Subgroups — C₂×C₇×#1715 #13196822

Order 1792.

1030 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#492	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#486	C₂	1	2	yes	maximal, complemented
5	C₇×#1715	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×SD₁₆	—	2	4	no	—
8	C₂²×C₇×D₈	—	2	2	no	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₇×#127	C₂²	1	1	yes	characteristic
11	C₂×C₇×#127	C₂²	1	1	yes	characteristic
12	C₇×#498	—	2	2	no	—
13	C₇×#498	C₂²	1	4	yes	complemented
14	C₇×#493	—	2	2	no	—
15	C₇×#493	C₂²	1	4	yes	—
16	C₇×#492	—	2	4	no	—
17	C₇×#492	C₂²	1	8	yes	complemented
18	C₇×#486	—	2	4	no	—
19	C₇×#486	C₂²	1	8	yes	complemented
20	C₂×C₇×#106	—	2	2	no	—
21	C₂×C₇×#102	—	2	4	no	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
24	C₇×#322	C₂²	1	4	yes	complemented
25	C₂×#1715	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₇×Q₈	—	2	2	no	—
27	C₂²×C₇×D₄	—	2	2	no	—
28	C₂²×C₇×D₄	—	2	1	no	—
29	C₂²×C₇×D₄	—	2	1	no	—
30	C₂²×C₇×D₄	—	2	2	no	—
31	C₂×C₇×SD₁₆	—	2	16	no	—
32	C₂×C₇×SD₁₆	—	4	16	no	—
33	C₂×C₇×D₈	—	2	8	no	—
34	C₂×C₇×D₈	—	4	8	no	—
35	C₂²×C₇×C₈	D₄	1	4	yes	—
36	C₂²×C₇×C₈	D₄	1	2	yes	—
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×C₄⋊Q₈	—	2	2	no	—
39	C₇×C₄⋊Q₈	—	2	2	no	—
40	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₇×#127	—	2	2	no	—
42	C₇×#127	—	2	2	no	—
43	C₇×#127	—	2	1	no	—
44	C₇×#127	—	2	1	no	—
45	C₇×#127	C₂³	1	2	yes	—
46	C₇×#127	C₂³	1	2	yes	—
47	C₂×C₇×C₄⋊C₄	—	2	2	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₇×#106	—	2	8	no	—
51	C₇×#102	—	2	16	no	—
52	C₄×C₇×C₈	D₄	1	4	yes	—
53	C₄×C₇×C₈	D₄	1	2	yes	—
54	C₄×C₇×C₈	C₂³	1	4	yes	—
55	C₄×C₇×C₈	C₂³	1	2	yes	—
56	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂×#493	C₂×C₇	1	1	yes	characteristic
58	C₂×#492	C₂×C₇	1	2	yes	complemented
59	C₂×#486	C₂×C₇	1	2	yes	complemented
60	#1715	C₂×C₇	1	8	yes	direct product factor
61	C₂×#322	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂⁴×C₇	—	4	1	no	—
63	C₂⁴×C₇	—	4	1	no	—
64	C₂×C₇×Q₈	—	2	4	no	—
65	C₂×C₇×Q₈	—	4	2	no	—
66	C₂×C₇×Q₈	—	4	8	no	—
67	C₂×C₇×D₄	—	2	4	no	—
68	C₂×C₇×D₄	—	2	2	no	—
69	C₂×C₇×D₄	—	2	2	no	—
70	C₂×C₇×D₄	—	2	4	no	—
71	C₂×C₇×D₄	—	4	1	no	—
72	C₂×C₇×D₄	—	4	1	no	—
73	C₂×C₇×D₄	—	4	2	no	—
74	C₂×C₇×D₄	—	4	2	no	—
75	C₂×C₇×D₄	—	4	4	no	—
76	C₂×C₇×D₄	—	4	4	no	—
77	C₂×C₇×D₄	—	4	8	no	—
78	C₂×C₇×D₄	—	4	8	no	—
79	C₂²×C₄×C₇	—	4	2	no	—
80	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₇×SD₁₆	—	4	32	no	—
83	C₇×D₈	—	4	16	no	—
84	C₂×C₇×C₈	—	2	8	no	—
85	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
86	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
87	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	4	2	no	—
91	C₇×C₄⋊C₄	—	4	2	no	—
92	C₄²×C₇	C₂×D₄	1	2	yes	—
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇	C₂⁴	1	1	yes	characteristic
95	C₂²×SD₁₆	—	2	4	no	—
96	C₂²×D₈	—	2	2	no	—
97	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
98	C₂×#127	C₂²×C₇	1	1	yes	characteristic
99	C₂×#127	C₂²×C₇	1	1	yes	characteristic
100	#498	—	2	2	no	—
101	#498	C₂²×C₇	1	4	yes	complemented
102	#493	—	2	2	no	—
103	#493	C₂²×C₇	1	4	yes	—
104	#492	—	2	4	no	—
105	#492	C₂²×C₇	1	8	yes	complemented
106	#486	—	2	4	no	—
107	#486	C₂²×C₇	1	8	yes	complemented
108	C₂×#106	—	2	2	no	—
109	C₂×#102	—	2	4	no	—
110	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
112	#322	C₂²×C₇	1	4	yes	complemented
113	C₂³×C₇	—	4	2	no	—
114	C₂³×C₇	—	4	2	no	—
115	C₂³×C₇	—	8	2	no	—
116	C₂³×C₇	—	8	2	no	—
117	C₂³×C₇	—	8	4	no	—
118	C₂³×C₇	—	8	4	no	—
119	C₂³×C₇	#127	1	1	yes	characteristic
120	C₇×Q₈	—	4	8	no	—
121	C₇×Q₈	—	8	4	no	—
122	C₇×D₄	—	4	4	no	—
123	C₇×D₄	—	4	4	no	—
124	C₇×D₄	—	4	8	no	—
125	C₇×D₄	—	4	8	no	—
126	C₇×D₄	—	8	2	no	—
127	C₇×D₄	—	8	2	no	—
128	C₇×D₄	—	8	4	no	—
129	C₇×D₄	—	8	4	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	8	4	no	—
132	C₂×C₄×C₇	#127	1	1	yes	characteristic
133	C₂×C₄×C₇	#127	1	2	yes	—
134	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄×C₇	AGL(1,ℤ/8)	1	8	yes	—
136	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
138	C₇×C₈	—	2	8	no	—
139	C₇×C₈	C₂×SD₁₆	1	8	yes	complemented
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×D₄	—	2	1	no	—
142	C₂²×D₄	—	2	1	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	—	2	2	no	—
145	C₂×SD₁₆	—	2	16	no	—
146	C₂×SD₁₆	—	4	16	no	—
147	C₂×D₈	—	2	8	no	—
148	C₂×D₈	—	4	8	no	—
149	C₂²×C₈	C₇×D₄	1	4	yes	—
150	C₂²×C₈	C₇×D₄	1	2	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
155	#127	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	2	1	no	—
158	#127	—	2	1	no	—
159	#127	C₂³×C₇	1	2	yes	—
160	#127	C₂³×C₇	1	2	yes	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
164	#106	—	2	8	no	—
165	#102	—	2	16	no	—
166	C₄×C₈	C₇×D₄	1	4	yes	—
167	C₄×C₈	C₇×D₄	1	2	yes	—
168	C₄×C₈	C₂³×C₇	1	4	yes	—
169	C₄×C₈	C₂³×C₇	1	2	yes	—
170	C₂²×C₇	—	8	2	no	—
171	C₂²×C₇	—	8	2	no	—
172	C₂²×C₇	—	8	4	no	—
173	C₂²×C₇	—	8	4	no	—
174	C₂²×C₇	—	16	4	no	—
175	C₂²×C₇	—	16	4	no	—
176	C₂²×C₇	#492	1	2	yes	—
177	C₂²×C₇	#496	1	4	yes	—
178	C₂²×C₇	C₂×#127	1	1	yes	characteristic
179	C₄×C₇	—	8	4	no	—
180	C₄×C₇	#492	1	2	yes	—
181	C₄×C₇	#496	1	4	yes	—
182	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
183	C₄×C₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
184	C₂⁴	—	4	1	no	—
185	C₂⁴	—	4	1	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	8	no	—
200	C₂×D₄	—	4	8	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
204	SD₁₆	—	4	32	no	—
205	D₈	—	4	16	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
208	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
209	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄²	C₂×C₇×D₄	1	2	yes	—
215	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
216	C₄²	C₂⁴×C₇	1	1	yes	characteristic
217	C₂×C₇	—	16	2	no	—
218	C₂×C₇	—	16	2	no	—
219	C₂×C₇	#1715	1	4	yes	direct product factor
220	C₂×C₇	C₂×#492	1	1	yes	characteristic
221	C₂×C₇	C₂×#496	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	8	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	4	no	—
227	C₂³	—	8	4	no	—
228	C₂³	C₇×#127	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	Q₈	—	8	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	8	no	—
234	D₄	—	4	8	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	D₄	—	8	4	no	—
238	D₄	—	8	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	8	4	no	—
241	C₂×C₄	C₇×#127	1	1	yes	characteristic
242	C₂×C₄	C₇×#127	1	2	yes	—
243	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
244	C₂×C₄	C₇×AGL(1,ℤ/8)	1	8	yes	—
245	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
247	C₈	—	2	8	no	—
248	C₈	C₂×C₇×SD₁₆	1	8	yes	complemented
249	C₇	C₂×#1715	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	4	no	—
255	C₂²	—	16	4	no	—
256	C₂²	C₇×#492	1	2	yes	—
257	C₂²	C₇×#496	1	4	yes	—
258	C₂²	C₂×C₇×#127	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	C₇×#492	1	2	yes	—
261	C₄	C₇×#496	1	4	yes	—
262	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
263	C₄	C₂×C₇×AGL(1,ℤ/8)	1	4	yes	—
264	C₂	—	16	2	no	—
265	C₂	—	16	2	no	—
266	C₂	C₇×#1715	1	4	yes	direct product factor
267	C₂	C₂×C₇×#492	1	1	yes	characteristic
268	C₂	C₂×C₇×#496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#498	C₂	semidirect
3. C₂×C₇×#492	C₂	semidirect
4. C₂×C₇×#486	C₂	semidirect
5. C₇×#1715	C₂	direct (N × H)
6. C₂×C₇×#322	C₂	semidirect
13. C₇×#498	C₂²	semidirect
17. C₇×#492	C₂²	semidirect
19. C₇×#486	C₂²	semidirect
24. C₇×#322	C₂²	semidirect
25. C₂×#1715	C₇	direct (N × H)
56. C₂×#498	C₂×C₇	semidirect
58. C₂×#492	C₂×C₇	semidirect
59. C₂×#486	C₂×C₇	semidirect
60. #1715	C₂×C₇	direct (N × H)
61. C₂×#322	C₂×C₇	semidirect
85. C₂×C₇×C₈	SD₁₆	semidirect
101. #498	C₂²×C₇	semidirect
105. #492	C₂²×C₇	semidirect
107. #486	C₂²×C₇	semidirect
112. #322	C₂²×C₇	semidirect
139. C₇×C₈	C₂×SD₁₆	semidirect
207. C₂×C₈	C₇×SD₁₆	semidirect
219. C₂×C₇	#1715	direct (N × H)
248. C₈	C₂×C₇×SD₁₆	semidirect
249. C₇	C₂×#1715	direct (N × H)
266. C₂	C₇×#1715	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×#498	2
C₂×C₇×#493	2
C₂×C₇×#492	2
C₂×C₇×#486	2
C₇×#1715	2
C₂×C₇×#322	2
C₂×#1715	7

Subgroups — C2×C7×#1715 #13196822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1715 #13196822