no	—
72	C₄²×C₇	—	3	1	no	—
73	C₂²⋊Dic₇	—	3	2	no	—
74	C₂²⋊Dic₇	—	3	2	no	—
75	C₂²⋊Dic₇	C₂×D₃	1	2	yes	—
76	C₄⋊Dic₇	—	3	1	no	—
77	C₄⋊Dic₇	—	3	1	no	—
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	2	no	—
82	Dic₇⋊C₄	—	3	2	no	—
83	Dic₇⋊C₄	—	3	2	no	—
84	Dic₇⋊C₄	C₂×D₃	1	2	yes	—
85	C₄×Dic₇	—	3	2	no	—
86	C₄×Dic₇	—	3	2	no	—
87	C₃×C₂²⋊Q₈	—	7	1	no	—
88	#946	—	7	1	no	—
89	Q₈×Dic₃	—	7	1	no	—
90	C₄×#67	D₇	1	1	yes	characteristic
91	C₂×C₄⋊Dic₃	—	7	1	no	—
92	#898	—	7	2	no	—
93	#892	—	7	1	no	—
94	#890	—	7	2	no	—
95	#890	—	7	1	no	—
96	#886	—	7	2	no	—
97	#877	—	7	2	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	—	2	1	no	—
100	C₂²×C₃×C₇	#41	1	2	yes	—
101	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
102	C₂×D₃×C₇	—	2	1	no	—
103	C₂×D₃×C₇	—	2	1	no	—
104	C₂×D₃×C₇	—	2	1	no	—
105	C₃×C₄×C₇	—	2	1	no	—
106	C₃×C₄×C₇	#41	1	2	yes	—
107	Dic₂₁	—	2	2	no	—
108	Dic₂₁	—	2	2	no	—
109	C₃×Dic₇	—	2	2	no	—
110	C₃×Dic₇	—	2	2	no	—
111	C₇×Dic₃	—	2	1	no	—
112	C₇×Dic₃	—	2	1	no	—
113	C₇×Dic₃	—	2	1	no	—
114	#552	—	21	1	no	Sylow
115	C₂³×C₇	—	3	1	no	—
116	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
117	C₇×D₄	—	6	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	3	1	no	—
120	C₂×C₄×C₇	—	3	1	no	—
121	C₂×C₄×C₇	—	3	1	no	—
122	C₂×C₄×C₇	—	6	2	no	—
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	—	6	4	no	—
128	C₂×Dic₇	C₂²×D₃	1	2	yes	—
129	C₂×Dic₇	C₂²×D₃	1	2	yes	—
130	Dic₁₄	—	2	2	no	—
131	C₂×C₃×Q₈	—	7	1	no	—
132	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
133	C₂×#67	C₂×D₇	1	1	yes	characteristic
134	C₂²×Dic₃	—	7	2	no	—
135	C₂×C₄×D₃	C₂×D₇	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	7	2	no	—
137	C₃×C₄⋊C₄	—	7	1	no	—
138	C₃×C₂²⋊C₄	—	7	2	no	—
139	#217	C₂×D₇	1	1	yes	characteristic
140	(C₂×D₃)⋊C₄	—	7	2	no	—
141	(C₂×D₃)⋊C₄	—	7	2	no	—
142	(C₂×D₃)⋊C₄	C₂×D₇	1	1	yes	characteristic
143	C₄⋊Dic₃	—	7	2	no	—
144	C₄⋊Dic₃	—	7	1	no	—
145	C₄⋊Dic₃	—	7	1	no	—
146	C₄⋊Dic₃	—	7	2	no	—
147	C₄⋊Dic₃	—	7	2	no	—
148	Dic₃⋊C₄	—	7	2	no	—
149	Dic₃⋊C₄	—	7	2	no	—
150	Dic₃⋊C₄	C₂×D₇	1	1	yes	characteristic
151	C₄×Dic₃	—	7	2	no	—
152	C₄×Dic₃	—	7	2	no	—
153	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
154	C₂×C₃×C₇	—	2	2	no	—
155	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
156	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
158	D₃×C₇	—	4	1	no	—
159	#126	—	21	2	no	—
160	#125	—	21	1	no	—
161	#125	—	21	2	no	—
162	#123	—	21	2	no	—
163	C₂²⋊Q₈	—	7	1	no	—
164	C₂²⋊Q₈	—	21	1	no	—
165	C₄×Q₈	—	21	1	no	—
166	C₄×D₄	—	3	1	no	—
167	#117	—	21	2	no	—
168	#117	—	21	1	no	—
169	C₂×C₄⋊C₄	—	21	1	no	—
170	C₂²×C₇	—	2	1	no	—
171	C₂²×C₇	—	2	1	no	—
172	C₂²×C₇	—	6	1	no	—
173	C₂²×C₇	—	6	1	no	—
174	C₂²×C₇	—	6	1	no	—
175	C₂²×C₇	#239	1	2	yes	—
176	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	6	1	no	—
179	C₄×C₇	—	6	1	no	—
180	C₄×C₇	—	6	1	no	—
181	C₄×C₇	#237	1	2	yes	—
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	2	no	—
184	Dic₇	—	6	2	no	—
185	Dic₇	—	6	2	no	—
186	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
187	C₂²×D₃	C₂²×D₇	1	1	yes	characteristic
188	C₃×Q₈	—	14	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	7	2	no	—
191	C₂×C₃×C₄	—	7	2	no	—
192	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
194	#67	—	2	2	no	—
195	C₂×Dic₃	—	7	2	no	—
196	C₂×Dic₃	—	7	2	no	—
197	C₂×Dic₃	—	14	4	no	—
198	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
199	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
200	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
201	C₄×D₃	—	2	2	no	—
202	C₃×C₇	#552	1	1	yes	characteristic, complemented, Hall
203	C₂×Q₈	—	7	1	no	—
204	C₂×D₄	—	3	1	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	21	2	no	—
207	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	7	2	no	—
210	C₄⋊C₄	—	7	1	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	—	21	1	no	—
217	C₄⋊C₄	—	21	2	no	—
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	3	1	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	21	2	no	—
222	C₂²⋊C₄	—	21	2	no	—
223	C₄²	—	3	1	no	—
224	C₄²	—	21	2	no	—
225	C₄²	—	21	2	no	—
226	C₂×C₇	—	2	2	no	—
227	C₂×C₇	—	12	1	no	—
228	C₂×C₇	C₂×#237	1	1	yes	characteristic
229	C₂×C₇	C₂×#239	1	1	yes	characteristic
230	C₂×C₇	#1010	1	1	yes	characteristic
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	#1157	1	2	yes	—
234	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	—	2	1	no	—
237	C₂×D₃	—	2	1	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	14	2	no	—
240	C₃×C₄	—	14	2	no	—
241	C₃×C₄	#1159	1	2	yes	—
242	Dic₃	—	2	1	no	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	14	2	no	—
246	Dic₃	—	14	2	no	—
247	C₂³	—	3	1	no	—
248	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
249	Q₈	—	14	2	no	—
250	D₄	—	6	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	1	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	—	7	2	no	—
257	C₂×C₄	—	7	2	no	—
258	C₂×C₄	—	21	2	no	—
259	C₂×C₄	—	21	2	no	—
260	C₂×C₄	—	42	4	no	—
261	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
263	C₇	#5642	1	1	yes	characteristic, complemented, Sylow
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	C₂×#1157	1	1	yes	characteristic
266	C₂×C₃	C₂×#1159	1	1	yes	characteristic
267	C₂×C₃	#6554	1	1	yes	characteristic
268	D₃	—	4	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	6	1	no	—
272	C₂²	—	6	1	no	—
273	C₂²	—	6	1	no	—
274	C₂²	#66673	1	2	yes	—
275	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	6	1	no	—
278	C₄	—	6	1	no	—
279	C₄	—	6	1	no	—
280	C₄	—	14	2	no	—
281	C₄	—	14	2	no	—
282	C₄	—	42	2	no	—
283	C₄	—	42	2	no	—
284	C₄	#66684	1	2	yes	—
285	C₃	#90232	1	1	yes	characteristic, complemented, Sylow
286	C₂	—	2	2	no	—
287	C₂	—	12	1	no	—
288	C₂	#10621548	1	1	yes	characteristic
289	C₂	C₂×#66673	1	1	yes	characteristic
290	C₂	C₂×#66684	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×C₇×#67	C₂²
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#552
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. C₂×C₄⋊Dic₂₁	C₂	semidirect
3. C₃×#6487	C₂	semidirect
5. #10621094	C₂	semidirect
6. #10620943	C₂	semidirect
7. #10620931	C₂	semidirect
8. #10620904	C₂	semidirect
9. #10620900	C₂	semidirect
10. #10620896	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
41. #6487	D₃	semidirect
202. C₃×C₇	#552	semidirect
263. C₇	#5642	semidirect
285. C₃	#90232	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₁	2
C₄×C₇×#67	2
C₃×#6487	2
#10621105	2
#10621094	2
#10620943	2
#10620931	2
#10620904	2
#10620900	2
#10620896	2
Dic₃×Dic₁₄	2
#90232	3
#5642	7