Subgroups — C₂²×C₇×#540 #13239893

Order 1792.

8838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#540	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×#123	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×#121	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×#121	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
8	C₂²×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₇×D₄	C₂²	1	2	yes	—
10	C₂³×C₇×D₄	C₂²	1	4	yes	—
11	C₂³×C₇×D₄	C₂²	1	4	yes	—
12	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₇	C₂²	1	4	yes	—
14	C₇×#540	C₂²	1	256	yes	direct product factor
15	C₂×C₇×#127	C₂²	1	24	yes	complemented
16	C₂×C₇×#125	C₂²	1	24	yes	complemented
17	C₂×C₇×#123	C₂²	1	96	yes	complemented
18	C₂×C₇×#121	C₂²	1	96	yes	complemented
19	C₂×C₇×#121	C₂²	1	48	yes	complemented
20	C₂×C₄×C₇×D₄	C₂²	1	48	yes	complemented
21	C₂×C₇×#117	C₂²	1	24	yes	complemented
22	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂²×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
28	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₂²×#540	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂⁵×C₇	C₂³	1	1	yes	characteristic
32	C₂⁵×C₇	C₂³	1	4	yes	—
33	C₂²×C₇×D₄	—	2	16	no	—
34	C₂²×C₇×D₄	—	4	16	no	—
35	C₂²×C₇×D₄	—	4	16	no	—
36	C₂²×C₇×D₄	C₂³	1	24	yes	—
37	C₂²×C₇×D₄	C₂³	1	48	yes	—
38	C₂²×C₇×D₄	C₂³	1	48	yes	—
39	C₂³×C₄×C₇	—	2	8	no	—
40	C₂³×C₄×C₇	—	4	16	no	—
41	C₂³×C₄×C₇	C₂³	1	12	yes	—
42	C₂³×C₄×C₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₇	C₂³	1	48	yes	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×#127	C₂³	1	64	yes	complemented
49	C₇×#125	C₂³	1	64	yes	complemented
50	C₇×#123	C₂³	1	256	yes	complemented
51	C₇×#121	C₂³	1	256	yes	complemented
52	C₇×#121	C₂³	1	128	yes	complemented
53	C₄×C₇×D₄	C₂³	1	128	yes	complemented
54	C₇×#117	C₂³	1	64	yes	complemented
55	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
56	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
57	C₂×C₇×C₄⋊C₄	C₂³	1	48	yes	—
58	C₂×C₇×C₂²⋊C₄	C₂³	1	24	yes	—
59	C₂×C₇×C₂²⋊C₄	C₂³	1	48	yes	—
60	C₂×C₇×C₂²⋊C₄	C₂³	1	48	yes	—
61	C₂×C₄²×C₇	C₂³	1	12	yes	—
62	C₂×C₄²×C₇	C₂³	1	12	yes	—
63	C₂×#540	C₂×C₇	1	48	yes	direct product factor
64	C₂²×#127	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂²×#125	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂²×#123	C₂×C₇	1	4	yes	complemented
67	C₂²×#121	C₂×C₇	1	4	yes	complemented
68	C₂²×#121	C₂×C₇	1	2	yes	complemented
69	C₂²×C₄×D₄	C₂×C₇	1	2	yes	complemented
70	C₂²×#117	C₂×C₇	1	1	yes	characteristic, complemented
71	C₂⁴×C₇	—	2	4	no	—
72	C₂⁴×C₇	—	2	16	no	—
73	C₂⁴×C₇	—	2	16	no	—
74	C₂⁴×C₇	—	2	8	no	—
75	C₂⁴×C₇	—	2	16	no	—
76	C₂⁴×C₇	C₂⁴	1	6	yes	—
77	C₂⁴×C₇	C₂⁴	1	24	yes	—
78	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₇×D₄	—	2	96	no	—
80	C₂×C₇×D₄	—	4	96	no	—
81	C₂×C₇×D₄	—	4	96	no	—
82	C₂×C₇×D₄	C₂⁴	1	32	yes	—
83	C₂×C₇×D₄	C₂⁴	1	64	yes	—
84	C₂×C₇×D₄	C₂⁴	1	64	yes	—
85	C₂²×C₄×C₇	—	2	4	no	—
86	C₂²×C₄×C₇	—	2	8	no	—
87	C₂²×C₄×C₇	—	2	16	no	—
88	C₂²×C₄×C₇	—	2	8	no	—
89	C₂²×C₄×C₇	—	2	48	no	—
90	C₂²×C₄×C₇	—	4	96	no	—
91	C₂²×C₄×C₇	#41	1	8	yes	—
92	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
93	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
94	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
95	C₂²×C₄×C₇	C₂⁴	1	24	yes	—
96	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
97	C₂²×C₄×C₇	C₂⁴	1	64	yes	—
98	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
99	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
100	C₇×C₄⋊C₄	C₂⁴	1	32	yes	—
101	C₇×C₄⋊C₄	C₂⁴	1	64	yes	—
102	C₇×C₂²⋊C₄	C₂⁴	1	32	yes	—
103	C₇×C₂²⋊C₄	C₂⁴	1	64	yes	—
104	C₇×C₂²⋊C₄	C₂⁴	1	64	yes	—
105	C₄²×C₇	C₂⁴	1	16	yes	—
106	C₄²×C₇	C₂⁴	1	16	yes	—
107	C₂³×D₄	C₂²×C₇	1	2	yes	—
108	C₂³×D₄	C₂²×C₇	1	4	yes	—
109	C₂³×D₄	C₂²×C₇	1	4	yes	—
110	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂⁴×C₄	C₂²×C₇	1	4	yes	—
112	#540	C₂²×C₇	1	256	yes	direct product factor
113	C₂×#127	C₂²×C₇	1	24	yes	complemented
114	C₂×#125	C₂²×C₇	1	24	yes	complemented
115	C₂×#123	C₂²×C₇	1	96	yes	complemented
116	C₂×#121	C₂²×C₇	1	96	yes	complemented
117	C₂×#121	C₂²×C₇	1	48	yes	complemented
118	C₂×C₄×D₄	C₂²×C₇	1	48	yes	complemented
119	C₂×#117	C₂²×C₇	1	24	yes	complemented
120	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
123	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
124	C₂²×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
125	C₂²×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
126	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
127	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
128	C₂³×C₇	—	2	12	no	—
129	C₂³×C₇	—	2	48	no	—
130	C₂³×C₇	—	2	48	no	—
131	C₂³×C₇	—	2	24	no	—
132	C₂³×C₇	—	2	48	no	—
133	C₂³×C₇	—	4	16	no	—
134	C₂³×C₇	—	4	64	no	—
135	C₂³×C₇	C₂×#41	1	4	yes	—
136	C₂³×C₇	H(5,𝔽₂)	1	8	yes	—
137	C₂³×C₇	C₂⁵	1	4	yes	—
138	C₂³×C₇	C₂⁵	1	16	yes	—
139	C₂³×C₇	C₂⁵	1	3	yes	—
140	C₇×D₄	—	2	64	no	—
141	C₇×D₄	—	4	64	no	—
142	C₇×D₄	—	4	64	no	—
143	C₂×C₄×C₇	—	2	12	no	—
144	C₂×C₄×C₇	—	2	24	no	—
145	C₂×C₄×C₇	—	2	48	no	—
146	C₂×C₄×C₇	—	2	24	no	—
147	C₂×C₄×C₇	—	2	32	no	—
148	C₂×C₄×C₇	—	4	64	no	—
149	C₂×C₄×C₇	C₂×#41	1	24	yes	—
150	C₂×C₄×C₇	C₂⁵	1	8	yes	—
151	C₂×C₄×C₇	C₂⁵	1	8	yes	—
152	C₂×C₄×C₇	C₂⁵	1	4	yes	—
153	C₂×C₄×C₇	C₂⁵	1	16	yes	—
154	C₂×C₄×C₇	C₂⁵	1	4	yes	—
155	C₂⁵	C₂³×C₇	1	1	yes	characteristic
156	C₂⁵	C₂³×C₇	1	4	yes	—
157	C₂²×D₄	—	2	16	no	—
158	C₂²×D₄	—	4	16	no	—
159	C₂²×D₄	—	4	16	no	—
160	C₂²×D₄	C₂³×C₇	1	24	yes	—
161	C₂²×D₄	C₂³×C₇	1	48	yes	—
162	C₂²×D₄	C₂³×C₇	1	48	yes	—
163	C₂³×C₄	—	2	8	no	—
164	C₂³×C₄	—	4	16	no	—
165	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₇	1	2	yes	—
167	C₂³×C₄	C₂³×C₇	1	4	yes	—
168	C₂³×C₄	C₂³×C₇	1	2	yes	—
169	C₂³×C₄	C₂³×C₇	1	12	yes	—
170	C₂³×C₄	C₂³×C₇	1	48	yes	—
171	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
172	#127	C₂³×C₇	1	64	yes	complemented
173	#125	C₂³×C₇	1	64	yes	complemented
174	#123	C₂³×C₇	1	256	yes	complemented
175	#121	C₂³×C₇	1	256	yes	complemented
176	#121	C₂³×C₇	1	128	yes	complemented
177	C₄×D₄	C₂³×C₇	1	128	yes	complemented
178	#117	C₂³×C₇	1	64	yes	complemented
179	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
180	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	48	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₇	1	24	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₇	1	48	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₇	1	48	yes	—
185	C₂×C₄²	C₂³×C₇	1	12	yes	—
186	C₂×C₄²	C₂³×C₇	1	12	yes	—
187	C₂²×C₇	—	2	4	no	—
188	C₂²×C₇	—	2	16	no	—
189	C₂²×C₇	—	2	16	no	—
190	C₂²×C₇	—	2	8	no	—
191	C₂²×C₇	—	2	16	no	—
192	C₂²×C₇	—	4	24	no	—
193	C₂²×C₇	—	4	96	no	—
194	C₂²×C₇	#540	1	16	yes	direct product factor
195	C₂²×C₇	C₂²×#41	1	6	yes	—
196	C₂²×C₇	C₂×H(5,𝔽₂)	1	12	yes	—
197	C₂²×C₇	C₂⁶	1	1	yes	characteristic
198	C₄×C₇	—	2	4	no	—
199	C₄×C₇	—	2	8	no	—
200	C₄×C₇	—	2	16	no	—
201	C₄×C₇	—	2	8	no	—
202	C₄×C₇	C₂²×#41	1	8	yes	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	—	2	16	no	—
205	C₂⁴	—	2	16	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	—	2	16	no	—
208	C₂⁴	C₂⁴×C₇	1	6	yes	—
209	C₂⁴	C₂⁴×C₇	1	24	yes	—
210	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
211	C₂×D₄	—	2	96	no	—
212	C₂×D₄	—	4	96	no	—
213	C₂×D₄	—	4	96	no	—
214	C₂×D₄	C₂⁴×C₇	1	32	yes	—
215	C₂×D₄	C₂⁴×C₇	1	64	yes	—
216	C₂×D₄	C₂⁴×C₇	1	64	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	48	no	—
222	C₂²×C₄	—	4	96	no	—
223	C₂²×C₄	C₇×#41	1	8	yes	—
224	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
225	C₂²×C₄	C₂⁴×C₇	1	12	yes	—
226	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
227	C₂²×C₄	C₂⁴×C₇	1	24	yes	—
228	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
229	C₂²×C₄	C₂⁴×C₇	1	64	yes	—
230	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
231	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
232	C₄⋊C₄	C₂⁴×C₇	1	32	yes	—
233	C₄⋊C₄	C₂⁴×C₇	1	64	yes	—
234	C₂²⋊C₄	C₂⁴×C₇	1	32	yes	—
235	C₂²⋊C₄	C₂⁴×C₇	1	64	yes	—
236	C₂²⋊C₄	C₂⁴×C₇	1	64	yes	—
237	C₄²	C₂⁴×C₇	1	16	yes	—
238	C₄²	C₂⁴×C₇	1	16	yes	—
239	C₂×C₇	—	4	4	no	—
240	C₂×C₇	—	4	16	no	—
241	C₂×C₇	C₂×#540	1	12	yes	direct product factor
242	C₂×C₇	C₂³×#41	1	1	yes	characteristic
243	C₂×C₇	C₂²×H(5,𝔽₂)	1	2	yes	—
244	C₂³	—	2	12	no	—
245	C₂³	—	2	48	no	—
246	C₂³	—	2	48	no	—
247	C₂³	—	2	24	no	—
248	C₂³	—	2	48	no	—
249	C₂³	—	4	16	no	—
250	C₂³	—	4	64	no	—
251	C₂³	C₂×C₇×#41	1	4	yes	—
252	C₂³	C₇×H(5,𝔽₂)	1	8	yes	—
253	C₂³	C₂⁵×C₇	1	4	yes	—
254	C₂³	C₂⁵×C₇	1	16	yes	—
255	C₂³	C₂⁵×C₇	1	3	yes	—
256	D₄	—	2	64	no	—
257	D₄	—	4	64	no	—
258	D₄	—	4	64	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	48	no	—
262	C₂×C₄	—	2	24	no	—
263	C₂×C₄	—	2	32	no	—
264	C₂×C₄	—	4	64	no	—
265	C₂×C₄	C₂×C₇×#41	1	24	yes	—
266	C₂×C₄	C₂⁵×C₇	1	8	yes	—
267	C₂×C₄	C₂⁵×C₇	1	8	yes	—
268	C₂×C₄	C₂⁵×C₇	1	4	yes	—
269	C₂×C₄	C₂⁵×C₇	1	16	yes	—
270	C₂×C₄	C₂⁵×C₇	1	4	yes	—
271	C₇	C₂²×#540	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	2	4	no	—
273	C₂²	—	2	16	no	—
274	C₂²	—	2	16	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	16	no	—
277	C₂²	—	4	24	no	—
278	C₂²	—	4	96	no	—
279	C₂²	C₇×#540	1	16	yes	direct product factor
280	C₂²	C₂²×C₇×#41	1	6	yes	—
281	C₂²	C₂×C₇×H(5,𝔽₂)	1	12	yes	—
282	C₂²	C₂⁶×C₇	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	8	no	—
285	C₄	—	2	16	no	—
286	C₄	—	2	8	no	—
287	C₄	C₂²×C₇×#41	1	8	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	4	16	no	—
290	C₂	C₂×C₇×#540	1	12	yes	direct product factor
291	C₂	C₂³×C₇×#41	1	1	yes	characteristic
292	C₂	C₂²×C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#540	C₂	direct (N × H)
2. C₂²×C₇×#127	C₂	semidirect
3. C₂²×C₇×#125	C₂	semidirect
4. C₂²×C₇×#123	C₂	semidirect
5. C₂²×C₇×#121	C₂	semidirect
6. C₂²×C₇×#121	C₂	semidirect
7. C₂²×C₄×C₇×D₄	C₂	semidirect
8. C₂²×C₇×#117	C₂	semidirect
14. C₇×#540	C₂²	direct (N × H)
15. C₂×C₇×#127	C₂²	semidirect
16. C₂×C₇×#125	C₂²	semidirect
17. C₂×C₇×#123	C₂²	semidirect
18. C₂×C₇×#121	C₂²	semidirect
19. C₂×C₇×#121	C₂²	semidirect
20. C₂×C₄×C₇×D₄	C₂²	semidirect
21. C₂×C₇×#117	C₂²	semidirect
30. C₂²×#540	C₇	direct (N × H)
48. C₇×#127	C₂³	semidirect
49. C₇×#125	C₂³	semidirect
50. C₇×#123	C₂³	semidirect
51. C₇×#121	C₂³	semidirect
52. C₇×#121	C₂³	semidirect
53. C₄×C₇×D₄	C₂³	semidirect
54. C₇×#117	C₂³	semidirect
63. C₂×#540	C₂×C₇	direct (N × H)
64. C₂²×#127	C₂×C₇	semidirect
65. C₂²×#125	C₂×C₇	semidirect
66. C₂²×#123	C₂×C₇	semidirect
67. C₂²×#121	C₂×C₇	semidirect
68. C₂²×#121	C₂×C₇	semidirect
69. C₂²×C₄×D₄	C₂×C₇	semidirect
70. C₂²×#117	C₂×C₇	semidirect
112. #540	C₂²×C₇	direct (N × H)
113. C₂×#127	C₂²×C₇	semidirect
114. C₂×#125	C₂²×C₇	semidirect
115. C₂×#123	C₂²×C₇	semidirect
116. C₂×#121	C₂²×C₇	semidirect
117. C₂×#121	C₂²×C₇	semidirect
118. C₂×C₄×D₄	C₂²×C₇	semidirect
119. C₂×#117	C₂²×C₇	semidirect
172. #127	C₂³×C₇	semidirect
173. #125	C₂³×C₇	semidirect
174. #123	C₂³×C₇	semidirect
175. #121	C₂³×C₇	semidirect
176. #121	C₂³×C₇	semidirect
177. C₄×D₄	C₂³×C₇	semidirect
178. #117	C₂³×C₇	semidirect
194. C₂²×C₇	#540	direct (N × H)
241. C₂×C₇	C₂×#540	direct (N × H)
271. C₇	C₂²×#540	direct (N × H)
279. C₂²	C₇×#540	direct (N × H)
290. C₂	C₂×C₇×#540	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×#540	2
C₂²×C₇×#127	2
C₂²×C₇×#125	2
C₂²×C₇×#123	2
C₂²×C₇×#121	2
C₂²×C₄×C₇×D₄	2
C₂²×C₇×#117	2
C₂²×#540	7

Subgroups — C22×C7×#540 #13239893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#540 #13239893