Subgroups — C₇×#21284 #13198500

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2319	C₂	1	2	yes	maximal, complemented
3	C₇×#1972	C₂	1	2	yes	maximal
4	C₇×#1943	C₂	1	2	yes	maximal, complemented
5	C₇×#1934	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1762	C₂	1	2	yes	maximal, complemented
8	C₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₄×C₇×D₈	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	C₂²	1	2	yes	—
14	C₂×C₇×#102	C₂²	1	4	yes	—
15	C₂×C₇×#102	C₂²	1	2	yes	—
16	C₂×C₇×#97	C₂²	1	2	yes	—
17	C₂×C₇×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#390	C₂²	1	4	yes	—
21	C₇×#389	C₂²	1	4	yes	—
22	C₇×#388	—	2	4	no	—
23	C₇×#381	—	2	2	no	—
24	C₇×#380	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#376	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21284	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	4	no	—
33	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₇×D₄	—	2	4	no	—
38	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₇×#107	C₂×C₄	1	4	yes	—
49	C₇×#106	—	2	4	no	—
50	C₇×#102	—	2	8	no	—
51	C₇×#102	C₂×C₄	1	8	yes	—
52	C₇×#97	—	2	4	no	—
53	C₇×#97	—	2	2	no	—
54	C₄×C₇×C₈	—	2	4	no	—
55	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
56	C₇×#95	—	2	2	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	D₄	1	2	yes	—
59	C₇×#95	D₄	1	2	yes	—
60	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
61	#2319	C₂×C₇	1	2	yes	complemented
62	#1972	C₂×C₇	1	2	yes	—
63	#1943	C₂×C₇	1	2	yes	complemented
64	#1934	C₂×C₇	1	4	yes	complemented
65	C₄×#106	C₂×C₇	1	1	yes	characteristic, complemented
66	#1762	C₂×C₇	1	2	yes	complemented
67	#1726	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂⁴×C₇	—	2	2	no	—
69	C₂×C₇×D₄	—	2	4	no	—
70	C₂×C₇×D₄	—	2	4	no	—
71	C₂×C₇×D₄	—	2	2	no	—
72	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	8	no	—
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	—	8	4	no	—
77	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×D₈	—	2	8	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	#41	1	4	yes	—
88	C₂×C₇×C₈	#41	1	4	yes	—
89	C₇×C₄⋊C₄	—	2	2	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	8	4	no	—
93	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
94	C₇×C₂²⋊C₄	—	2	8	no	—
95	C₇×C₂²⋊C₄	—	8	4	no	—
96	C₇×C₂²⋊C₄	—	8	4	no	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	2	2	no	—
99	C₄²×C₇	—	8	2	no	—
100	C₄²×C₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₇	#41	1	2	yes	—
102	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
104	C₄×D₈	C₄×C₇	1	8	yes	complemented
105	C₂×#107	C₂²×C₇	1	1	yes	characteristic
106	C₂×#106	C₂²×C₇	1	2	yes	—
107	C₂×#102	C₂²×C₇	1	4	yes	—
108	C₂×#102	C₂²×C₇	1	2	yes	—
109	C₂×#97	C₂²×C₇	1	2	yes	—
110	C₂×#97	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
112	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
113	#390	C₂²×C₇	1	4	yes	—
114	#389	C₂²×C₇	1	4	yes	—
115	#388	—	2	4	no	—
116	#381	—	2	2	no	—
117	#380	—	2	2	no	—
118	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
119	#376	C₂²×C₇	1	1	yes	characteristic
120	#340	C₂²×C₇	1	2	yes	—
121	#340	C₂²×C₇	1	4	yes	—
122	C₂³×C₇	—	2	4	no	—
123	C₂³×C₇	—	4	4	no	—
124	C₂³×C₇	—	4	4	no	—
125	C₂³×C₇	—	4	2	no	—
126	C₂³×C₇	—	4	2	no	—
127	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
128	C₇×D₄	—	2	4	no	—
129	C₇×D₄	—	2	4	no	—
130	C₇×D₄	—	4	4	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	4	no	—
134	C₂×C₄×C₇	—	2	4	no	—
135	C₂×C₄×C₇	—	2	4	no	—
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	4	4	no	—
138	C₂×C₄×C₇	—	4	4	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	4	4	no	—
141	C₂×C₄×C₇	—	8	8	no	—
142	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₇	C₂×#41	1	2	yes	—
147	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
148	C₇×C₈	—	2	2	no	—
149	C₇×C₈	—	2	4	no	—
150	C₇×C₈	#117	1	4	yes	—
151	C₂²×D₄	C₂³×C₇	1	2	yes	—
152	C₂³×C₄	—	2	2	no	—
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
155	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₇	1	2	yes	—
157	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
158	C₄×D₄	—	2	4	no	—
159	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
162	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂³×C₇	1	2	yes	—
168	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
169	#107	C₂×C₄×C₇	1	4	yes	—
170	#106	—	2	4	no	—
171	#102	—	2	8	no	—
172	#102	C₂×C₄×C₇	1	8	yes	—
173	#97	—	2	4	no	—
174	#97	—	2	2	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
177	#95	—	2	2	no	—
178	#95	—	2	4	no	—
179	#95	C₇×D₄	1	2	yes	—
180	#95	C₇×D₄	1	2	yes	—
181	C₂²×C₇	—	4	4	no	—
182	C₂²×C₇	—	4	4	no	—
183	C₂²×C₇	—	4	4	no	—
184	C₂²×C₇	—	8	4	no	—
185	C₂²×C₇	—	8	4	no	—
186	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₇	#532	1	2	yes	—
188	C₂²×C₇	#576	1	2	yes	—
189	C₂²×C₇	#577	1	2	yes	—
190	C₄×C₇	—	2	2	no	—
191	C₄×C₇	—	2	2	no	—
192	C₄×C₇	—	4	4	no	—
193	C₄×C₇	—	4	4	no	—
194	C₄×C₇	—	8	4	no	—
195	C₄×C₇	C₂×#117	1	1	yes	characteristic
196	C₄×C₇	C₄×#41	1	1	yes	characteristic
197	C₄×C₇	#529	1	2	yes	—
198	C₂⁴	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	8	4	no	—
207	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
208	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
211	D₈	—	2	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
217	C₂×C₈	C₇×#41	1	4	yes	—
218	C₂×C₈	C₇×#41	1	4	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	8	4	no	—
223	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
224	C₂²⋊C₄	—	2	8	no	—
225	C₂²⋊C₄	—	8	4	no	—
226	C₂²⋊C₄	—	8	4	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	8	2	no	—
230	C₄²	C₂²×C₄×C₇	1	2	yes	—
231	C₄²	C₇×#41	1	2	yes	—
232	C₂×C₇	—	8	4	no	—
233	C₂×C₇	#2293	1	1	yes	characteristic
234	C₂×C₇	#2948	1	1	yes	characteristic
235	C₂×C₇	#2952	1	1	yes	characteristic
236	C₂×C₇	#3179	1	4	yes	—
237	C₂³	—	2	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
243	D₄	—	2	4	no	—
244	D₄	—	2	4	no	—
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	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	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
258	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
259	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₇×#41	1	2	yes	—
261	C₂×C₄	C₂×C₇×#41	1	2	yes	—
262	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	C₇×#117	1	4	yes	—
266	C₇	#21284	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
273	C₂²	C₇×#532	1	2	yes	—
274	C₂²	C₇×#576	1	2	yes	—
275	C₂²	C₇×#577	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂×C₇×#117	1	1	yes	characteristic
282	C₄	C₄×C₇×#41	1	1	yes	characteristic
283	C₄	C₇×#529	1	2	yes	—
284	C₂	—	8	4	no	—
285	C₂	C₇×#2293	1	1	yes	characteristic
286	C₂	C₇×#2948	1	1	yes	characteristic
287	C₂	C₇×#2952	1	1	yes	characteristic
288	C₂	C₇×#3179	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2319	C₂	semidirect
4. C₇×#1943	C₂	semidirect
5. C₇×#1934	C₂	semidirect
6. C₄×C₇×#106	C₂	semidirect
7. C₇×#1762	C₂	semidirect
8. C₇×#1726	C₂	semidirect
11. C₄×C₇×D₈	C₄	semidirect
29. #21284	C₇	direct (N × H)
61. #2319	C₂×C₇	semidirect
63. #1943	C₂×C₇	semidirect
64. #1934	C₂×C₇	semidirect
65. C₄×#106	C₂×C₇	semidirect
66. #1762	C₂×C₇	semidirect
67. #1726	C₂×C₇	semidirect
104. C₄×D₈	C₄×C₇	semidirect
266. C₇	#21284	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×C₄×C₇×D₈	2
C₇×#2319	2
C₇×#1972	2
C₇×#1943	2
C₇×#1934	2
C₄×C₇×#106	2
C₇×#1762	2
C₇×#1726	2
#21284	7

Subgroups — C7×#21284 #13198500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21284 #13198500