Subgroups — C₇×#7898 #13185114

Order 1792.

710 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₇×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#432	—	2	2	no	—
8	C₇×#432	C₄	1	4	yes	complemented
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
11	C₂×C₇×#97	—	2	1	no	—
12	C₂×C₄×C₇×C₈	—	2	1	no	—
13	C₂×C₄×C₇×C₈	—	2	2	no	—
14	C₂×C₄×C₇×C₈	C₄	1	4	yes	complemented
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#383	—	2	1	no	—
17	C₇×#382	C₄	1	2	yes	—
18	C₇×#381	—	2	1	no	—
19	C₇×#380	C₄	1	2	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
22	C₇×#376	—	2	1	no	—
23	#7898	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	—	2	8	no	—
28	C₂²×C₇×C₈	—	2	4	no	—
29	C₂²×C₇×C₈	—	2	2	no	—
30	C₂²×C₇×C₈	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₇	—	2	2	no	—
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	2	yes	—
40	C₂×C₄²×C₇	Q₈	1	2	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₇×C₄⋊C₈	—	2	2	no	—
43	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₇×C₂²⋊C₈	—	2	4	no	—
45	C₇×#97	—	2	4	no	—
46	C₄×C₇×C₈	—	2	4	no	—
47	C₄×C₇×C₈	—	2	2	no	—
48	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
49	C₇×#95	—	2	2	no	—
50	C₇×#95	—	2	2	no	—
51	C₇×#95	—	2	2	no	—
52	C₇×#95	—	2	2	no	—
53	C₂×#432	C₂×C₇	1	1	yes	characteristic
54	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
55	#2283	C₂×C₇	1	1	yes	characteristic
56	#1279	C₂×C₇	1	4	yes	complemented
57	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	2	4	no	—
60	C₂²×C₄×C₇	—	2	4	no	—
61	C₂²×C₄×C₇	—	2	4	no	—
62	C₂²×C₄×C₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₇×C₈	—	2	16	no	—
75	C₂×C₇×C₈	—	2	8	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	—	2	4	no	—
78	C₂×C₇×C₈	—	4	4	no	—
79	C₇×C₄⋊C₄	—	2	8	no	—
80	C₇×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₇	—	4	4	no	—
82	C₄²×C₇	C₄²	1	2	yes	—
83	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
92	C₂³×C₈	—	2	1	no	—
93	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
94	#432	—	2	2	no	—
95	#432	C₄×C₇	1	4	yes	complemented
96	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
97	C₂×C₂²⋊C₈	—	2	1	no	—
98	C₂×#97	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂×C₄×C₈	—	2	2	no	—
101	C₂×C₄×C₈	C₄×C₇	1	4	yes	complemented
102	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
103	#383	—	2	1	no	—
104	#382	C₄×C₇	1	2	yes	—
105	#381	—	2	1	no	—
106	#380	C₄×C₇	1	2	yes	—
107	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
108	C₄×C₂²⋊C₄	—	2	1	no	—
109	#376	—	2	1	no	—
110	C₂³×C₇	—	2	2	no	—
111	C₂³×C₇	—	2	2	no	—
112	C₂³×C₇	#95	1	2	yes	—
113	C₂³×C₇	#104	1	4	yes	—
114	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	—	2	4	no	—
116	C₂×C₄×C₇	—	2	4	no	—
117	C₂×C₄×C₇	—	2	2	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	4	no	—
123	C₂×C₄×C₇	—	2	4	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	4	4	no	—
126	C₂×C₄×C₇	—	4	4	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	#95	1	2	yes	—
129	C₂×C₄×C₇	#95	1	2	yes	—
130	C₂×C₄×C₇	#95	1	2	yes	—
131	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇	#131	1	4	yes	—
138	C₂×C₄×C₇	#135	1	4	yes	—
139	C₇×C₈	—	2	8	no	—
140	C₇×C₈	—	4	4	no	—
141	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
142	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
143	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
144	C₂²×C₈	—	2	8	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
149	C₂×C₂²⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
154	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
155	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
156	C₂×C₄²	C₇×D₄	1	2	yes	—
157	C₂×C₄²	C₇×Q₈	1	2	yes	—
158	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
161	C₂²⋊C₈	—	2	4	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	8	no	—
173	C₂²×C₇	#339	1	4	yes	—
174	C₂²×C₇	C₂×#95	1	1	yes	characteristic
175	C₂²×C₇	#408	1	2	yes	—
176	C₂²×C₇	#416	1	2	yes	—
177	C₂²×C₇	C₂×#104	1	2	yes	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	8	4	no	—
184	C₄×C₇	#405	1	2	yes	—
185	C₄×C₇	#416	1	2	yes	—
186	C₄×C₇	#424	1	2	yes	—
187	C₄×C₇	#427	1	2	yes	—
188	C₂⁴	C₇×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	C₄²×C₇	1	2	yes	—
194	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
198	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
199	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
200	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
201	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
205	C₂×C₈	—	2	16	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	4	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₄²	C₄²×C₇	1	2	yes	—
214	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
215	C₄²	C₇×C₂²⋊C₄	1	2	yes	—
216	C₄²	C₇×C₄⋊C₄	1	2	yes	—
217	C₄²	C₇×C₄⋊C₄	1	2	yes	—
218	C₄²	C₇×C₄⋊C₄	1	2	yes	—
219	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
220	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
221	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
222	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
223	C₂×C₇	—	2	4	no	—
224	C₂×C₇	#1494	1	4	yes	—
225	C₂×C₇	#1729	1	1	yes	characteristic
226	C₂×C₇	C₂×#339	1	1	yes	characteristic
227	C₂×C₇	#1735	1	1	yes	characteristic
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₇×#95	1	2	yes	—
231	C₂³	C₇×#104	1	4	yes	—
232	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₇×#95	1	2	yes	—
247	C₂×C₄	C₇×#95	1	2	yes	—
248	C₂×C₄	C₇×#95	1	2	yes	—
249	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₇×#131	1	4	yes	—
256	C₂×C₄	C₇×#135	1	4	yes	—
257	C₈	—	2	8	no	—
258	C₈	—	4	4	no	—
259	C₇	#7898	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	8	no	—
263	C₂²	C₇×#339	1	4	yes	—
264	C₂²	C₂×C₇×#95	1	1	yes	characteristic
265	C₂²	C₇×#408	1	2	yes	—
266	C₂²	C₇×#416	1	2	yes	—
267	C₂²	C₂×C₇×#104	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₇×#405	1	2	yes	—
275	C₄	C₇×#416	1	2	yes	—
276	C₄	C₇×#424	1	2	yes	—
277	C₄	C₇×#427	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	C₇×#1494	1	4	yes	—
280	C₂	C₇×#1729	1	1	yes	characteristic
281	C₂	C₂×C₇×#339	1	1	yes	characteristic
282	C₂	C₇×#1735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂³×C₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇	C₂²⋊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×#1279	C₂	semidirect
8. C₇×#432	C₄	semidirect
14. C₂×C₄×C₇×C₈	C₄	semidirect
23. #7898	C₇	direct (N × H)
56. #1279	C₂×C₇	semidirect
95. #432	C₄×C₇	semidirect
101. C₂×C₄×C₈	C₄×C₇	semidirect
259. C₇	#7898	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×#432	2
C₂²×C₄×C₇×C₈	2
C₇×#2283	2
C₇×#1279	2
#7898	7

Subgroups — C7×#7898 #13185114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7898 #13185114