Subgroups — C₂×C₇×#1360 #13189139

Order 1792.

434 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1360	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×SD₁₆	—	2	1	no	—
7	C₂²×C₇×D₈	C₄	1	1	yes	characteristic
8	C₂×C₇×#124	—	2	1	no	—
9	C₂²×C₇×C₁₆	—	2	1	no	—
10	C₇×#495	—	2	1	no	—
11	C₇×#495	—	2	1	no	—
12	C₇×#495	C₄	1	2	yes	—
13	C₇×#495	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	—	2	1	no	—
17	C₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₇×#334	—	2	1	no	—
19	C₇×#334	C₂²	1	2	yes	complemented
20	C₂×#1360	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×D₄	—	2	1	no	—
23	C₂×C₇×Q₁₆	—	2	1	no	—
24	C₂×C₇×Q₁₆	—	4	2	no	—
25	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₇×SD₁₆	—	2	2	no	—
27	C₂×C₇×SD₁₆	—	2	2	no	—
28	C₂×C₇×SD₁₆	—	4	2	no	—
29	C₂×C₇×SD₁₆	—	4	2	no	—
30	C₂×C₇×D₈	—	2	1	no	—
31	C₂×C₇×D₈	—	4	2	no	—
32	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	—	4	1	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₇×#124	—	2	2	no	—
37	C₇×#124	—	2	2	no	—
38	C₇×#124	—	4	1	no	—
39	C₇×#124	—	4	1	no	—
40	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	C₂×C₇×C₁₆	—	2	2	no	—
44	C₂×C₇×C₁₆	—	4	2	no	—
45	C₇×C₄⋊C₈	—	2	2	no	—
46	C₇×C₄⋊C₈	—	4	1	no	—
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	D₄	1	1	yes	characteristic
49	C₄×C₇×C₈	D₄	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂×#495	C₂×C₇	1	1	yes	characteristic
52	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂×#334	C₂×C₇	1	1	yes	characteristic, complemented
54	#1360	C₂×C₇	1	4	yes	direct product factor
55	C₂⁴×C₇	—	4	1	no	—
56	C₂×C₇×Q₈	—	2	2	no	—
57	C₂×C₇×Q₈	—	4	1	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂×C₇×Q₈	—	4	2	no	—
60	C₂×C₇×D₄	—	2	2	no	—
61	C₂×C₇×D₄	—	4	2	no	—
62	C₂×C₇×D₄	—	4	2	no	—
63	C₂×C₇×D₄	—	4	1	no	—
64	C₂²×C₄×C₇	—	4	1	no	—
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
68	C₇×Q₁₆	—	4	2	no	—
69	C₇×Q₁₆	—	4	2	no	—
70	C₇×SD₁₆	—	4	4	no	—
71	C₇×SD₁₆	—	4	2	no	—
72	C₇×SD₁₆	—	4	2	no	—
73	C₇×D₈	—	4	2	no	—
74	C₇×D₈	—	4	2	no	—
75	C₂×C₇×C₈	—	2	2	no	—
76	C₂×C₇×C₈	—	4	2	no	—
77	C₂×C₇×C₈	—	8	2	no	—
78	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₇×C₈	D₈	1	2	yes	—
81	C₂×C₇×C₈	SD₁₆	1	2	yes	—
82	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
84	C₇×C₂²⋊C₄	—	4	2	no	—
85	C₇×C₂²⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	4	2	no	—
87	C₇×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₇	D₈	1	1	yes	characteristic
90	C₄²×C₇	SD₁₆	1	1	yes	characteristic
91	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
92	C₇×C₁₆	—	4	2	no	—
93	C₂²×Q₁₆	C₄×C₇	1	1	yes	characteristic
94	C₂²×SD₁₆	—	2	1	no	—
95	C₂²×D₈	C₄×C₇	1	1	yes	characteristic
96	C₂×#124	—	2	1	no	—
97	C₂²×C₁₆	—	2	1	no	—
98	#495	—	2	1	no	—
99	#495	—	2	1	no	—
100	#495	C₄×C₇	1	2	yes	—
101	#495	C₂²×C₇	1	2	yes	—
102	C₂×C₄⋊C₈	—	2	1	no	—
103	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
104	C₄⋊C₁₆	—	2	1	no	—
105	C₄⋊C₁₆	C₂²×C₇	1	2	yes	complemented
106	#334	—	2	1	no	—
107	#334	C₂²×C₇	1	2	yes	complemented
108	C₂³×C₇	—	4	2	no	—
109	C₂³×C₇	—	8	2	no	—
110	C₂³×C₇	—	8	2	no	—
111	C₂³×C₇	—	8	2	no	—
112	C₂³×C₇	#102	1	1	yes	characteristic
113	C₇×Q₈	—	4	2	no	—
114	C₇×Q₈	—	4	2	no	—
115	C₇×Q₈	—	8	2	no	—
116	C₇×D₄	—	4	2	no	—
117	C₇×D₄	—	4	2	no	—
118	C₇×D₄	—	8	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	—	8	2	no	—
123	C₂×C₄×C₇	#100	1	2	yes	—
124	C₂×C₄×C₇	#102	1	1	yes	characteristic
125	C₂×C₄×C₇	#102	1	1	yes	characteristic
126	C₂×C₄×C₇	#102	1	1	yes	characteristic
127	C₂×C₄×C₇	#102	1	2	yes	—
128	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
131	C₇×C₈	—	2	1	no	—
132	C₇×C₈	—	2	1	no	—
133	C₇×C₈	—	8	2	no	—
134	C₇×C₈	#102	1	2	yes	—
135	C₇×C₈	C₂×D₈	1	1	yes	characteristic
136	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
137	C₂²×Q₈	—	2	1	no	—
138	C₂²×D₄	—	2	1	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	—	4	2	no	—
141	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
142	C₂×SD₁₆	—	2	2	no	—
143	C₂×SD₁₆	—	2	2	no	—
144	C₂×SD₁₆	—	4	2	no	—
145	C₂×SD₁₆	—	4	2	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	—	4	2	no	—
148	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
149	C₂²×C₈	—	4	1	no	—
150	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
152	#124	—	2	2	no	—
153	#124	—	2	2	no	—
154	#124	—	4	1	no	—
155	#124	—	4	1	no	—
156	C₂×C₂²⋊C₄	—	4	1	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	4	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	4	1	no	—
163	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
164	C₄×C₈	C₇×D₄	1	1	yes	characteristic
165	C₄×C₈	C₇×D₄	1	1	yes	characteristic
166	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
167	C₂²×C₇	—	8	2	no	—
168	C₂²×C₇	—	8	2	no	—
169	C₂²×C₇	—	8	2	no	—
170	C₂²×C₇	—	16	4	no	—
171	C₂²×C₇	#331	1	2	yes	—
172	C₂²×C₇	#359	1	2	yes	—
173	C₂²×C₇	#360	1	2	yes	—
174	C₂²×C₇	C₂×#102	1	1	yes	characteristic
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	8	2	no	—
180	C₄×C₇	#331	1	2	yes	—
181	C₄×C₇	C₂×#100	1	1	yes	characteristic
182	C₄×C₇	C₂×#102	1	1	yes	characteristic
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
195	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	2	no	—
200	SD₁₆	—	4	2	no	—
201	D₈	—	4	2	no	—
202	D₈	—	4	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	8	2	no	—
206	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₈	C₇×D₈	1	2	yes	—
209	C₂×C₈	C₇×SD₁₆	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
211	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
212	C₂²⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
217	C₄²	C₇×D₈	1	1	yes	characteristic
218	C₄²	C₇×SD₁₆	1	1	yes	characteristic
219	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
220	C₁₆	—	4	2	no	—
221	C₂×C₇	—	16	2	no	—
222	C₂×C₇	#1360	1	4	yes	direct product factor
223	C₂×C₇	C₂×#331	1	1	yes	characteristic
224	C₂×C₇	C₂×#359	1	1	yes	characteristic
225	C₂×C₇	C₂×#360	1	1	yes	characteristic
226	C₂³	—	4	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	C₇×#102	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	8	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	C₇×#100	1	2	yes	—
242	C₂×C₄	C₇×#102	1	1	yes	characteristic
243	C₂×C₄	C₇×#102	1	1	yes	characteristic
244	C₂×C₄	C₇×#102	1	1	yes	characteristic
245	C₂×C₄	C₇×#102	1	2	yes	—
246	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	8	2	no	—
252	C₈	C₇×#102	1	2	yes	—
253	C₈	C₂×C₇×D₈	1	1	yes	characteristic
254	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
255	C₇	C₂×#1360	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	16	4	no	—
260	C₂²	C₇×#331	1	2	yes	—
261	C₂²	C₇×#359	1	2	yes	—
262	C₂²	C₇×#360	1	2	yes	—
263	C₂²	C₂×C₇×#102	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	2	no	—
269	C₄	C₇×#331	1	2	yes	—
270	C₄	C₂×C₇×#100	1	1	yes	characteristic
271	C₄	C₂×C₇×#102	1	1	yes	characteristic
272	C₂	—	16	2	no	—
273	C₂	C₇×#1360	1	4	yes	direct product factor
274	C₂	C₂×C₇×#331	1	1	yes	characteristic
275	C₂	C₂×C₇×#359	1	1	yes	characteristic
276	C₂	C₂×C₇×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₈	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₇×#334	C₂	semidirect
4. C₇×#1360	C₂	direct (N × H)
17. C₇×C₄⋊C₁₆	C₂²	semidirect
19. C₇×#334	C₂²	semidirect
20. C₂×#1360	C₇	direct (N × H)
52. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
53. C₂×#334	C₂×C₇	semidirect
54. #1360	C₂×C₇	direct (N × H)
105. C₄⋊C₁₆	C₂²×C₇	semidirect
107. #334	C₂²×C₇	semidirect
222. C₂×C₇	#1360	direct (N × H)
255. C₇	C₂×#1360	direct (N × H)
273. C₂	C₇×#1360	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#495	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×#334	2
C₇×#1360	2
C₂×#1360	7

Subgroups — C2×C7×#1360 #13189139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1360 #13189139