Subgroups — C₂²×C₇×#481 #13237720

Order 1792.

3006 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#121	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#481	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#106	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×D₄	—	2	1	no	—
8	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₇	—	2	1	no	—
10	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂³×C₇×C₈	C₂²	1	2	yes	—
12	C₂×C₇×#121	—	2	6	no	—
13	C₂×C₇×#121	C₂²	1	12	yes	—
14	C₂²×C₇×C₄⋊C₄	—	2	6	no	—
15	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₇×C₄⋊C₄	C₂²	1	12	yes	complemented
19	C₂²×C₇×C₂²⋊C₄	—	2	1	no	—
20	C₇×#481	C₂²	1	64	yes	direct product factor
21	C₂×C₇×#106	C₂²	1	24	yes	complemented
22	C₂×C₇×#102	C₂²	1	24	yes	complemented
23	C₂×C₇×C₂²⋊C₈	C₂²	1	12	yes	complemented
24	C₂²×#481	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₇	—	2	1	no	—
26	C₂⁵×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₇×D₄	—	2	12	no	—
28	C₂²×C₇×D₄	—	2	3	no	—
29	C₂²×C₇×D₄	—	2	8	no	—
30	C₂²×C₇×D₄	—	4	8	no	—
31	C₂²×C₇×D₄	C₂³	1	6	yes	—
32	C₂³×C₄×C₇	—	2	8	no	—
33	C₂³×C₄×C₇	—	2	12	no	—
34	C₂³×C₄×C₇	—	2	1	no	—
35	C₂³×C₄×C₇	—	2	2	no	—
36	C₂³×C₄×C₇	—	4	8	no	—
37	C₂³×C₄×C₇	D₄	1	6	yes	—
38	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇	C₂³	1	6	yes	—
40	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₇×C₈	—	2	8	no	—
42	C₂²×C₇×C₈	C₂³	1	12	yes	—
43	C₇×#121	—	2	24	no	—
44	C₇×#121	C₂³	1	16	yes	—
45	C₂×C₇×C₄⋊C₄	—	2	6	no	—
46	C₂×C₇×C₄⋊C₄	—	2	24	no	—
47	C₂×C₇×C₄⋊C₄	—	2	12	no	—
48	C₂×C₇×C₄⋊C₄	—	2	3	no	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	complemented
52	C₂×C₇×C₂²⋊C₄	—	2	12	no	—
53	C₇×#106	C₂³	1	32	yes	complemented
54	C₇×#102	C₂³	1	32	yes	complemented
55	C₇×C₂²⋊C₈	C₂³	1	16	yes	complemented
56	C₂²×#121	C₂×C₇	1	1	yes	characteristic
57	C₂³×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂×#481	C₂×C₇	1	24	yes	direct product factor
59	C₂²×#106	C₂×C₇	1	2	yes	complemented
60	C₂²×#102	C₂×C₇	1	2	yes	complemented
61	C₂²×C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂⁴×C₇	—	2	4	no	—
63	C₂⁴×C₇	—	2	4	no	—
64	C₂⁴×C₇	—	2	6	no	—
65	C₂⁴×C₇	—	4	4	no	—
66	C₂⁴×C₇	—	4	4	no	—
67	C₂⁴×C₇	—	4	4	no	—
68	C₂⁴×C₇	SD₁₆	1	8	yes	—
69	C₂⁴×C₇	C₂×D₄	1	6	yes	—
70	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₇×D₄	—	2	16	no	—
72	C₂×C₇×D₄	—	2	6	no	—
73	C₂×C₇×D₄	—	2	24	no	—
74	C₂×C₇×D₄	—	4	48	no	—
75	C₂×C₇×D₄	—	4	12	no	—
76	C₂×C₇×D₄	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₇	—	2	4	no	—
78	C₂²×C₄×C₇	—	2	24	no	—
79	C₂²×C₄×C₇	—	2	24	no	—
80	C₂²×C₄×C₇	—	2	6	no	—
81	C₂²×C₄×C₇	—	2	12	no	—
82	C₂²×C₄×C₇	—	2	16	no	—
83	C₂²×C₄×C₇	—	4	48	no	—
84	C₂²×C₄×C₇	—	4	4	no	—
85	C₂²×C₄×C₇	—	4	8	no	—
86	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
88	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₇	#41	1	8	yes	—
90	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
92	C₂×C₇×C₈	—	2	24	no	—
93	C₂×C₇×C₈	C₂⁴	1	8	yes	—
94	C₇×C₄⋊C₄	—	2	12	no	—
95	C₇×C₄⋊C₄	—	2	16	no	—
96	C₇×C₄⋊C₄	—	2	6	no	—
97	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
98	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
99	C₇×C₂²⋊C₄	—	2	16	no	—
100	C₂³×D₄	—	2	1	no	—
101	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
102	C₂⁴×C₄	—	2	1	no	—
103	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₇	1	2	yes	—
105	C₂×#121	—	2	6	no	—
106	C₂×#121	C₂²×C₇	1	12	yes	—
107	C₂²×C₄⋊C₄	—	2	6	no	—
108	C₂²×C₄⋊C₄	—	2	1	no	—
109	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂²×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂²×C₄⋊C₄	C₂²×C₇	1	12	yes	complemented
112	C₂²×C₂²⋊C₄	—	2	1	no	—
113	#481	C₂²×C₇	1	64	yes	direct product factor
114	C₂×#106	C₂²×C₇	1	24	yes	complemented
115	C₂×#102	C₂²×C₇	1	24	yes	complemented
116	C₂×C₂²⋊C₈	C₂²×C₇	1	12	yes	complemented
117	C₂³×C₇	—	2	12	no	—
118	C₂³×C₇	—	2	12	no	—
119	C₂³×C₇	—	2	32	no	—
120	C₂³×C₇	—	2	4	no	—
121	C₂³×C₇	—	4	12	no	—
122	C₂³×C₇	—	4	12	no	—
123	C₂³×C₇	—	4	12	no	—
124	C₂³×C₇	—	8	16	no	—
125	C₂³×C₇	#123	1	4	yes	—
126	C₂³×C₇	C₂×SD₁₆	1	24	yes	—
127	C₂³×C₇	C₂×SD₁₆	1	4	yes	—
128	C₂³×C₇	AGL(1,ℤ/8)	1	4	yes	—
129	C₂³×C₇	C₂²×D₄	1	4	yes	—
130	C₂³×C₇	C₂²×D₄	1	3	yes	—
131	C₇×D₄	—	2	8	no	—
132	C₇×D₄	—	4	32	no	—
133	C₇×D₄	—	4	12	no	—
134	C₂×C₄×C₇	—	2	12	no	—
135	C₂×C₄×C₇	—	2	24	no	—
136	C₂×C₄×C₇	—	2	8	no	—
137	C₂×C₄×C₇	—	2	4	no	—
138	C₂×C₄×C₇	—	2	8	no	—
139	C₂×C₄×C₇	—	4	32	no	—
140	C₂×C₄×C₇	—	4	12	no	—
141	C₂×C₄×C₇	—	4	24	no	—
142	C₂×C₄×C₇	#123	1	12	yes	—
143	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
144	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
145	C₂×C₄×C₇	C₂×#41	1	12	yes	—
146	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
147	C₇×C₈	—	2	8	no	—
148	C₂⁵	—	2	1	no	—
149	C₂⁵	C₇×D₄	1	1	yes	characteristic
150	C₂²×D₄	—	2	12	no	—
151	C₂²×D₄	—	2	3	no	—
152	C₂²×D₄	—	2	8	no	—
153	C₂²×D₄	—	4	8	no	—
154	C₂²×D₄	C₂³×C₇	1	6	yes	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	—	2	2	no	—
158	C₂³×C₄	—	2	12	no	—
159	C₂³×C₄	—	4	8	no	—
160	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
161	C₂³×C₄	C₇×D₄	1	6	yes	—
162	C₂³×C₄	C₂³×C₇	1	6	yes	—
163	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
164	C₂²×C₈	—	2	8	no	—
165	C₂²×C₈	C₂³×C₇	1	12	yes	—
166	#121	—	2	24	no	—
167	#121	C₂³×C₇	1	16	yes	—
168	C₂×C₄⋊C₄	—	2	6	no	—
169	C₂×C₄⋊C₄	—	2	24	no	—
170	C₂×C₄⋊C₄	—	2	12	no	—
171	C₂×C₄⋊C₄	—	2	3	no	—
172	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
173	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
174	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	complemented
175	C₂×C₂²⋊C₄	—	2	12	no	—
176	#106	C₂³×C₇	1	32	yes	complemented
177	#102	C₂³×C₇	1	32	yes	complemented
178	C₂²⋊C₈	C₂³×C₇	1	16	yes	complemented
179	C₂²×C₇	—	2	48	no	—
180	C₂²×C₇	—	2	4	no	—
181	C₂²×C₇	—	2	4	no	—
182	C₂²×C₇	—	4	4	no	—
183	C₂²×C₇	—	4	4	no	—
184	C₂²×C₇	—	4	4	no	—
185	C₂²×C₇	—	8	24	no	—
186	C₂²×C₇	#481	1	16	yes	direct product factor
187	C₂²×C₇	C₂×#123	1	6	yes	—
188	C₂²×C₇	C₂²×SD₁₆	1	8	yes	—
189	C₂²×C₇	C₂²×SD₁₆	1	6	yes	—
190	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
191	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
192	C₄×C₇	—	2	4	no	—
193	C₄×C₇	—	4	4	no	—
194	C₄×C₇	—	4	8	no	—
195	C₄×C₇	C₂×#123	1	6	yes	—
196	C₄×C₇	C₂²×#41	1	2	yes	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	4	no	—
199	C₂⁴	—	2	6	no	—
200	C₂⁴	—	4	4	no	—
201	C₂⁴	—	4	4	no	—
202	C₂⁴	—	4	4	no	—
203	C₂⁴	C₇×SD₁₆	1	8	yes	—
204	C₂⁴	C₂×C₇×D₄	1	6	yes	—
205	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
206	C₂×D₄	—	2	16	no	—
207	C₂×D₄	—	2	6	no	—
208	C₂×D₄	—	2	24	no	—
209	C₂×D₄	—	4	48	no	—
210	C₂×D₄	—	4	12	no	—
211	C₂×D₄	C₂⁴×C₇	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	24	no	—
214	C₂²×C₄	—	2	24	no	—
215	C₂²×C₄	—	2	6	no	—
216	C₂²×C₄	—	2	12	no	—
217	C₂²×C₄	—	2	16	no	—
218	C₂²×C₄	—	4	48	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	8	no	—
221	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
222	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
223	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
224	C₂²×C₄	C₇×#41	1	8	yes	—
225	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
227	C₂×C₈	—	2	24	no	—
228	C₂×C₈	C₂⁴×C₇	1	8	yes	—
229	C₄⋊C₄	—	2	12	no	—
230	C₄⋊C₄	—	2	16	no	—
231	C₄⋊C₄	—	2	6	no	—
232	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
233	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
234	C₂²⋊C₄	—	2	16	no	—
235	C₂×C₇	—	2	8	no	—
236	C₂×C₇	—	8	4	no	—
237	C₂×C₇	C₂×#481	1	12	yes	direct product factor
238	C₂×C₇	C₂²×#123	1	1	yes	characteristic
239	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
240	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₂³	—	2	32	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	12	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	12	no	—
246	C₂³	—	4	12	no	—
247	C₂³	—	4	12	no	—
248	C₂³	—	8	16	no	—
249	C₂³	C₇×#123	1	4	yes	—
250	C₂³	C₂×C₇×SD₁₆	1	24	yes	—
251	C₂³	C₂×C₇×SD₁₆	1	4	yes	—
252	C₂³	C₇×AGL(1,ℤ/8)	1	4	yes	—
253	C₂³	C₂²×C₇×D₄	1	4	yes	—
254	C₂³	C₂²×C₇×D₄	1	3	yes	—
255	D₄	—	2	8	no	—
256	D₄	—	4	32	no	—
257	D₄	—	4	12	no	—
258	C₂×C₄	—	2	12	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	32	no	—
264	C₂×C₄	—	4	12	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	C₇×#123	1	12	yes	—
267	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
268	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
269	C₂×C₄	C₂×C₇×#41	1	12	yes	—
270	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₇	C₂²×#481	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	48	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	8	24	no	—
280	C₂²	C₇×#481	1	16	yes	direct product factor
281	C₂²	C₂×C₇×#123	1	6	yes	—
282	C₂²	C₂²×C₇×SD₁₆	1	8	yes	—
283	C₂²	C₂²×C₇×SD₁₆	1	6	yes	—
284	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
285	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	8	no	—
289	C₄	C₂×C₇×#123	1	6	yes	—
290	C₄	C₂²×C₇×#41	1	2	yes	—
291	C₂	—	2	8	no	—
292	C₂	—	8	4	no	—
293	C₂	C₂×C₇×#481	1	12	yes	direct product factor
294	C₂	C₂²×C₇×#123	1	1	yes	characteristic
295	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
296	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂²×C₇×#121	C₂
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. C₂³×C₇×C₄⋊C₄	C₂	semidirect
3. C₂×C₇×#481	C₂	direct (N × H)
4. C₂²×C₇×#106	C₂	semidirect
5. C₂²×C₇×#102	C₂	semidirect
6. C₂²×C₇×C₂²⋊C₈	C₂	semidirect
18. C₂²×C₇×C₄⋊C₄	C₂²	semidirect
20. C₇×#481	C₂²	direct (N × H)
21. C₂×C₇×#106	C₂²	semidirect
22. C₂×C₇×#102	C₂²	semidirect
23. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
24. C₂²×#481	C₇	direct (N × H)
51. C₂×C₇×C₄⋊C₄	C₂³	semidirect
53. C₇×#106	C₂³	semidirect
54. C₇×#102	C₂³	semidirect
55. C₇×C₂²⋊C₈	C₂³	semidirect
57. C₂³×C₄⋊C₄	C₂×C₇	semidirect
58. C₂×#481	C₂×C₇	direct (N × H)
59. C₂²×#106	C₂×C₇	semidirect
60. C₂²×#102	C₂×C₇	semidirect
61. C₂²×C₂²⋊C₈	C₂×C₇	semidirect
111. C₂²×C₄⋊C₄	C₂²×C₇	semidirect
113. #481	C₂²×C₇	direct (N × H)
114. C₂×#106	C₂²×C₇	semidirect
115. C₂×#102	C₂²×C₇	semidirect
116. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
174. C₂×C₄⋊C₄	C₂³×C₇	semidirect
176. #106	C₂³×C₇	semidirect
177. #102	C₂³×C₇	semidirect
178. C₂²⋊C₈	C₂³×C₇	semidirect
186. C₂²×C₇	#481	direct (N × H)
237. C₂×C₇	C₂×#481	direct (N × H)
272. C₇	C₂²×#481	direct (N × H)
280. C₂²	C₇×#481	direct (N × H)
293. C₂	C₂×C₇×#481	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×#121	2
C₂³×C₇×C₄⋊C₄	2
C₂×C₇×#481	2
C₂²×C₇×#106	2
C₂²×C₇×#102	2
C₂²×C₇×C₂²⋊C₈	2
C₂²×#481	7

Subgroups — C22×C7×#481 #13237720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#481 #13237720