Subgroups — C₇×#13290 #13190506

Order 1792.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#1848	C₂	1	1	yes	maximal, characteristic
4	C₇×#1387	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₇×#500	—	2	1	no	—
8	C₇×#500	—	2	1	no	—
9	C₇×#500	C₄	1	2	yes	—
10	C₇×#500	C₄	1	2	yes	—
11	C₂×C₇×#108	—	2	1	no	—
12	C₂×C₇×#108	C₂²	1	2	yes	—
13	C₂×C₇×#107	—	2	1	no	—
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₇×#423	—	2	1	no	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#346	—	2	1	no	—
18	C₄×C₇×C₁₆	—	2	1	no	—
19	#13290	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	4	1	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₇×C₄⋊Q₈	—	2	2	no	—
25	C₇×C₄⋊Q₈	—	2	2	no	—
26	C₇×C₄⋊Q₈	—	4	1	no	—
27	C₇×C₄⋊Q₈	—	4	1	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	4	1	no	—
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₇×M₃₂	—	2	2	no	—
32	C₇×M₃₂	Q₈	1	4	yes	complemented
33	C₂×C₇×C₁₆	—	2	2	no	—
34	C₇×#108	—	2	4	no	—
35	C₇×#108	—	4	2	no	—
36	C₇×#107	—	2	2	no	—
37	C₇×#107	—	2	2	no	—
38	C₇×#107	—	2	2	no	—
39	C₇×#107	C₂×C₄	1	4	yes	—
40	C₇×C₄⋊C₈	—	4	1	no	—
41	C₇×C₄⋊C₈	—	4	1	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₂×#500	C₂×C₇	1	1	yes	characteristic
47	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
48	#1848	C₂×C₇	1	1	yes	characteristic
49	#1387	C₂×C₇	1	4	yes	—
50	C₂×C₇×Q₈	—	4	2	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	8	4	no	—
53	C₂²×C₄×C₇	—	4	2	no	—
54	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₇×M₁₆	—	4	2	no	—
58	C₇×M₁₆	—	4	2	no	—
59	C₂×C₇×C₈	—	4	2	no	—
60	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	SD₁₆	1	2	yes	—
63	C₂×C₇×C₈	Q₁₆	1	2	yes	—
64	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₇×C₈	#41	1	2	yes	—
68	C₇×C₄⋊C₄	—	2	4	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇	D₈	1	1	yes	characteristic
75	C₄²×C₇	D₈	1	1	yes	characteristic
76	C₇×C₁₆	—	2	4	no	—
77	C₂×C₄⋊Q₈	—	2	1	no	—
78	C₂×M₃₂	C₂²×C₇	1	2	yes	—
79	#500	—	2	1	no	—
80	#500	—	2	1	no	—
81	#500	C₄×C₇	1	2	yes	—
82	#500	C₄×C₇	1	2	yes	—
83	C₂×#108	—	2	1	no	—
84	C₂×#108	C₂²×C₇	1	2	yes	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	C₂²×C₇	1	2	yes	—
87	#423	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
89	#346	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₇×Q₈	—	8	4	no	—
93	C₇×Q₈	—	8	4	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	8	4	no	—
98	C₂×C₄×C₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₇	#102	1	1	yes	characteristic
100	C₂×C₄×C₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₇×C₈	—	8	2	no	—
109	C₇×C₈	#103	1	2	yes	—
110	C₇×C₈	#103	1	2	yes	—
111	C₇×C₈	C₄×Q₈	1	2	yes	—
112	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
113	C₂²×Q₈	—	4	1	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	4	1	no	—
120	C₄⋊Q₈	—	4	1	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
124	M₃₂	—	2	2	no	—
125	M₃₂	C₇×Q₈	1	4	yes	complemented
126	C₂×C₁₆	—	2	2	no	—
127	#108	—	2	4	no	—
128	#108	—	4	2	no	—
129	#107	—	2	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	C₂×C₄×C₇	1	4	yes	—
133	C₄⋊C₈	—	4	1	no	—
134	C₄⋊C₈	—	4	1	no	—
135	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
137	C₄×C₈	C₇×D₄	1	1	yes	characteristic
138	C₄×C₈	C₇×D₄	1	1	yes	characteristic
139	C₂²×C₇	—	2	1	no	—
140	C₂²×C₇	—	2	1	no	—
141	C₂²×C₇	C₂×#102	1	1	yes	characteristic
142	C₂²×C₇	#487	1	1	yes	characteristic
143	C₂²×C₇	#498	1	1	yes	characteristic
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	—	2	1	no	—
146	C₄×C₇	—	8	4	no	—
147	C₄×C₇	#361	1	2	yes	—
148	C₄×C₇	#362	1	2	yes	—
149	C₄×C₇	#391	1	1	yes	characteristic
150	C₄×C₇	C₂×#103	1	1	yes	characteristic
151	C₄×C₇	#486	1	1	yes	characteristic
152	C₄×C₇	#500	1	1	yes	characteristic
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	8	4	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
160	M₁₆	—	4	2	no	—
161	M₁₆	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₇×SD₁₆	1	2	yes	—
166	C₂×C₈	C₇×Q₁₆	1	2	yes	—
167	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
168	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
170	C₂×C₈	C₇×#41	1	2	yes	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
176	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
177	C₄²	C₇×D₈	1	1	yes	characteristic
178	C₄²	C₇×D₈	1	1	yes	characteristic
179	C₁₆	—	2	4	no	—
180	C₂×C₇	—	2	1	no	—
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	#1990	1	1	yes	characteristic
183	C₂×C₇	C₂×#361	1	1	yes	characteristic
184	C₂×C₇	C₂×#362	1	1	yes	characteristic
185	C₂³	C₂×C₇×SD₁₆	1	1	yes	characteristic
186	Q₈	—	8	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	C₇×#102	1	1	yes	characteristic
193	C₂×C₄	C₇×#102	1	1	yes	characteristic
194	C₂×C₄	C₇×#102	1	1	yes	characteristic
195	C₂×C₄	C₇×#102	1	1	yes	characteristic
196	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₇×#124	1	1	yes	characteristic
198	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
199	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
201	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
202	C₈	—	8	2	no	—
203	C₈	C₇×#103	1	2	yes	—
204	C₈	C₇×#103	1	2	yes	—
205	C₈	C₄×C₇×Q₈	1	2	yes	—
206	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
207	C₇	#13290	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×C₇×#102	1	1	yes	characteristic
211	C₂²	C₇×#487	1	1	yes	characteristic
212	C₂²	C₇×#498	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	C₇×#361	1	2	yes	—
217	C₄	C₇×#362	1	2	yes	—
218	C₄	C₇×#391	1	1	yes	characteristic
219	C₄	C₂×C₇×#103	1	1	yes	characteristic
220	C₄	C₇×#486	1	1	yes	characteristic
221	C₄	C₇×#500	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	C₇×#1990	1	1	yes	characteristic
225	C₂	C₂×C₇×#361	1	1	yes	characteristic
226	C₂	C₂×C₇×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
19. #13290	C₇	direct (N × H)
32. C₇×M₃₂	Q₈	semidirect
125. M₃₂	C₇×Q₈	semidirect
207. C₇	#13290	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×#500	2
C₄×C₇×M₃₂	2
C₇×#1848	2
C₇×#1387	2
#13290	7

Subgroups — C7×#13290 #13190506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13290 #13190506