Subgroups — C₇×#8928 #13186144

Order 1792.

476 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2369	C₂	1	1	yes	maximal, characteristic
2	C₇×#1956	C₂	1	1	yes	maximal, characteristic
3	C₇×#1948	C₂	1	2	yes	maximal
4	C₂×C₇×#333	C₂	1	1	yes	maximal, characteristic
5	C₇×#1279	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×#106	—	2	2	no	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₇×#103	—	2	2	no	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×#391	—	2	2	no	—
12	C₇×#389	—	2	2	no	—
13	C₇×#384	C₂²	1	1	yes	characteristic
14	C₄×C₇×C₄⋊C₄	—	2	2	no	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₇×#340	—	2	2	no	—
17	C₇×#333	C₄	1	4	yes	complemented
18	#8928	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×Q₈	D₄	1	2	yes	—
20	C₂²×C₇×C₈	—	2	2	no	—
21	C₂²×C₇×C₈	—	2	1	no	—
22	C₇×C₄⋊Q₈	—	2	2	no	—
23	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₇×C₄⋊Q₈	Q₈	1	2	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	1	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
30	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
31	C₂×C₄²×C₇	—	2	2	no	—
32	C₂×C₄²×C₇	—	2	2	no	—
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₇×#106	—	4	4	no	—
35	C₇×C₄⋊C₈	—	2	2	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
37	C₇×#103	—	4	4	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₇×#95	—	2	4	no	—
40	C₇×#95	—	2	2	no	—
41	#2369	C₂×C₇	1	1	yes	characteristic
42	#1956	C₂×C₇	1	1	yes	characteristic
43	#1948	C₂×C₇	1	2	yes	—
44	C₂×#333	C₂×C₇	1	1	yes	characteristic
45	#1279	C₂×C₇	1	2	yes	—
46	C₂×C₇×Q₈	—	2	2	no	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂×C₇×Q₈	—	4	2	no	—
49	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	4	1	no	—
54	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	4	no	—
58	C₂×C₇×C₈	—	4	1	no	—
59	C₂×C₇×C₈	—	4	1	no	—
60	C₂×C₇×C₈	—	4	2	no	—
61	C₂×C₇×C₈	—	4	2	no	—
62	C₇×C₄⋊C₄	—	2	4	no	—
63	C₇×C₄⋊C₄	—	2	4	no	—
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₄⋊C₄	—	2	2	no	—
67	C₇×C₄⋊C₄	—	4	2	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	8	2	no	—
71	C₇×C₄⋊C₄	SD₁₆	1	4	yes	—
72	C₄²×C₇	—	4	2	no	—
73	C₄²×C₇	—	4	2	no	—
74	C₄²×C₇	—	8	2	no	—
75	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₇	#41	1	2	yes	—
78	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
79	C₂×#106	—	2	2	no	—
80	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
81	C₂×#103	—	2	2	no	—
82	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
83	#391	—	2	2	no	—
84	#389	—	2	2	no	—
85	#384	C₂²×C₇	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	2	2	no	—
87	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
88	#340	—	2	2	no	—
89	#333	C₄×C₇	1	4	yes	complemented
90	C₂³×C₇	#120	1	1	yes	characteristic
91	C₇×Q₈	—	4	4	no	—
92	C₇×Q₈	—	8	2	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	4	2	no	—
98	C₂×C₄×C₇	—	4	2	no	—
99	C₂×C₄×C₇	—	4	2	no	—
100	C₂×C₄×C₇	—	4	2	no	—
101	C₂×C₄×C₇	—	4	2	no	—
102	C₂×C₄×C₇	—	4	2	no	—
103	C₂×C₄×C₇	—	4	2	no	—
104	C₂×C₄×C₇	—	8	2	no	—
105	C₂×C₄×C₇	—	8	1	no	—
106	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
111	C₂×C₄×C₇	#135	1	2	yes	—
112	C₂×C₄×C₇	#137	1	2	yes	—
113	C₂×C₄×C₇	#137	1	2	yes	—
114	C₇×C₈	—	4	2	no	—
115	C₇×C₈	—	4	4	no	—
116	C₂²×Q₈	C₇×D₄	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
121	C₄⋊Q₈	C₇×Q₈	1	2	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
127	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
131	#106	—	4	4	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
134	#103	—	4	4	no	—
135	C₄×C₈	—	2	2	no	—
136	#95	—	2	4	no	—
137	#95	—	2	2	no	—
138	C₂²×C₇	#385	1	1	yes	characteristic
139	C₂²×C₇	#448	1	2	yes	—
140	C₂²×C₇	#452	1	2	yes	—
141	C₂²×C₇	#453	1	1	yes	characteristic
142	C₂²×C₇	#455	1	1	yes	characteristic
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	—	4	4	no	—
146	C₄×C₇	—	4	4	no	—
147	C₄×C₇	—	8	2	no	—
148	C₄×C₇	—	16	1	no	—
149	C₄×C₇	C₄×SD₁₆	1	2	yes	—
150	C₄×C₇	#441	1	2	yes	—
151	C₄×C₇	#475	1	2	yes	—
152	C₄×C₇	#479	1	2	yes	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	8	2	no	—
178	C₄⋊C₄	C₇×SD₁₆	1	4	yes	—
179	C₄²	—	4	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	8	2	no	—
182	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
183	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
184	C₄²	C₇×#41	1	2	yes	—
185	C₂×C₇	#1623	1	1	yes	characteristic
186	C₂×C₇	#1634	1	1	yes	characteristic
187	C₂×C₇	#1643	1	2	yes	—
188	C₂×C₇	#1856	1	1	yes	characteristic
189	C₂×C₇	#1865	1	2	yes	—
190	C₂³	C₇×#120	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	1	no	—
206	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
208	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
209	C₂×C₄	C₇×#123	1	1	yes	characteristic
210	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
211	C₂×C₄	C₇×#135	1	2	yes	—
212	C₂×C₄	C₇×#137	1	2	yes	—
213	C₂×C₄	C₇×#137	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	4	4	no	—
216	C₇	#8928	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	C₇×#385	1	1	yes	characteristic
218	C₂²	C₇×#448	1	2	yes	—
219	C₂²	C₇×#452	1	2	yes	—
220	C₂²	C₇×#453	1	1	yes	characteristic
221	C₂²	C₇×#455	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	4	4	no	—
226	C₄	—	8	2	no	—
227	C₄	—	16	1	no	—
228	C₄	C₄×C₇×SD₁₆	1	2	yes	—
229	C₄	C₇×#441	1	2	yes	—
230	C₄	C₇×#475	1	2	yes	—
231	C₄	C₇×#479	1	2	yes	—
232	C₂	C₇×#1623	1	1	yes	characteristic
233	C₂	C₇×#1634	1	1	yes	characteristic
234	C₂	C₇×#1643	1	2	yes	—
235	C₂	C₇×#1856	1	1	yes	characteristic
236	C₂	C₇×#1865	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₇×#333	C₄	semidirect
18. #8928	C₇	direct (N × H)
89. #333	C₄×C₇	semidirect
216. C₇	#8928	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2369	2
C₇×#1956	2
C₇×#1948	2
C₂×C₇×#333	2
C₇×#1279	2
#8928	7

Subgroups — C7×#8928 #13186144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8928 #13186144