Subgroups — C₂×C₇×#1673 #13196319

Order 1792.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#461	C₂	1	2	yes	maximal
3	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#1673	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×SD₁₆	—	2	2	no	—
8	C₂³×C₇×C₈	—	2	1	no	—
9	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₇×#500	—	2	2	no	—
12	C₇×#500	C₂²	1	4	yes	complemented
13	C₇×#461	—	2	4	no	—
14	C₇×#461	C₂²	1	8	yes	—
15	C₇×C₈×D₄	—	2	2	no	—
16	C₇×C₈×D₄	C₂²	1	4	yes	—
17	C₂×C₇×#107	—	2	1	no	—
18	C₂×C₇×#107	—	2	1	no	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×#103	—	2	2	no	—
22	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
24	C₇×#335	C₂²	1	4	yes	complemented
25	C₇×#332	C₂²	1	8	yes	complemented
26	C₂×#1673	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×Q₈	—	2	2	no	—
28	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	—	2	1	no	—
30	C₂×C₇×SD₁₆	—	2	8	no	—
31	C₂×C₇×SD₁₆	—	4	8	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	2	no	—
34	C₂²×C₇×C₈	—	2	4	no	—
35	C₂²×C₇×C₈	—	4	4	no	—
36	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
38	C₇×C₄⋊Q₈	—	2	2	no	—
39	C₇×C₄⋊Q₈	—	2	2	no	—
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×C₇×D₄	—	2	1	no	—
43	C₄×C₇×D₄	—	2	1	no	—
44	C₄×C₇×D₄	D₄	1	2	yes	—
45	C₄×C₇×D₄	C₂³	1	2	yes	—
46	C₂×C₇×C₄⋊C₄	—	2	2	no	—
47	C₂×C₇×C₄⋊C₄	—	2	2	no	—
48	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
51	C₇×#107	—	2	4	no	—
52	C₇×#107	—	2	4	no	—
53	C₇×C₄⋊C₈	—	2	1	no	—
54	C₇×C₄⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	C₂³	1	2	yes	—
56	C₇×C₄⋊C₈	C₂³	1	4	yes	—
57	C₇×#103	—	2	8	no	—
58	C₇×C₂²⋊C₈	—	2	4	no	—
59	C₄×C₇×C₈	D₄	1	2	yes	—
60	C₄×C₇×C₈	C₂³	1	2	yes	—
61	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂×#461	C₂×C₇	1	2	yes	—
63	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
64	#1673	C₂×C₇	1	8	yes	direct product factor
65	C₂×#335	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂×#332	C₂×C₇	1	2	yes	complemented
67	C₂⁴×C₇	—	2	1	no	—
68	C₂×C₇×Q₈	—	2	4	no	—
69	C₂×C₇×Q₈	—	4	2	no	—
70	C₂×C₇×Q₈	—	4	4	no	—
71	C₂×C₇×Q₈	—	4	4	no	—
72	C₂×C₇×D₄	—	2	1	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	D₈	1	4	yes	—
75	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇	—	2	1	no	—
77	C₂²×C₄×C₇	—	2	2	no	—
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	4	2	no	—
80	C₂²×C₄×C₇	—	4	2	no	—
81	C₂²×C₄×C₇	—	4	2	no	—
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	#41	1	1	yes	characteristic
85	C₇×SD₁₆	—	4	16	no	—
86	C₂×C₇×C₈	—	2	2	no	—
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	4	4	no	—
90	C₂×C₇×C₈	—	4	2	no	—
91	C₂×C₇×C₈	—	4	8	no	—
92	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
93	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
95	C₇×C₄⋊C₄	—	2	1	no	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
101	C₇×C₂²⋊C₄	—	2	2	no	—
102	C₇×C₂²⋊C₄	—	2	2	no	—
103	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	#41	1	1	yes	characteristic
106	C₄²×C₇	C₂⁴	1	1	yes	characteristic
107	C₂²×SD₁₆	—	2	2	no	—
108	C₂³×C₈	—	2	1	no	—
109	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
110	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
111	#500	—	2	2	no	—
112	#500	C₂²×C₇	1	4	yes	complemented
113	#461	—	2	4	no	—
114	#461	C₂²×C₇	1	8	yes	—
115	C₈×D₄	—	2	2	no	—
116	C₈×D₄	C₂²×C₇	1	4	yes	—
117	C₂×#107	—	2	1	no	—
118	C₂×#107	—	2	1	no	—
119	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
121	C₂×#103	—	2	2	no	—
122	C₂×C₂²⋊C₈	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
124	#335	C₂²×C₇	1	4	yes	complemented
125	#332	C₂²×C₇	1	8	yes	complemented
126	C₂³×C₇	—	2	2	no	—
127	C₂³×C₇	—	2	4	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	—	4	2	no	—
130	C₂³×C₇	#121	1	1	yes	characteristic
131	C₇×Q₈	—	4	4	no	—
132	C₇×Q₈	—	4	4	no	—
133	C₇×Q₈	—	8	4	no	—
134	C₇×D₄	—	2	2	no	—
135	C₇×D₄	—	2	4	no	—
136	C₇×D₄	C₂×D₈	1	4	yes	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	2	no	—
139	C₂×C₄×C₇	—	4	2	no	—
140	C₂×C₄×C₇	—	4	2	no	—
141	C₂×C₄×C₇	—	4	2	no	—
142	C₂×C₄×C₇	—	4	4	no	—
143	C₂×C₄×C₇	—	4	4	no	—
144	C₂×C₄×C₇	—	8	4	no	—
145	C₂×C₄×C₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₇	#135	1	2	yes	—
151	C₂×C₄×C₇	#137	1	2	yes	—
152	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
155	C₇×C₈	—	2	2	no	—
156	C₇×C₈	—	4	4	no	—
157	C₇×C₈	—	4	2	no	—
158	C₇×C₈	C₂×SD₁₆	1	4	yes	complemented
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂×SD₁₆	—	2	8	no	—
163	C₂×SD₁₆	—	4	8	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	4	4	no	—
168	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
174	C₄×D₄	—	2	1	no	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	C₇×D₄	1	2	yes	—
177	C₄×D₄	C₂³×C₇	1	2	yes	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
183	#107	—	2	4	no	—
184	#107	—	2	4	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	C₂³×C₇	1	2	yes	—
188	C₄⋊C₈	C₂³×C₇	1	4	yes	—
189	#103	—	2	8	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₄×C₈	C₇×D₄	1	2	yes	—
192	C₄×C₈	C₂³×C₇	1	2	yes	—
193	C₂²×C₇	—	2	4	no	—
194	C₂²×C₇	—	4	8	no	—
195	C₂²×C₇	—	4	2	no	—
196	C₂²×C₇	—	4	2	no	—
197	C₂²×C₇	#461	1	2	yes	—
198	C₂²×C₇	#466	1	2	yes	—
199	C₂²×C₇	#473	1	2	yes	—
200	C₂²×C₇	C₂×#121	1	1	yes	characteristic
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	—	2	1	no	—
203	C₄×C₇	—	4	2	no	—
204	C₄×C₇	—	8	4	no	—
205	C₄×C₇	#461	1	2	yes	—
206	C₄×C₇	#466	1	2	yes	—
207	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
208	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
209	C₄×C₇	C₂×#135	1	1	yes	characteristic
210	C₄×C₇	C₂×#137	1	1	yes	characteristic
211	C₂⁴	—	2	1	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₇×D₈	1	4	yes	—
219	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₇×#41	1	1	yes	characteristic
229	SD₁₆	—	4	16	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	8	no	—
236	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
237	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
238	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
248	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
249	C₄²	C₇×#41	1	1	yes	characteristic
250	C₄²	C₂⁴×C₇	1	1	yes	characteristic
251	C₂×C₇	—	4	4	no	—
252	C₂×C₇	#1673	1	4	yes	direct product factor
253	C₂×C₇	C₂×#461	1	1	yes	characteristic
254	C₂×C₇	C₂×#466	1	1	yes	characteristic
255	C₂×C₇	C₂×#473	1	1	yes	characteristic
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₇×#121	1	1	yes	characteristic
261	Q₈	—	4	4	no	—
262	Q₈	—	4	4	no	—
263	Q₈	—	8	4	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	2	no	—
266	D₄	C₂×C₇×D₈	1	4	yes	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	C₇×#121	1	1	yes	characteristic
276	C₂×C₄	C₇×#121	1	1	yes	characteristic
277	C₂×C₄	C₇×#121	1	1	yes	characteristic
278	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
279	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
280	C₂×C₄	C₇×#135	1	2	yes	—
281	C₂×C₄	C₇×#137	1	2	yes	—
282	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	4	4	no	—
287	C₈	—	4	2	no	—
288	C₈	C₂×C₇×SD₁₆	1	4	yes	complemented
289	C₇	C₂×#1673	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	8	no	—
294	C₂²	C₇×#461	1	2	yes	—
295	C₂²	C₇×#466	1	2	yes	—
296	C₂²	C₇×#473	1	2	yes	—
297	C₂²	C₂×C₇×#121	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₇×#461	1	2	yes	—
303	C₄	C₇×#466	1	2	yes	—
304	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
305	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
306	C₄	C₂×C₇×#135	1	1	yes	characteristic
307	C₄	C₂×C₇×#137	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	C₇×#1673	1	4	yes	direct product factor
310	C₂	C₂×C₇×#461	1	1	yes	characteristic
311	C₂	C₂×C₇×#466	1	1	yes	characteristic
312	C₂	C₂×C₇×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	D₄
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×#500	C₂	semidirect
4. C₇×#1673	C₂	direct (N × H)
5. C₂×C₇×#335	C₂	semidirect
6. C₂×C₇×#332	C₂	semidirect
12. C₇×#500	C₂²	semidirect
24. C₇×#335	C₂²	semidirect
25. C₇×#332	C₂²	semidirect
26. C₂×#1673	C₇	direct (N × H)
61. C₂×#500	C₂×C₇	semidirect
64. #1673	C₂×C₇	direct (N × H)
65. C₂×#335	C₂×C₇	semidirect
66. C₂×#332	C₂×C₇	semidirect
92. C₂×C₇×C₈	SD₁₆	semidirect
112. #500	C₂²×C₇	semidirect
124. #335	C₂²×C₇	semidirect
125. #332	C₂²×C₇	semidirect
158. C₇×C₈	C₂×SD₁₆	semidirect
236. C₂×C₈	C₇×SD₁₆	semidirect
252. C₂×C₇	#1673	direct (N × H)
288. C₈	C₂×C₇×SD₁₆	semidirect
289. C₇	C₂×#1673	direct (N × H)
309. C₂	C₇×#1673	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×#500	2
C₂×C₇×#461	2
C₂×C₇×C₈×D₄	2
C₇×#1673	2
C₂×C₇×#335	2
C₂×C₇×#332	2
C₂×#1673	7

Subgroups — C2×C7×#1673 #13196319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1673 #13196319