Subgroups — C₃×#89759 #13055422

Order 1344.

468 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
4	C₂×C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₇⋊#95	C₂	1	2	yes	maximal, complemented
6	#89759	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×D₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
11	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₇	—	2	2	no	—
13	C₂×C₃×C₄×D₇	—	2	2	no	—
14	C₂×C₃×C₄×D₇	C₄	1	4	yes	complemented
15	C₃×C₇×C₄⋊C₄	—	2	2	no	—
16	C₃×(C₂×D₇)⋊C₄	—	2	4	no	—
17	C₃×C₄⋊Dic₇	—	2	2	no	—
18	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	—
21	C₂×C₄⋊Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
22	C₇⋊#95	C₂×C₃	1	2	yes	complemented
23	C₃×#380	—	7	1	no	maximal, Hall
24	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₃×D₇	—	2	1	no	—
26	C₂²×C₃×D₇	—	2	1	no	—
27	C₂²×C₃×D₇	—	2	1	no	—
28	C₂²×C₃×D₇	—	2	1	no	—
29	C₂²×C₃×D₇	C₂×C₄	1	2	yes	—
30	C₂²×C₃×D₇	D₄	1	2	yes	—
31	C₂²×C₃×D₇	Q₈	1	2	yes	—
32	C₂×C₃×C₄×C₇	—	2	2	no	—
33	C₂×C₃×C₄×C₇	—	2	2	no	—
34	C₂×C₃×C₄×C₇	—	2	2	no	—
35	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
38	C₂×C₃×Dic₇	—	2	1	no	—
39	C₂×C₃×Dic₇	—	2	1	no	—
40	C₂×C₃×Dic₇	—	2	2	no	—
41	C₂×C₃×Dic₇	—	2	2	no	—
42	C₂×C₃×Dic₇	—	2	2	no	—
43	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
44	C₃×C₄×D₇	—	2	8	no	—
45	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
47	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
48	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
49	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₄×D₇	—	2	2	no	—
51	C₂×C₄×D₇	—	2	2	no	—
52	C₂×C₄×D₇	C₃×C₄	1	4	yes	complemented
53	C₇×C₄⋊C₄	—	2	2	no	—
54	(C₂×D₇)⋊C₄	—	2	4	no	—
55	C₄⋊Dic₇	—	2	2	no	—
56	C₂³×C₃×C₄	—	7	1	no	—
57	C₂×C₃×C₄⋊C₄	—	7	1	no	—
58	C₂×C₃×C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
59	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
60	C₃×#95	—	7	2	no	—
61	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₃×C₇	#41	1	1	yes	characteristic
67	C₂²×C₃×C₇	#41	1	1	yes	characteristic
68	C₂×C₃×D₇	—	2	2	no	—
69	C₂×C₃×D₇	—	2	2	no	—
70	C₂×C₃×D₇	—	2	2	no	—
71	C₂×C₃×D₇	—	2	2	no	—
72	C₂×C₃×D₇	—	2	2	no	—
73	C₂×C₃×D₇	—	2	2	no	—
74	C₂×C₃×D₇	C₄⋊C₄	1	4	yes	complemented
75	C₃×C₄×C₇	—	4	2	no	—
76	C₃×C₄×C₇	C₂²⋊C₄	1	4	yes	complemented
77	C₃×Dic₇	—	2	2	no	—
78	C₃×Dic₇	—	4	2	no	—
79	#380	—	7	1	no	Sylow
80	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
81	C₂²×D₇	—	2	1	no	—
82	C₂²×D₇	—	2	1	no	—
83	C₂²×D₇	—	2	1	no	—
84	C₂²×D₇	—	2	1	no	—
85	C₂²×D₇	C₂×C₃×C₄	1	2	yes	—
86	C₂²×D₇	C₃×D₄	1	2	yes	—
87	C₂²×D₇	C₃×Q₈	1	2	yes	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
93	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
94	C₂×Dic₇	—	2	1	no	—
95	C₂×Dic₇	—	2	1	no	—
96	C₂×Dic₇	—	2	2	no	—
97	C₂×Dic₇	—	2	2	no	—
98	C₂×Dic₇	—	2	2	no	—
99	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
100	C₄×D₇	—	2	8	no	—
101	C₂⁴×C₃	—	7	1	no	—
102	C₂²×C₃×C₄	—	7	1	no	—
103	C₂²×C₃×C₄	—	7	4	no	—
104	C₂²×C₃×C₄	—	7	2	no	—
105	C₂²×C₃×C₄	—	14	2	no	—
106	C₂²×C₃×C₄	—	14	2	no	—
107	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
108	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	14	2	no	—
111	C₃×C₂²⋊C₄	—	14	4	no	—
112	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₇	#117	1	1	yes	characteristic
115	C₂×C₃×C₇	#121	1	1	yes	characteristic
116	C₂×C₃×C₇	#121	1	1	yes	characteristic
117	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₃×C₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₃×D₇	—	2	4	no	—
120	C₂³×C₄	—	7	1	no	—
121	C₂×C₄⋊C₄	—	7	1	no	—
122	C₂×C₄⋊C₄	C₃×D₇	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊C₄	—	7	2	no	—
124	#95	—	7	2	no	—
125	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
127	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
130	C₂²×C₇	C₃×#41	1	1	yes	characteristic
131	C₂²×C₇	C₃×#41	1	1	yes	characteristic
132	C₂×D₇	—	2	2	no	—
133	C₂×D₇	—	2	2	no	—
134	C₂×D₇	—	2	2	no	—
135	C₂×D₇	—	2	2	no	—
136	C₂×D₇	—	2	2	no	—
137	C₂×D₇	—	2	2	no	—
138	C₂×D₇	C₃×C₄⋊C₄	1	4	yes	complemented
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	C₃×C₂²⋊C₄	1	4	yes	complemented
141	Dic₇	—	2	2	no	—
142	Dic₇	—	4	2	no	—
143	C₂³×C₃	—	7	2	no	—
144	C₂³×C₃	—	7	2	no	—
145	C₂³×C₃	—	7	2	no	—
146	C₂³×C₃	—	14	1	no	—
147	C₂³×C₃	—	14	1	no	—
148	C₂³×C₃	—	14	1	no	—
149	C₂³×C₃	—	14	1	no	—
150	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	7	2	no	—
155	C₂×C₃×C₄	—	14	1	no	—
156	C₂×C₃×C₄	—	14	1	no	—
157	C₂×C₃×C₄	—	14	8	no	—
158	C₂×C₃×C₄	—	14	2	no	—
159	C₂×C₃×C₄	—	14	2	no	—
160	C₂×C₃×C₄	—	14	2	no	—
161	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
162	C₂×C₃×C₄	D₂₈	1	2	yes	—
163	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
164	C₃×C₇	#380	1	1	yes	characteristic, complemented, Hall
165	C₂⁴	—	7	1	no	—
166	C₂²×C₄	—	7	1	no	—
167	C₂²×C₄	—	7	4	no	—
168	C₂²×C₄	—	7	2	no	—
169	C₂²×C₄	—	14	2	no	—
170	C₂²×C₄	—	14	2	no	—
171	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
172	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	14	2	no	—
175	C₂²⋊C₄	—	14	4	no	—
176	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₇	C₃×#117	1	1	yes	characteristic
179	C₂×C₇	C₃×#121	1	1	yes	characteristic
180	C₂×C₇	C₃×#121	1	1	yes	characteristic
181	C₂×C₇	C₃×C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₇	C₃×C₂²⋊Q₈	1	1	yes	characteristic
183	D₇	—	2	4	no	—
184	C₂²×C₃	—	7	4	no	—
185	C₂²×C₃	—	14	2	no	—
186	C₂²×C₃	—	14	2	no	—
187	C₂²×C₃	—	14	2	no	—
188	C₂²×C₃	—	14	2	no	—
189	C₂²×C₃	—	14	2	no	—
190	C₂²×C₃	—	14	2	no	—
191	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
192	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
193	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
194	C₂²×C₃	#1159	1	1	yes	characteristic
195	C₂²×C₃	Q₈×D₇	1	1	yes	characteristic
196	C₂²×C₃	#1161	1	1	yes	characteristic
197	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	14	2	no	—
200	C₃×C₄	—	28	2	no	—
201	C₃×C₄	(C₂×D₇)⋊C₄	1	4	yes	complemented
202	C₂³	—	7	2	no	—
203	C₂³	—	7	2	no	—
204	C₂³	—	7	2	no	—
205	C₂³	—	14	1	no	—
206	C₂³	—	14	1	no	—
207	C₂³	—	14	1	no	—
208	C₂³	—	14	1	no	—
209	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	7	2	no	—
214	C₂×C₄	—	14	1	no	—
215	C₂×C₄	—	14	1	no	—
216	C₂×C₄	—	14	8	no	—
217	C₂×C₄	—	14	2	no	—
218	C₂×C₄	—	14	2	no	—
219	C₂×C₄	—	14	2	no	—
220	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
221	C₂×C₄	C₃×D₂₈	1	2	yes	—
222	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
223	C₇	C₃×#380	1	1	yes	characteristic, complemented, Sylow
224	C₂×C₃	—	14	4	no	—
225	C₂×C₃	D₇×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₃	#6455	1	1	yes	characteristic
227	C₂×C₃	#6458	1	1	yes	characteristic
228	C₂×C₃	#6460	1	1	yes	characteristic
229	C₂×C₃	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
230	C₂×C₃	#6501	1	1	yes	characteristic
231	C₂×C₃	#6509	1	1	yes	characteristic
232	C₂²	—	7	4	no	—
233	C₂²	—	14	2	no	—
234	C₂²	—	14	2	no	—
235	C₂²	—	14	2	no	—
236	C₂²	—	14	2	no	—
237	C₂²	—	14	2	no	—
238	C₂²	—	14	2	no	—
239	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
240	C₂²	C₂×C₃×D₂₈	1	1	yes	characteristic
241	C₂²	C₃×D₄×D₇	1	1	yes	characteristic
242	C₂²	C₃×#1159	1	1	yes	characteristic
243	C₂²	C₃×Q₈×D₇	1	1	yes	characteristic
244	C₂²	C₃×#1161	1	1	yes	characteristic
245	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	14	2	no	—
248	C₄	—	28	2	no	—
249	C₄	C₃×(C₂×D₇)⋊C₄	1	4	yes	complemented
250	C₃	#89759	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	14	4	no	—
252	C₂	C₃×D₇×C₄⋊C₄	1	1	yes	characteristic
253	C₂	C₃×#6455	1	1	yes	characteristic
254	C₂	C₃×#6458	1	1	yes	characteristic
255	C₂	C₃×#6460	1	1	yes	characteristic
256	C₂	C₂×C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
257	C₂	C₃×#6501	1	1	yes	characteristic
258	C₂	C₃×#6509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₄	D₇
hypocenter	C₇	C₃×#380
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₄⋊C₄	C₂	semidirect
4. C₂×C₃×C₄⋊Dic₇	C₂	semidirect
5. C₃×C₇⋊#95	C₂	semidirect
6. #89759	C₃	direct (N × H)
14. C₂×C₃×C₄×D₇	C₄	semidirect
19. C₂×C₇×C₄⋊C₄	C₂×C₃	semidirect
21. C₂×C₄⋊Dic₇	C₂×C₃	semidirect
22. C₇⋊#95	C₂×C₃	semidirect
52. C₂×C₄×D₇	C₃×C₄	semidirect
58. C₂×C₃×C₄⋊C₄	D₇	semidirect
74. C₂×C₃×D₇	C₄⋊C₄	semidirect
76. C₃×C₄×C₇	C₂²⋊C₄	semidirect
122. C₂×C₄⋊C₄	C₃×D₇	semidirect
138. C₂×D₇	C₃×C₄⋊C₄	semidirect
140. C₄×C₇	C₃×C₂²⋊C₄	semidirect
164. C₃×C₇	#380	semidirect
201. C₃×C₄	(C₂×D₇)⋊C₄	semidirect
223. C₇	C₃×#380	semidirect
249. C₄	C₃×(C₂×D₇)⋊C₄	semidirect
250. C₃	#89759	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×C₄×D₇	2
C₂×C₃×C₇×C₄⋊C₄	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×C₃×C₄⋊Dic₇	2
C₃×C₇⋊#95	2
#89759	3
C₃×#380	7

Subgroups — C3×#89759 #13055422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89759 #13055422