Subgroups — C₂×C₇×#1807 #13197872

Order 1792.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×#106	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#380	C₂	1	2	yes	maximal
4	C₇×#1807	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#340	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂³×C₇×C₈	—	2	8	no	—
8	C₂³×C₇×C₈	C₄	1	8	yes	complemented
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₇×C₈	C₂²	1	4	yes	—
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
14	C₂×C₇×#106	—	2	8	no	—
15	C₂×C₇×#106	C₂²	1	8	yes	complemented
16	C₇×#380	—	2	4	no	—
17	C₇×#380	C₂²	1	8	yes	—
18	C₂×C₇×#95	—	2	2	no	—
19	C₇×#340	C₂²	1	8	yes	complemented
20	C₂×#1807	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂⁵×C₇	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	4	no	—
23	C₂³×C₄×C₇	—	2	4	no	—
24	C₂³×C₄×C₇	—	2	4	no	—
25	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₇	D₄	1	2	yes	—
27	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₇	Q₈	1	4	yes	—
29	C₂³×C₄×C₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₇×C₈	—	2	4	no	—
32	C₂²×C₇×C₈	—	2	32	no	—
33	C₂²×C₇×C₈	—	2	16	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₇×C₈	C₂×C₄	1	16	yes	complemented
37	C₂²×C₇×C₈	D₄	1	8	yes	—
38	C₂²×C₇×C₈	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	—	2	8	no	—
41	C₂×C₇×C₄⋊C₄	—	2	8	no	—
42	C₂×C₇×C₄⋊C₄	—	2	4	no	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
44	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
45	C₂×C₇×C₂²⋊C₄	—	4	8	no	—
46	C₇×#106	—	2	16	no	—
47	C₇×#95	—	2	8	no	—
48	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic
49	C₂²×#106	C₂×C₇	1	2	yes	complemented
50	C₂×#380	C₂×C₇	1	2	yes	—
51	#1807	C₂×C₇	1	8	yes	direct product factor
52	C₂×#340	C₂×C₇	1	2	yes	complemented
53	C₂⁴×C₇	—	2	4	no	—
54	C₂⁴×C₇	—	2	4	no	—
55	C₂⁴×C₇	C₄⋊C₄	1	4	yes	—
56	C₂⁴×C₇	SD₁₆	1	8	yes	—
57	C₂⁴×C₇	C₂×D₄	1	2	yes	—
58	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	—	2	4	no	—
60	C₂²×C₄×C₇	—	2	8	no	—
61	C₂²×C₄×C₇	—	2	8	no	—
62	C₂²×C₄×C₇	—	2	8	no	—
63	C₂²×C₄×C₇	—	2	16	no	—
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	8	no	—
67	C₂²×C₄×C₇	—	4	8	no	—
68	C₂²×C₄×C₇	—	4	8	no	—
69	C₂²×C₄×C₇	—	4	8	no	—
70	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
72	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇	#41	1	4	yes	—
80	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₇×C₈	—	2	8	no	—
82	C₂×C₇×C₈	—	2	4	no	—
83	C₂×C₇×C₈	—	2	32	no	—
84	C₂×C₇×C₈	C₂²⋊C₄	1	16	yes	complemented
85	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
88	C₇×C₄⋊C₄	—	2	8	no	—
89	C₇×C₄⋊C₄	—	2	8	no	—
90	C₇×C₄⋊C₄	—	4	8	no	—
91	C₇×C₂²⋊C₄	—	4	16	no	—
92	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
93	C₂³×C₈	—	2	8	no	—
94	C₂³×C₈	C₄×C₇	1	8	yes	complemented
95	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
96	C₂³×C₈	C₂²×C₇	1	4	yes	—
97	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
99	C₂²×C₂²⋊C₄	—	2	2	no	—
100	C₂×#106	—	2	8	no	—
101	C₂×#106	C₂²×C₇	1	8	yes	complemented
102	#380	—	2	4	no	—
103	#380	C₂²×C₇	1	8	yes	—
104	C₂×#95	—	2	2	no	—
105	#340	C₂²×C₇	1	8	yes	complemented
106	C₂³×C₇	—	2	4	no	—
107	C₂³×C₇	—	2	4	no	—
108	C₂³×C₇	—	2	8	no	—
109	C₂³×C₇	—	2	8	no	—
110	C₂³×C₇	—	2	32	no	—
111	C₂³×C₇	#106	1	16	yes	complemented
112	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂³×C₇	#121	1	4	yes	—
115	C₂³×C₇	C₂×SD₁₆	1	8	yes	—
116	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
117	C₂³×C₇	#135	1	4	yes	—
118	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇	—	2	4	no	—
120	C₂×C₄×C₇	—	2	8	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	8	no	—
124	C₂×C₄×C₇	—	2	16	no	—
125	C₂×C₄×C₇	—	4	8	no	—
126	C₂×C₄×C₇	—	4	8	no	—
127	C₂×C₄×C₇	—	4	8	no	—
128	C₂×C₄×C₇	—	8	16	no	—
129	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	4	yes	—
130	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₇	#117	1	4	yes	—
133	C₂×C₄×C₇	#121	1	4	yes	—
134	C₂×C₄×C₇	C₂²⋊Q₈	1	8	yes	—
135	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×#41	1	2	yes	—
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	C₂×C₂²⋊C₄	1	8	yes	complemented
141	C₂⁵	C₇×D₄	1	1	yes	characteristic
142	C₂³×C₄	—	2	4	no	—
143	C₂³×C₄	—	2	4	no	—
144	C₂³×C₄	—	2	4	no	—
145	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
146	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
147	C₂³×C₄	C₇×D₄	1	2	yes	—
148	C₂³×C₄	C₇×Q₈	1	4	yes	—
149	C₂³×C₄	C₂³×C₇	1	2	yes	—
150	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	—	2	32	no	—
153	C₂²×C₈	—	2	16	no	—
154	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
155	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
156	C₂²×C₈	C₂×C₄×C₇	1	16	yes	complemented
157	C₂²×C₈	C₇×D₄	1	8	yes	—
158	C₂²×C₈	C₂³×C₇	1	2	yes	—
159	C₂²×C₈	C₂³×C₇	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	8	no	—
161	C₂×C₄⋊C₄	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
164	C₂×C₂²⋊C₄	—	2	8	no	—
165	C₂×C₂²⋊C₄	—	4	8	no	—
166	#106	—	2	16	no	—
167	#95	—	2	8	no	—
168	C₂²×C₇	—	2	32	no	—
169	C₂²×C₇	—	2	4	no	—
170	C₂²×C₇	—	2	4	no	—
171	C₂²×C₇	—	2	16	no	—
172	C₂²×C₇	#380	1	4	yes	—
173	C₂²×C₇	C₂×#106	1	4	yes	—
174	C₂²×C₇	C₂×#106	1	8	yes	complemented
175	C₂²×C₇	#427	1	4	yes	—
176	C₂²×C₇	#465	1	16	yes	—
177	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₇	C₂×#121	1	2	yes	—
179	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
180	C₂²×C₇	C₂×#135	1	2	yes	—
181	C₄×C₇	—	2	4	no	—
182	C₄×C₇	—	8	8	no	—
183	C₄×C₇	#380	1	4	yes	—
184	C₄×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₄×C₇	C₂×#117	1	1	yes	characteristic
186	C₄×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	C₇×C₄⋊C₄	1	4	yes	—
190	C₂⁴	C₇×SD₁₆	1	8	yes	—
191	C₂⁴	C₂×C₇×D₄	1	2	yes	—
192	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	16	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	4	8	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	—	4	8	no	—
204	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
205	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
208	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
209	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
210	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
212	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
213	C₂²×C₄	C₇×#41	1	4	yes	—
214	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	32	no	—
218	C₂×C₈	C₇×C₂²⋊C₄	1	16	yes	complemented
219	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
220	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
221	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	16	no	—
226	C₂×C₇	—	2	8	no	—
227	C₂×C₇	#1807	1	8	yes	direct product factor
228	C₂×C₇	C₂×#380	1	1	yes	characteristic
229	C₂×C₇	C₂²×#106	1	1	yes	characteristic
230	C₂×C₇	C₂×#427	1	1	yes	characteristic
231	C₂×C₇	C₂×#465	1	4	yes	—
232	C₂³	—	2	8	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	32	no	—
237	C₂³	C₇×#106	1	16	yes	complemented
238	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
239	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
240	C₂³	C₇×#121	1	4	yes	—
241	C₂³	C₂×C₇×SD₁₆	1	8	yes	—
242	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
243	C₂³	C₇×#135	1	4	yes	—
244	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	8	16	no	—
255	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	4	yes	—
256	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
257	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
258	C₂×C₄	C₇×#117	1	4	yes	—
259	C₂×C₄	C₇×#121	1	4	yes	—
260	C₂×C₄	C₇×C₂²⋊Q₈	1	8	yes	—
261	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
263	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₇×#41	1	2	yes	—
265	C₈	—	2	4	no	—
266	C₈	C₂×C₇×C₂²⋊C₄	1	8	yes	complemented
267	C₇	C₂×#1807	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	4	no	—
269	C₂²	—	2	32	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	16	no	—
272	C₂²	C₇×#380	1	4	yes	—
273	C₂²	C₂×C₇×#106	1	4	yes	—
274	C₂²	C₂×C₇×#106	1	8	yes	complemented
275	C₂²	C₇×#427	1	4	yes	—
276	C₂²	C₇×#465	1	16	yes	—
277	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
278	C₂²	C₂×C₇×#121	1	2	yes	—
279	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
280	C₂²	C₂×C₇×#135	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	8	no	—
283	C₄	C₇×#380	1	4	yes	—
284	C₄	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
285	C₄	C₂×C₇×#117	1	1	yes	characteristic
286	C₄	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
287	C₂	—	2	8	no	—
288	C₂	C₇×#1807	1	8	yes	direct product factor
289	C₂	C₂×C₇×#380	1	1	yes	characteristic
290	C₂	C₂²×C₇×#106	1	1	yes	characteristic
291	C₂	C₂×C₇×#427	1	1	yes	characteristic
292	C₂	C₂×C₇×#465	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×#106	C₂	semidirect
4. C₇×#1807	C₂	direct (N × H)
5. C₂×C₇×#340	C₂	semidirect
8. C₂³×C₇×C₈	C₄	semidirect
15. C₂×C₇×#106	C₂²	semidirect
19. C₇×#340	C₂²	semidirect
20. C₂×#1807	C₇	direct (N × H)
36. C₂²×C₇×C₈	C₂×C₄	semidirect
49. C₂²×#106	C₂×C₇	semidirect
51. #1807	C₂×C₇	direct (N × H)
52. C₂×#340	C₂×C₇	semidirect
84. C₂×C₇×C₈	C₂²⋊C₄	semidirect
94. C₂³×C₈	C₄×C₇	semidirect
101. C₂×#106	C₂²×C₇	semidirect
105. #340	C₂²×C₇	semidirect
111. C₂³×C₇	#106	semidirect
140. C₇×C₈	C₂×C₂²⋊C₄	semidirect
156. C₂²×C₈	C₂×C₄×C₇	semidirect
174. C₂²×C₇	C₂×#106	semidirect
218. C₂×C₈	C₇×C₂²⋊C₄	semidirect
227. C₂×C₇	#1807	direct (N × H)
237. C₂³	C₇×#106	semidirect
266. C₈	C₂×C₇×C₂²⋊C₄	semidirect
267. C₇	C₂×#1807	direct (N × H)
274. C₂²	C₂×C₇×#106	semidirect
288. C₂	C₇×#1807	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×C₈	2
C₂²×C₇×#106	2
C₂×C₇×#380	2
C₇×#1807	2
C₂×C₇×#340	2
C₂×#1807	7

Subgroups — C2×C7×#1807 #13197872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1807 #13197872