Subgroups — C₂×C₇×#1689 #13196348

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#474	C₂	1	2	yes	maximal
3	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic
4	C₇×#1689	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×Q₁₆	—	2	1	no	—
8	C₂²×C₇×Q₁₆	—	2	1	no	—
9	C₂²×C₇×M₁₆	—	2	1	no	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₇×#494	—	2	2	no	—
13	C₇×#494	C₂²	1	4	yes	complemented
14	C₇×#474	—	2	4	no	—
15	C₇×#474	C₂²	1	8	yes	—
16	C₇×#436	—	2	2	no	—
17	C₇×#436	C₂²	1	4	yes	—
18	C₂×C₇×#107	—	2	2	no	—
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₇×#102	—	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×#1689	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×Q₁₆	—	2	4	no	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	—	4	4	no	—
33	C₂×C₇×Q₁₆	—	4	4	no	—
34	C₂×C₇×M₁₆	—	2	4	no	—
35	C₂×C₇×M₁₆	—	4	4	no	—
36	C₂²×C₇×C₈	—	2	1	no	—
37	C₂²×C₇×C₈	—	2	2	no	—
38	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	C₇×C₄⋊Q₈	—	2	2	no	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
44	C₄×C₇×D₄	—	2	1	no	—
45	C₄×C₇×D₄	—	2	1	no	—
46	C₄×C₇×D₄	D₄	1	2	yes	—
47	C₄×C₇×D₄	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	—	2	2	no	—
50	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
53	C₇×#107	—	2	8	no	—
54	C₇×C₄⋊C₈	—	2	2	no	—
55	C₇×C₄⋊C₈	—	2	1	no	—
56	C₇×C₄⋊C₈	C₂³	1	4	yes	—
57	C₇×C₄⋊C₈	C₂³	1	2	yes	—
58	C₇×#102	—	2	8	no	—
59	C₇×C₂²⋊C₈	—	2	4	no	—
60	C₄×C₇×C₈	D₄	1	2	yes	—
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂×#474	C₂×C₇	1	2	yes	—
64	C₂×#436	C₂×C₇	1	1	yes	characteristic
65	#1689	C₂×C₇	1	8	yes	direct product factor
66	C₂×#334	C₂×C₇	1	1	yes	characteristic, complemented
67	C₂×#332	C₂×C₇	1	2	yes	complemented
68	C₂⁴×C₇	—	2	1	no	—
69	C₂×C₇×Q₈	—	2	4	no	—
70	C₂×C₇×Q₈	—	4	2	no	—
71	C₂×C₇×Q₈	—	4	4	no	—
72	C₂×C₇×Q₈	—	4	4	no	—
73	C₂×C₇×D₄	—	2	1	no	—
74	C₂×C₇×D₄	—	2	4	no	—
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×Q₁₆	—	4	8	no	—
86	C₇×Q₁₆	—	4	8	no	—
87	C₇×M₁₆	—	4	8	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	4	no	—
91	C₂×C₇×C₈	—	4	2	no	—
92	C₂×C₇×C₈	—	4	4	no	—
93	C₂×C₇×C₈	D₈	1	4	yes	—
94	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
96	C₇×C₄⋊C₄	—	2	1	no	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₇×C₄⋊C₄	—	4	2	no	—
101	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
102	C₇×C₂²⋊C₄	—	2	2	no	—
103	C₇×C₂²⋊C₄	—	2	2	no	—
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇	#41	1	1	yes	characteristic
107	C₄²×C₇	C₂⁴	1	1	yes	characteristic
108	C₂²×Q₁₆	—	2	1	no	—
109	C₂²×Q₁₆	—	2	1	no	—
110	C₂²×M₁₆	—	2	1	no	—
111	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
112	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
113	#494	—	2	2	no	—
114	#494	C₂²×C₇	1	4	yes	complemented
115	#474	—	2	4	no	—
116	#474	C₂²×C₇	1	8	yes	—
117	#436	—	2	2	no	—
118	#436	C₂²×C₇	1	4	yes	—
119	C₂×#107	—	2	2	no	—
120	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
122	C₂×#102	—	2	2	no	—
123	C₂×C₂²⋊C₈	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
125	#334	C₂²×C₇	1	4	yes	complemented
126	#332	C₂²×C₇	1	8	yes	complemented
127	C₂³×C₇	—	2	2	no	—
128	C₂³×C₇	—	4	2	no	—
129	C₂³×C₇	—	4	2	no	—
130	C₂³×C₇	—	4	2	no	—
131	C₂³×C₇	#121	1	1	yes	characteristic
132	C₇×Q₈	—	4	4	no	—
133	C₇×Q₈	—	4	4	no	—
134	C₇×Q₈	—	8	4	no	—
135	C₇×D₄	—	2	4	no	—
136	C₇×D₄	—	4	2	no	—
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	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₇	#137	1	4	yes	—
151	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
154	C₇×C₈	—	2	2	no	—
155	C₇×C₈	—	4	2	no	—
156	C₇×C₈	—	4	4	no	—
157	C₇×C₈	C₂×D₈	1	4	yes	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	4	4	no	—
164	C₂×Q₁₆	—	4	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
170	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	C₇×D₄	1	2	yes	—
178	C₄×D₄	C₂³×C₇	1	2	yes	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
184	#107	—	2	8	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	#102	—	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	—	4	2	no	—
194	C₂²×C₇	—	4	2	no	—
195	C₂²×C₇	—	4	2	no	—
196	C₂²×C₇	—	8	4	no	—
197	C₂²×C₇	#459	1	2	yes	—
198	C₂²×C₇	#470	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	#459	1	2	yes	—
206	C₄×C₇	#470	1	2	yes	—
207	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
208	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₄×C₇	C₂×#137	1	2	yes	—
210	C₂⁴	—	2	1	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₇×#41	1	1	yes	characteristic
227	Q₁₆	—	4	8	no	—
228	Q₁₆	—	4	8	no	—
229	M₁₆	—	4	8	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	C₇×D₈	1	4	yes	—
236	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
237	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
247	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
248	C₄²	C₇×#41	1	1	yes	characteristic
249	C₄²	C₂⁴×C₇	1	1	yes	characteristic
250	C₂×C₇	—	8	2	no	—
251	C₂×C₇	#1689	1	4	yes	direct product factor
252	C₂×C₇	C₂×#459	1	1	yes	characteristic
253	C₂×C₇	C₂×#470	1	1	yes	characteristic
254	C₂×C₇	C₂×#473	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	C₇×#121	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	8	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	4	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	C₇×#121	1	1	yes	characteristic
274	C₂×C₄	C₇×#121	1	1	yes	characteristic
275	C₂×C₄	C₇×#121	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
277	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
278	C₂×C₄	C₇×#137	1	4	yes	—
279	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	4	no	—
285	C₈	C₂×C₇×D₈	1	4	yes	—
286	C₇	C₂×#1689	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₇×#459	1	2	yes	—
292	C₂²	C₇×#470	1	2	yes	—
293	C₂²	C₇×#473	1	2	yes	—
294	C₂²	C₂×C₇×#121	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	8	4	no	—
299	C₄	C₇×#459	1	2	yes	—
300	C₄	C₇×#470	1	2	yes	—
301	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
302	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₄	C₂×C₇×#137	1	2	yes	—
304	C₂	—	8	2	no	—
305	C₂	C₇×#1689	1	4	yes	direct product factor
306	C₂	C₂×C₇×#459	1	1	yes	characteristic
307	C₂	C₂×C₇×#470	1	1	yes	characteristic
308	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×#494	C₂	semidirect
4. C₇×#1689	C₂	direct (N × H)
5. C₂×C₇×#334	C₂	semidirect
6. C₂×C₇×#332	C₂	semidirect
13. C₇×#494	C₂²	semidirect
24. C₇×#334	C₂²	semidirect
25. C₇×#332	C₂²	semidirect
26. C₂×#1689	C₇	direct (N × H)
62. C₂×#494	C₂×C₇	semidirect
65. #1689	C₂×C₇	direct (N × H)
66. C₂×#334	C₂×C₇	semidirect
67. C₂×#332	C₂×C₇	semidirect
114. #494	C₂²×C₇	semidirect
125. #334	C₂²×C₇	semidirect
126. #332	C₂²×C₇	semidirect
251. C₂×C₇	#1689	direct (N × H)
286. C₇	C₂×#1689	direct (N × H)
305. C₂	C₇×#1689	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×#494	2
C₂×C₇×#474	2
C₂×C₇×#436	2
C₇×#1689	2
C₂×C₇×#334	2
C₂×C₇×#332	2
C₂×#1689	7

Subgroups — C2×C7×#1689 #13196348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1689 #13196348