Subgroups — #13059351

Order 1344.

384 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621407	C₂	1	1	yes	maximal, characteristic
2	C₇×#926	C₂	1	1	yes	maximal, characteristic
3	C₃×#6487	C₂	1	1	yes	maximal, characteristic
4	#10621093	C₂	1	2	yes	maximal
5	#10621092	C₂	1	2	yes	maximal
6	#10620951	C₂	1	1	yes	maximal, characteristic, complemented
7	#10620943	C₂	1	1	yes	maximal, characteristic, complemented
8	#10620902	C₂	1	1	yes	maximal, characteristic, complemented
9	#10620898	C₂	1	2	yes	maximal, complemented
10	#10620897	C₂	1	2	yes	maximal, complemented
11	Dic₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#90209	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
16	#66636	C₂²	1	2	yes	—
17	C₄⋊Dic₂₁	C₂²	1	1	yes	characteristic
18	#66629	C₂²	1	2	yes	—
19	C₇×#217	C₂²	1	1	yes	characteristic
20	C₇×#217	C₂²	1	1	yes	characteristic
21	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
22	C₇×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₄×C₇×Dic₃	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	Dic₃⋊Dic₇	C₂²	1	2	yes	—
29	Dic₃⋊Dic₇	C₂²	1	2	yes	—
30	#66577	C₂²	1	2	yes	—
31	#66577	C₂²	1	2	yes	—
32	Dic₇⋊Dic₃	C₂²	1	2	yes	—
33	Dic₇⋊Dic₃	C₂²	1	2	yes	—
34	Dic₃×Dic₇	C₂²	1	2	yes	—
35	Dic₃×Dic₇	C₂²	1	2	yes	—
36	C₇×#117	—	3	1	no	—
37	#6487	—	3	1	no	—
38	#6487	D₃	1	1	yes	characteristic
39	#6452	—	3	2	no	—
40	#6452	—	3	2	no	—
41	#6442	—	3	2	no	—
42	#6442	—	3	2	no	—
43	#6433	—	3	1	no	—
44	#6432	—	3	1	no	—
45	C₄×Dic₁₄	—	3	1	no	—
46	C₄×Dic₁₄	—	3	1	no	—
47	#5625	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₇	—	2	2	no	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₁	C₂³	1	2	yes	—
53	C₂×Dic₂₁	C₂³	1	2	yes	—
54	Dic₄₂	—	2	2	no	—
55	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₇×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	C₂×C₃×Dic₇	C₂³	1	2	yes	—
61	C₃×Dic₁₄	—	2	2	no	—
62	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
63	C₂×Dic₁₄	—	3	1	no	—
64	C₂×Dic₁₄	C₂×D₃	1	1	yes	characteristic
65	C₇×C₄⋊C₄	—	3	1	no	—
66	C₇×C₄⋊C₄	—	3	1	no	—
67	C₇×C₂²⋊C₄	—	3	1	no	—
68	C₇×C₂²⋊C₄	—	3	1	no	—
69	C₄²×C₇	—	3	1	no	—
70	C₄²×C₇	—	3	1	no	—
71	C₂²⋊Dic₇	—	3	2	no	—
72	C₂²⋊Dic₇	C₂×D₃	1	2	yes	—
73	C₄⋊Dic₇	—	3	1	no	—
74	C₄⋊Dic₇	—	3	2	no	—
75	C₄⋊Dic₇	—	3	2	no	—
76	C₄⋊Dic₇	C₂×D₃	1	1	yes	characteristic
77	Dic₇⋊C₄	—	3	2	no	—
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	C₂×D₃	1	2	yes	—
83	C₄×Dic₇	—	3	2	no	—
84	C₄×Dic₇	—	3	2	no	—
85	C₃×C₂²⋊Q₈	—	7	1	no	—
86	Q₈×Dic₃	—	7	1	no	—
87	#944	—	7	1	no	—
88	#926	D₇	1	1	yes	characteristic
89	#924	—	7	1	no	—
90	#890	—	7	2	no	—
91	#890	—	7	1	no	—
92	#889	—	7	2	no	—
93	#887	—	7	1	no	—
94	#879	—	7	2	no	—
95	#879	—	7	2	no	—
96	C₂²×C₃×C₇	—	2	1	no	—
97	C₂²×C₃×C₇	—	2	1	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
100	C₃×C₄×C₇	—	2	1	no	—
101	C₃×C₄×C₇	#41	1	2	yes	—
102	Dic₂₁	—	2	2	no	—
103	Dic₂₁	—	2	2	no	—
104	C₃×Dic₇	—	2	2	no	—
105	C₃×Dic₇	—	2	2	no	—
106	C₇×Dic₃	—	2	1	no	—
107	C₇×Dic₃	—	2	1	no	—
108	C₇×Dic₃	—	2	1	no	—
109	C₇×Dic₃	—	2	1	no	—
110	#541	—	21	1	no	Sylow
111	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	3	1	no	—
114	C₂×C₄×C₇	—	3	1	no	—
115	C₂×C₄×C₇	—	3	1	no	—
116	C₂×C₄×C₇	—	3	1	no	—
117	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
119	C₂×Dic₇	—	3	2	no	—
120	C₂×Dic₇	—	3	2	no	—
121	C₂×Dic₇	C₂²×D₃	1	2	yes	—
122	C₂×Dic₇	C₂²×D₃	1	2	yes	—
123	Dic₁₄	—	2	2	no	—
124	Dic₁₄	—	6	2	no	—
125	C₂×C₃×Q₈	—	7	1	no	—
126	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
127	C₂×Dic₆	—	7	1	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	1	no	—
130	C₃×C₂²⋊C₄	—	7	2	no	—
131	#217	—	7	2	no	—
132	#217	C₂×D₇	1	1	yes	characteristic
133	#217	C₂×D₇	1	1	yes	characteristic
134	C₄⋊Dic₃	—	7	1	no	—
135	C₄⋊Dic₃	—	7	2	no	—
136	C₄⋊Dic₃	—	7	2	no	—
137	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
138	C₄⋊Dic₃	C₂×D₇	1	1	yes	characteristic
139	Dic₃⋊C₄	—	7	2	no	—
140	Dic₃⋊C₄	—	7	2	no	—
141	Dic₃⋊C₄	—	7	2	no	—
142	Dic₃⋊C₄	—	7	2	no	—
143	Dic₃⋊C₄	—	7	2	no	—
144	C₄×Dic₃	—	7	2	no	—
145	C₄×Dic₃	—	7	2	no	—
146	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
147	C₄×Dic₃	C₂×D₇	1	1	yes	characteristic
148	C₂×C₃×C₇	—	4	1	no	—
149	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄⋊Q₈	—	21	1	no	—
153	#126	—	21	2	no	—
154	#126	—	21	2	no	—
155	#125	—	21	1	no	—
156	#125	—	21	2	no	—
157	#125	—	21	2	no	—
158	C₂²⋊Q₈	—	7	1	no	—
159	C₂²⋊Q₈	—	21	1	no	—
160	C₄×Q₈	—	21	1	no	—
161	C₄×Q₈	—	21	1	no	—
162	#117	—	3	1	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	6	1	no	—
169	C₄×C₇	—	6	1	no	—
170	C₄×C₇	—	6	1	no	—
171	C₄×C₇	—	6	1	no	—
172	C₄×C₇	#237	1	2	yes	—
173	Dic₇	—	2	2	no	—
174	Dic₇	—	2	2	no	—
175	Dic₇	—	6	2	no	—
176	Dic₇	—	6	2	no	—
177	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
178	C₃×Q₈	—	14	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	7	2	no	—
181	C₂×C₃×C₄	—	7	2	no	—
182	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
184	C₂×Dic₃	—	7	2	no	—
185	C₂×Dic₃	—	7	2	no	—
186	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
188	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
189	C₂×Dic₃	C₂²×D₇	1	1	yes	characteristic
190	Dic₆	—	14	2	no	—
191	C₃×C₇	#541	1	1	yes	characteristic, complemented, Hall
192	C₂×Q₈	—	7	1	no	—
193	C₂×Q₈	—	21	1	no	—
194	C₂²×C₄	D₃×D₇	1	1	yes	characteristic
195	C₄⋊C₄	—	3	1	no	—
196	C₄⋊C₄	—	3	1	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	—	7	1	no	—
199	C₄⋊C₄	—	21	2	no	—
200	C₄⋊C₄	—	21	2	no	—
201	C₄⋊C₄	—	21	2	no	—
202	C₄⋊C₄	—	21	2	no	—
203	C₄⋊C₄	—	21	2	no	—
204	C₄⋊C₄	—	21	2	no	—
205	C₄⋊C₄	—	21	2	no	—
206	C₄⋊C₄	—	21	1	no	—
207	C₂²⋊C₄	—	3	1	no	—
208	C₂²⋊C₄	—	3	1	no	—
209	C₂²⋊C₄	—	7	2	no	—
210	C₂²⋊C₄	—	21	2	no	—
211	C₄²	—	3	1	no	—
212	C₄²	—	3	1	no	—
213	C₄²	—	21	2	no	—
214	C₄²	—	21	2	no	—
215	C₂×C₇	—	4	1	no	—
216	C₂×C₇	C₂×#237	1	1	yes	characteristic
217	C₂×C₇	#1007	1	1	yes	characteristic
218	C₂×C₇	#1010	1	1	yes	characteristic
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	14	2	no	—
225	C₃×C₄	—	14	2	no	—
226	C₃×C₄	#1157	1	2	yes	—
227	Dic₃	—	2	1	no	—
228	Dic₃	—	2	1	no	—
229	Dic₃	—	2	1	no	—
230	Dic₃	—	2	1	no	—
231	Dic₃	—	14	2	no	—
232	Dic₃	—	14	2	no	—
233	C₂³	C₂×D₃×D₇	1	1	yes	characteristic
234	Q₈	—	14	2	no	—
235	Q₈	—	42	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	3	1	no	—
238	C₂×C₄	—	3	1	no	—
239	C₂×C₄	—	3	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	7	2	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	—	21	2	no	—
244	C₂×C₄	—	21	2	no	—
245	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₃×D₇	1	1	yes	characteristic
247	C₇	#5625	1	1	yes	characteristic, complemented, Sylow
248	C₂×C₃	—	4	1	no	—
249	C₂×C₃	C₂×#1157	1	1	yes	characteristic
250	C₂×C₃	#6554	1	1	yes	characteristic
251	C₂×C₃	#6554	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂²×D₃×D₇	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	6	1	no	—
258	C₄	—	6	1	no	—
259	C₄	—	6	1	no	—
260	C₄	—	6	1	no	—
261	C₄	—	14	2	no	—
262	C₄	—	14	2	no	—
263	C₄	—	42	2	no	—
264	C₄	—	42	2	no	—
265	C₄	#66683	1	2	yes	—
266	C₃	#90209	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	4	1	no	—
268	C₂	#10621548	1	1	yes	characteristic
269	C₂	#10621551	1	1	yes	characteristic
270	C₂	C₂×#66683	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	C₂³×C₃×C₇	C₂³
hypercenter	C₂²×C₄	D₃×D₇
hypocenter	C₃×C₇	#541
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
6. #10620951	C₂	semidirect
7. #10620943	C₂	semidirect
8. #10620902	C₂	semidirect
9. #10620898	C₂	semidirect
10. #10620897	C₂	semidirect
11. Dic₃×Dic₁₄	C₂	semidirect
191. C₃×C₇	#541	semidirect
247. C₇	#5625	semidirect
266. C₃	#90209	semidirect

Maximal subgroups

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

subgroup	index
#10621407	2
C₇×#926	2
C₃×#6487	2
#10621093	2
#10621092	2
#10620951	2
#10620943	2
#10620902	2
#10620898	2
#10620897	2
Dic₃×Dic₁₄	2
#90209	3
#5625	7