Subgroups — C₂×C₇×#1574 #13195021

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
4	C₇×#1574	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#335	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×C₇×M₁₆	—	2	1	no	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#431	—	2	2	no	—
10	C₇×#431	C₄	1	4	yes	—
11	C₇×#431	C₂²	1	4	yes	—
12	C₇×#431	C₂²	1	4	yes	—
13	C₇×#423	—	2	2	no	—
14	C₇×#423	C₂²	1	4	yes	—
15	C₂×C₇×C₄⋊C₈	—	2	1	no	—
16	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
21	C₂×C₇×C₂²⋊C₈	C₄	1	2	yes	—
22	C₂×C₄×C₇×C₈	—	2	1	no	—
23	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
24	C₇×#335	C₂²	1	8	yes	complemented
25	C₇×#334	C₂²	1	8	yes	complemented
26	C₂×#1574	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
30	C₂×C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×M₁₆	—	4	4	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂²×C₇×C₈	—	2	1	no	—
34	C₂²×C₇×C₈	—	2	2	no	—
35	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇	D₄	1	2	yes	—
39	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
41	C₂×C₄²×C₇	Q₈	1	2	yes	—
42	C₂×C₄²×C₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₇×C₄⋊C₈	—	2	4	no	—
45	C₇×C₄⋊C₈	—	2	2	no	—
46	C₇×C₄⋊C₈	—	2	1	no	—
47	C₇×C₄⋊C₈	—	2	1	no	—
48	C₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₇×C₄⋊C₈	C₂³	1	2	yes	—
50	C₇×C₄⋊C₈	C₂³	1	4	yes	—
51	C₇×C₄⋊C₈	C₂³	1	2	yes	—
52	C₇×C₂²⋊C₈	—	2	4	no	—
53	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
56	C₄×C₇×C₈	C₂³	1	4	yes	—
57	C₂×#431	C₂×C₇	1	1	yes	characteristic
58	C₂×#431	C₂×C₇	1	1	yes	characteristic
59	C₂×#423	C₂×C₇	1	1	yes	characteristic
60	#1574	C₂×C₇	1	8	yes	direct product factor
61	C₂×#335	C₂×C₇	1	2	yes	complemented
62	C₂×#334	C₂×C₇	1	2	yes	complemented
63	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	2	no	—
67	C₂²×C₄×C₇	—	2	2	no	—
68	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
80	C₇×M₁₆	—	4	8	no	—
81	C₂×C₇×C₈	—	2	8	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	4	no	—
85	C₂×C₇×C₈	—	4	2	no	—
86	C₂×C₇×C₈	—	4	4	no	—
87	C₂×C₇×C₈	—	4	2	no	—
88	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
89	C₄²×C₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₇	C₂×D₄	1	2	yes	—
95	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇	C₂×Q₈	1	2	yes	—
97	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₇	C₂⁴	1	1	yes	characteristic
99	C₂²×M₁₆	—	2	1	no	—
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	#431	—	2	2	no	—
102	#431	C₄×C₇	1	4	yes	—
103	#431	C₂²×C₇	1	4	yes	—
104	#431	C₂²×C₇	1	4	yes	—
105	#423	—	2	2	no	—
106	#423	C₂²×C₇	1	4	yes	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
109	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
111	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₂²⋊C₈	—	2	1	no	—
113	C₂×C₂²⋊C₈	C₄×C₇	1	2	yes	—
114	C₂×C₄×C₈	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
116	#335	C₂²×C₇	1	8	yes	complemented
117	#334	C₂²×C₇	1	8	yes	complemented
118	C₂³×C₇	—	2	2	no	—
119	C₂³×C₇	—	2	2	no	—
120	C₂³×C₇	—	2	2	no	—
121	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	4	no	—
131	C₂×C₄×C₇	—	2	4	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	#108	1	4	yes	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₇	#131	1	4	yes	—
141	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₇	#137	1	2	yes	—
143	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
146	C₇×C₈	—	2	8	no	—
147	C₇×C₈	—	4	2	no	—
148	C₇×C₈	—	4	4	no	—
149	C₇×C₈	—	4	2	no	—
150	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
151	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
152	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	4	4	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
159	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
161	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
162	C₂×C₄²	C₇×D₄	1	2	yes	—
163	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
164	C₂×C₄²	C₇×Q₈	1	2	yes	—
165	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₇	1	2	yes	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	C₂×C₄×C₇	1	8	yes	—
172	C₄⋊C₈	C₂³×C₇	1	2	yes	—
173	C₄⋊C₈	C₂³×C₇	1	4	yes	—
174	C₄⋊C₈	C₂³×C₇	1	2	yes	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
179	C₄×C₈	C₂³×C₇	1	4	yes	—
180	C₂²×C₇	—	2	2	no	—
181	C₂²×C₇	—	2	2	no	—
182	C₂²×C₇	—	2	2	no	—
183	C₂²×C₇	—	4	4	no	—
184	C₂²×C₇	#424	1	2	yes	—
185	C₂²×C₇	#428	1	2	yes	—
186	C₂²×C₇	C₂×#108	1	2	yes	—
187	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₇	—	2	2	no	—
189	C₄×C₇	—	2	2	no	—
190	C₄×C₇	—	2	1	no	—
191	C₄×C₇	—	2	1	no	—
192	C₄×C₇	#424	1	2	yes	—
193	C₄×C₇	#428	1	2	yes	—
194	C₄×C₇	C₂×#108	1	4	yes	—
195	C₄×C₇	C₂×#131	1	2	yes	—
196	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₇	C₂×#137	1	1	yes	characteristic
198	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
204	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
206	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
208	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
212	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
215	M₁₆	—	4	8	no	—
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
224	C₄²	C₇×C₄⋊C₄	1	2	yes	—
225	C₄²	C₇×C₄⋊C₄	1	2	yes	—
226	C₄²	C₇×C₄⋊C₄	1	2	yes	—
227	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
228	C₄²	C₂²×C₄×C₇	1	2	yes	—
229	C₄²	C₂×C₇×D₄	1	2	yes	—
230	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
231	C₄²	C₂×C₇×Q₈	1	2	yes	—
232	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
233	C₄²	C₂⁴×C₇	1	1	yes	characteristic
234	C₂×C₇	—	4	2	no	—
235	C₂×C₇	#1574	1	4	yes	direct product factor
236	C₂×C₇	C₂×#424	1	1	yes	characteristic
237	C₂×C₇	C₂×#428	1	1	yes	characteristic
238	C₂×C₇	C₂²×#108	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
243	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	C₇×#108	1	4	yes	—
255	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
257	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
258	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
261	C₂×C₄	C₇×#131	1	4	yes	—
262	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
263	C₂×C₄	C₇×#137	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
267	C₈	—	2	8	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₇	C₂×#1574	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₇×#424	1	2	yes	—
277	C₂²	C₇×#428	1	2	yes	—
278	C₂²	C₂×C₇×#108	1	2	yes	—
279	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	C₇×#424	1	2	yes	—
285	C₄	C₇×#428	1	2	yes	—
286	C₄	C₂×C₇×#108	1	4	yes	—
287	C₄	C₂×C₇×#131	1	2	yes	—
288	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₄	C₂×C₇×#137	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	C₇×#1574	1	4	yes	direct product factor
292	C₂	C₂×C₇×#424	1	1	yes	characteristic
293	C₂	C₂×C₇×#428	1	1	yes	characteristic
294	C₂	C₂²×C₇×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1574	C₂	direct (N × H)
5. C₂×C₇×#335	C₂	semidirect
6. C₂×C₇×#334	C₂	semidirect
24. C₇×#335	C₂²	semidirect
25. C₇×#334	C₂²	semidirect
26. C₂×#1574	C₇	direct (N × H)
60. #1574	C₂×C₇	direct (N × H)
61. C₂×#335	C₂×C₇	semidirect
62. C₂×#334	C₂×C₇	semidirect
116. #335	C₂²×C₇	semidirect
117. #334	C₂²×C₇	semidirect
235. C₂×C₇	#1574	direct (N × H)
271. C₇	C₂×#1574	direct (N × H)
291. C₂	C₇×#1574	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×#431	2
C₂×C₇×#423	2
C₇×#1574	2
C₂×C₇×#335	2
C₂×C₇×#334	2
C₂×#1574	7

Subgroups — C2×C7×#1574 #13195021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1574 #13195021