Subgroups — C₂×C₇×#1693 #13196351

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#436	C₂	1	1	yes	maximal, characteristic
4	C₇×#1693	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#334	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	C₂²×C₇×M₁₆	—	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×#436	—	2	2	no	—
16	C₇×#436	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×#334	C₂²	1	4	yes	complemented
25	C₇×#332	C₂²	1	8	yes	complemented
26	C₂×#1693	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	C₂×C₇×M₁₆	—	2	4	no	—
33	C₂×C₇×M₁₆	—	4	4	no	—
34	C₂²×C₇×C₈	—	2	1	no	—
35	C₂²×C₇×C₈	—	2	2	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×#436	C₂×C₇	1	1	yes	characteristic
64	#1693	C₂×C₇	1	8	yes	direct product factor
65	C₂×#334	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	4	no	—
74	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇	—	2	1	no	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	2	2	no	—
78	C₂²×C₄×C₇	—	4	2	no	—
79	C₂²×C₄×C₇	—	4	2	no	—
80	C₂²×C₄×C₇	—	4	2	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	#41	1	1	yes	characteristic
84	C₇×SD₁₆	—	4	16	no	—
85	C₇×M₁₆	—	4	8	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	2	no	—
90	C₂×C₇×C₈	—	4	4	no	—
91	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
92	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
94	C₇×C₄⋊C₄	—	2	1	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	4	2	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
100	C₇×C₂²⋊C₄	—	2	2	no	—
101	C₇×C₂²⋊C₄	—	2	2	no	—
102	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇	#41	1	1	yes	characteristic
105	C₄²×C₇	C₂⁴	1	1	yes	characteristic
106	C₂²×SD₁₆	—	2	2	no	—
107	C₂²×M₁₆	—	2	1	no	—
108	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
109	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
110	#500	—	2	2	no	—
111	#500	C₂²×C₇	1	4	yes	complemented
112	#461	—	2	4	no	—
113	#461	C₂²×C₇	1	8	yes	—
114	#436	—	2	2	no	—
115	#436	C₂²×C₇	1	4	yes	—
116	C₂×#107	—	2	1	no	—
117	C₂×#107	—	2	1	no	—
118	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
119	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
120	C₂×#103	—	2	2	no	—
121	C₂×C₂²⋊C₈	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
123	#334	C₂²×C₇	1	4	yes	complemented
124	#332	C₂²×C₇	1	8	yes	complemented
125	C₂³×C₇	—	2	2	no	—
126	C₂³×C₇	—	4	2	no	—
127	C₂³×C₇	—	4	2	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	#121	1	1	yes	characteristic
130	C₇×Q₈	—	4	4	no	—
131	C₇×Q₈	—	4	4	no	—
132	C₇×Q₈	—	8	4	no	—
133	C₇×D₄	—	2	4	no	—
134	C₇×D₄	—	4	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	2	2	no	—
137	C₂×C₄×C₇	—	4	2	no	—
138	C₂×C₄×C₇	—	4	2	no	—
139	C₂×C₄×C₇	—	4	2	no	—
140	C₂×C₄×C₇	—	4	4	no	—
141	C₂×C₄×C₇	—	4	4	no	—
142	C₂×C₄×C₇	—	8	4	no	—
143	C₂×C₄×C₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
148	C₂×C₄×C₇	#137	1	2	yes	—
149	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
152	C₇×C₈	—	2	2	no	—
153	C₇×C₈	—	4	2	no	—
154	C₇×C₈	—	4	4	no	—
155	C₇×C₈	C₂×SD₁₆	1	4	yes	complemented
156	C₂²×Q₈	—	2	2	no	—
157	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
158	C₂³×C₄	—	2	1	no	—
159	C₂×SD₁₆	—	2	8	no	—
160	C₂×SD₁₆	—	4	8	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
166	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₇×D₄	1	2	yes	—
174	C₄×D₄	C₂³×C₇	1	2	yes	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	—	2	1	no	—
179	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
180	#107	—	2	4	no	—
181	#107	—	2	4	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂³×C₇	1	2	yes	—
185	C₄⋊C₈	C₂³×C₇	1	4	yes	—
186	#103	—	2	8	no	—
187	C₂²⋊C₈	—	2	4	no	—
188	C₄×C₈	C₇×D₄	1	2	yes	—
189	C₄×C₈	C₂³×C₇	1	2	yes	—
190	C₂²×C₇	—	4	2	no	—
191	C₂²×C₇	—	4	2	no	—
192	C₂²×C₇	—	4	2	no	—
193	C₂²×C₇	—	8	4	no	—
194	C₂²×C₇	#461	1	2	yes	—
195	C₂²×C₇	#469	1	2	yes	—
196	C₂²×C₇	#473	1	2	yes	—
197	C₂²×C₇	C₂×#121	1	1	yes	characteristic
198	C₄×C₇	—	2	1	no	—
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	4	2	no	—
201	C₄×C₇	—	8	4	no	—
202	C₄×C₇	#461	1	2	yes	—
203	C₄×C₇	#469	1	2	yes	—
204	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
205	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
206	C₄×C₇	C₂×#137	1	1	yes	characteristic
207	C₂⁴	—	2	1	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₇×#41	1	1	yes	characteristic
224	SD₁₆	—	4	16	no	—
225	M₁₆	—	4	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
232	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
233	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
243	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
244	C₄²	C₇×#41	1	1	yes	characteristic
245	C₄²	C₂⁴×C₇	1	1	yes	characteristic
246	C₂×C₇	—	8	2	no	—
247	C₂×C₇	#1693	1	4	yes	direct product factor
248	C₂×C₇	C₂×#461	1	1	yes	characteristic
249	C₂×C₇	C₂×#469	1	1	yes	characteristic
250	C₂×C₇	C₂×#473	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	C₇×#121	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	8	4	no	—
259	D₄	—	2	4	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₇×#121	1	1	yes	characteristic
270	C₂×C₄	C₇×#121	1	1	yes	characteristic
271	C₂×C₄	C₇×#121	1	1	yes	characteristic
272	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
273	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
274	C₂×C₄	C₇×#137	1	2	yes	—
275	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	4	no	—
281	C₈	C₂×C₇×SD₁₆	1	4	yes	complemented
282	C₇	C₂×#1693	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₇×#461	1	2	yes	—
288	C₂²	C₇×#469	1	2	yes	—
289	C₂²	C₇×#473	1	2	yes	—
290	C₂²	C₂×C₇×#121	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₇×#461	1	2	yes	—
296	C₄	C₇×#469	1	2	yes	—
297	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
298	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
299	C₄	C₂×C₇×#137	1	1	yes	characteristic
300	C₂	—	8	2	no	—
301	C₂	C₇×#1693	1	4	yes	direct product factor
302	C₂	C₂×C₇×#461	1	1	yes	characteristic
303	C₂	C₂×C₇×#469	1	1	yes	characteristic
304	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×#1693	C₂	direct (N × H)
5. C₂×C₇×#334	C₂	semidirect
6. C₂×C₇×#332	C₂	semidirect
12. C₇×#500	C₂²	semidirect
24. C₇×#334	C₂²	semidirect
25. C₇×#332	C₂²	semidirect
26. C₂×#1693	C₇	direct (N × H)
61. C₂×#500	C₂×C₇	semidirect
64. #1693	C₂×C₇	direct (N × H)
65. C₂×#334	C₂×C₇	semidirect
66. C₂×#332	C₂×C₇	semidirect
91. C₂×C₇×C₈	SD₁₆	semidirect
111. #500	C₂²×C₇	semidirect
123. #334	C₂²×C₇	semidirect
124. #332	C₂²×C₇	semidirect
155. C₇×C₈	C₂×SD₁₆	semidirect
231. C₂×C₈	C₇×SD₁₆	semidirect
247. C₂×C₇	#1693	direct (N × H)
281. C₈	C₂×C₇×SD₁₆	semidirect
282. C₇	C₂×#1693	direct (N × H)
301. C₂	C₇×#1693	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×#436	2
C₇×#1693	2
C₂×C₇×#334	2
C₂×C₇×#332	2
C₂×#1693	7

Subgroups — C2×C7×#1693 #13196351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1693 #13196351