Subgroups — C₃×#89730 #13055391

Order 1344.

572 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×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₇⋊#95	C₂	1	2	yes	maximal, complemented
6	#89730	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
10	C₂×C₃×C₂²⋊D₇	—	2	2	no	—
11	C₂×C₃×C₂²⋊D₇	—	2	2	no	—
12	C₂×C₃×C₂²⋊D₇	C₄	1	4	yes	complemented
13	C₂²×C₃×Dic₇	—	2	2	no	—
14	C₂²×C₃×Dic₇	—	2	4	no	—
15	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
17	C₃×C₇×C₂²⋊C₄	—	2	2	no	—
18	C₃×(C₂×D₇)⋊C₄	—	2	4	no	—
19	C₂²×C₂²⋊D₇	C₂×C₃	1	1	yes	characteristic
20	C₂³×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	complemented
23	C₇⋊#95	C₂×C₃	1	2	yes	complemented
24	C₃×#386	—	7	1	no	maximal, Hall
25	C₂³×C₃×C₇	—	2	2	no	—
26	C₂³×C₃×C₇	—	2	2	no	—
27	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₃×C₇	D₄	1	2	yes	—
29	C₂³×C₃×C₇	D₄	1	2	yes	—
30	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₃×D₇	—	2	2	no	—
32	C₂²×C₃×D₇	—	2	1	no	—
33	C₂²×C₃×D₇	—	2	1	no	—
34	C₂²×C₃×D₇	—	2	2	no	—
35	C₂²×C₃×D₇	C₂×C₄	1	2	yes	complemented
36	C₂×C₃×C₄×C₇	—	2	2	no	—
37	C₂×C₃×C₄×C₇	—	2	2	no	—
38	C₂×C₃×C₄×C₇	—	2	2	no	—
39	C₃×C₂²⋊D₇	—	2	8	no	—
40	C₂×C₃×Dic₇	—	2	8	no	—
41	C₂×C₃×Dic₇	—	2	2	no	—
42	C₂×C₃×Dic₇	—	2	2	no	—
43	C₂×C₃×Dic₇	—	2	1	no	—
44	C₂×C₃×Dic₇	—	2	1	no	—
45	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	complemented
46	C₂×C₃×Dic₇	D₄	1	4	yes	complemented
47	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
50	C₂×C₂²⋊D₇	—	2	2	no	—
51	C₂×C₂²⋊D₇	—	2	2	no	—
52	C₂×C₂²⋊D₇	C₃×C₄	1	4	yes	complemented
53	C₂²×Dic₇	—	2	2	no	—
54	C₂²×Dic₇	—	2	4	no	—
55	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
56	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
57	C₇×C₂²⋊C₄	—	2	2	no	—
58	(C₂×D₇)⋊C₄	—	2	4	no	—
59	C₂²×C₃×D₄	—	7	1	no	—
60	C₂³×C₃×C₄	—	7	1	no	—
61	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
62	C₂×C₃×C₂²⋊C₄	D₇	1	1	yes	characteristic, complemented
63	C₃×#95	—	7	2	no	—
64	C₂²×C₃×C₇	—	2	4	no	—
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	—	2	4	no	—
67	C₂²×C₃×C₇	—	2	2	no	—
68	C₂²×C₃×C₇	C₂²⋊C₄	1	4	yes	complemented
69	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
72	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₃×C₇	#41	1	2	yes	—
74	C₂×C₃×D₇	—	2	2	no	—
75	C₂×C₃×D₇	—	2	4	no	—
76	C₂×C₃×D₇	—	4	1	no	—
77	C₂×C₃×D₇	—	4	2	no	—
78	C₂×C₃×D₇	—	4	1	no	—
79	C₃×C₄×C₇	—	4	2	no	—
80	C₃×Dic₇	—	2	2	no	—
81	C₃×Dic₇	—	2	4	no	—
82	#386	—	7	1	no	Sylow
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
86	C₂³×C₇	C₃×D₄	1	2	yes	—
87	C₂³×C₇	C₃×D₄	1	2	yes	—
88	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
89	C₂²×D₇	—	2	2	no	—
90	C₂²×D₇	—	2	1	no	—
91	C₂²×D₇	—	2	1	no	—
92	C₂²×D₇	—	2	2	no	—
93	C₂²×D₇	C₂×C₃×C₄	1	2	yes	complemented
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂²⋊D₇	—	2	8	no	—
98	C₂×Dic₇	—	2	8	no	—
99	C₂×Dic₇	—	2	2	no	—
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	—	2	1	no	—
102	C₂×Dic₇	—	2	1	no	—
103	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	complemented
104	C₂×Dic₇	C₃×D₄	1	4	yes	complemented
105	C₂⁴×C₃	—	7	1	no	—
106	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
107	C₂×C₃×D₄	—	7	4	no	—
108	C₂×C₃×D₄	—	14	2	no	—
109	C₂×C₃×D₄	—	14	2	no	—
110	C₂²×C₃×C₄	—	7	1	no	—
111	C₂²×C₃×C₄	—	7	2	no	—
112	C₂²×C₃×C₄	—	14	4	no	—
113	C₂²×C₃×C₄	—	14	2	no	—
114	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
115	C₃×C₂²⋊C₄	—	2	2	no	—
116	C₃×C₂²⋊C₄	—	14	4	no	—
117	C₂×C₃×C₇	—	2	4	no	—
118	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
120	C₂×C₃×C₇	#120	1	1	yes	characteristic
121	C₂×C₃×C₇	#121	1	2	yes	—
122	C₂×C₃×C₇	#123	1	1	yes	characteristic
123	C₃×D₇	—	4	2	no	—
124	C₂²×D₄	—	7	1	no	—
125	C₂³×C₄	—	7	1	no	—
126	C₂×C₂²⋊C₄	—	7	2	no	—
127	C₂×C₂²⋊C₄	C₃×D₇	1	1	yes	characteristic, complemented
128	#95	—	7	2	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	2	4	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	C₃×C₂²⋊C₄	1	4	yes	complemented
134	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
137	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×C₇	C₃×#41	1	2	yes	—
139	C₂×D₇	—	2	2	no	—
140	C₂×D₇	—	2	4	no	—
141	C₂×D₇	—	4	1	no	—
142	C₂×D₇	—	4	2	no	—
143	C₂×D₇	—	4	1	no	—
144	C₄×C₇	—	4	2	no	—
145	Dic₇	—	2	2	no	—
146	Dic₇	—	2	4	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	7	2	no	—
150	C₂³×C₃	—	14	1	no	—
151	C₂³×C₃	—	14	2	no	—
152	C₂³×C₃	—	14	2	no	—
153	C₂³×C₃	—	14	1	no	—
154	C₂³×C₃	C₄×D₇	1	2	yes	—
155	C₂³×C₃	D₂₈	1	2	yes	—
156	C₂³×C₃	C₂²⋊D₇	1	2	yes	—
157	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
158	C₃×D₄	—	14	8	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	7	2	no	—
163	C₂×C₃×C₄	—	7	4	no	—
164	C₂×C₃×C₄	—	14	8	no	—
165	C₂×C₃×C₄	—	14	2	no	—
166	C₂×C₃×C₄	—	14	1	no	—
167	C₂×C₃×C₄	—	14	2	no	—
168	C₂×C₃×C₄	—	14	1	no	—
169	C₃×C₇	#386	1	1	yes	characteristic, complemented, Hall
170	C₂⁴	—	7	1	no	—
171	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
172	C₂×D₄	—	7	4	no	—
173	C₂×D₄	—	14	2	no	—
174	C₂×D₄	—	14	2	no	—
175	C₂²×C₄	—	7	1	no	—
176	C₂²×C₄	—	7	2	no	—
177	C₂²×C₄	—	14	4	no	—
178	C₂²×C₄	—	14	2	no	—
179	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	14	4	no	—
182	C₂×C₇	—	2	4	no	—
183	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
185	C₂×C₇	C₃×#120	1	1	yes	characteristic
186	C₂×C₇	C₃×#121	1	2	yes	—
187	C₂×C₇	C₃×#123	1	1	yes	characteristic
188	D₇	—	4	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	14	4	no	—
194	C₂²×C₃	—	14	2	no	—
195	C₂²×C₃	—	28	1	no	—
196	C₂²×C₃	—	28	1	no	—
197	C₂²×C₃	—	28	2	no	—
198	C₂²×C₃	(C₂×D₇)⋊C₄	1	4	yes	complemented
199	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
200	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
201	C₂²×C₃	D₄×D₇	1	2	yes	—
202	C₂²×C₃	#1159	1	2	yes	—
203	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	14	4	no	—
206	C₃×C₄	—	14	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	7	2	no	—
210	C₂³	—	14	1	no	—
211	C₂³	—	14	2	no	—
212	C₂³	—	14	2	no	—
213	C₂³	—	14	1	no	—
214	C₂³	C₃×C₄×D₇	1	2	yes	—
215	C₂³	C₃×D₂₈	1	2	yes	—
216	C₂³	C₃×C₂²⋊D₇	1	2	yes	—
217	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
218	D₄	—	14	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	7	2	no	—
223	C₂×C₄	—	7	4	no	—
224	C₂×C₄	—	14	8	no	—
225	C₂×C₄	—	14	2	no	—
226	C₂×C₄	—	14	1	no	—
227	C₂×C₄	—	14	2	no	—
228	C₂×C₄	—	14	1	no	—
229	C₇	C₃×#386	1	1	yes	characteristic, complemented, Sylow
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	28	2	no	—
232	C₂×C₃	(C₂²⋊D₇)⋊C₄	1	2	yes	—
233	C₂×C₃	#6445	1	1	yes	characteristic
234	C₂×C₃	#6449	1	1	yes	characteristic
235	C₂×C₃	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₃	#6502	1	2	yes	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	14	4	no	—
242	C₂²	—	14	2	no	—
243	C₂²	—	28	1	no	—
244	C₂²	—	28	1	no	—
245	C₂²	—	28	2	no	—
246	C₂²	C₃×(C₂×D₇)⋊C₄	1	4	yes	complemented
247	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
248	C₂²	C₂×C₃×D₂₈	1	1	yes	characteristic
249	C₂²	C₃×D₄×D₇	1	2	yes	—
250	C₂²	C₃×#1159	1	2	yes	—
251	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
252	C₄	—	4	2	no	—
253	C₄	—	14	4	no	—
254	C₄	—	14	2	no	—
255	C₃	#89730	1	1	yes	characteristic, direct product factor, Sylow
256	C₂	—	2	4	no	—
257	C₂	—	28	2	no	—
258	C₂	C₃×(C₂²⋊D₇)⋊C₄	1	2	yes	—
259	C₂	C₃×#6445	1	1	yes	characteristic
260	C₂	C₃×#6449	1	1	yes	characteristic
261	C₂	C₂×C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
262	C₂	C₃×#6502	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂²×C₃×C₂²⋊D₇	C₂
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₂²⋊C₄	D₇
hypocenter	C₇	C₃×#386
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Dic₇	C₂	semidirect
3. C₂×C₃×C₇×C₂²⋊C₄	C₂	semidirect
4. C₂×C₃×(C₂×D₇)⋊C₄	C₂	semidirect
5. C₃×C₇⋊#95	C₂	semidirect
6. #89730	C₃	direct (N × H)
12. C₂×C₃×C₂²⋊D₇	C₄	semidirect
20. C₂³×Dic₇	C₂×C₃	semidirect
21. C₂×C₇×C₂²⋊C₄	C₂×C₃	semidirect
22. C₂×(C₂×D₇)⋊C₄	C₂×C₃	semidirect
23. C₇⋊#95	C₂×C₃	semidirect
35. C₂²×C₃×D₇	C₂×C₄	semidirect
45. C₂×C₃×Dic₇	C₂×C₄	semidirect
46. C₂×C₃×Dic₇	D₄	semidirect
52. C₂×C₂²⋊D₇	C₃×C₄	semidirect
62. C₂×C₃×C₂²⋊C₄	D₇	semidirect
68. C₂²×C₃×C₇	C₂²⋊C₄	semidirect
93. C₂²×D₇	C₂×C₃×C₄	semidirect
103. C₂×Dic₇	C₂×C₃×C₄	semidirect
104. C₂×Dic₇	C₃×D₄	semidirect
127. C₂×C₂²⋊C₄	C₃×D₇	semidirect
133. C₂²×C₇	C₃×C₂²⋊C₄	semidirect
169. C₃×C₇	#386	semidirect
198. C₂²×C₃	(C₂×D₇)⋊C₄	semidirect
229. C₇	C₃×#386	semidirect
246. C₂²	C₃×(C₂×D₇)⋊C₄	semidirect
255. C₃	#89730	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₂²⋊D₇	2
C₂³×C₃×Dic₇	2
C₂×C₃×C₇×C₂²⋊C₄	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₃×C₇⋊#95	2
#89730	3
C₃×#386	7

Subgroups — C3×#89730 #13055391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89730 #13055391