Finite Group Databasebeta

← C7×#12260 #13189476

Subgroups — C7×#12260 #13189476

Order 1792.

342 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2844C211yesmaximal, characteristic
2C7×#2061C211yesmaximal, characteristic
3C7×#1605C212yesmaximal
4C7×#1354C212yesmaximal
5C7×#1324C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C7×#451—22no—
8C7×#428—21no—
9C7×#400—21no—
10C7×#400—22no—
11C7×#395—22no—
12C7×#395C2212yes—
13C7×#382—21no—
14C7×#382C2211yescharacteristic
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12260C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8D412yes—
21C2×C7×Q16—42no—
22C2×C7×M16—21no—
23C7×C4⋊Q8—41no—
24C7×#125—41no—
25C7×C22⋊Q8—22no—
26C7×#117—21no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4C2311yescharacteristic
33C2×C42×C7D411yescharacteristic
34C7×#107—41no—
35C7×#106—41no—
36C7×#103—42no—
37C7×C22⋊C8—22no—
38C7×#95—22no—
39C7×#95—22no—
40C7×#95—21no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95D412yes—
44#2844C2×C711yescharacteristic
45#2061C2×C711yescharacteristic
46#1605C2×C712yes—
47#1354C2×C712yes—
48#1324C2×C711yescharacteristic
49C2×C7×Q8—22no—
50C2×C7×Q8—42no—
51C2×C7×Q8—82no—
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D412yes—
58C7×Q16—84no—
59C7×M16—22no—
60C7×M16—41no—
61C2×C7×C8—41no—
62C2×C7×C8—42no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—21no—
65C7×C4⋊C4—21no—
66C7×C4⋊C4—41no—
67C7×C4⋊C4—41no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—81no—
70C7×C4⋊C4—81no—
71C7×C4⋊C4—82no—
72C7×C4⋊C4—81no—
73C7×C22⋊C4—41no—
74C42×C7—21no—
75C42×C7—41no—
76C42×C7Q1612yescomplemented
77#544C22×C711yescharacteristic
78#451—22no—
79#428—21no—
80#400—21no—
81#400—22no—
82#395—22no—
83#395C22×C712yes—
84#382—21no—
85#382C22×C711yescharacteristic
86#343C22×C711yescharacteristic
87#339—21no—
88#328—22no—
89#324C22×C712yes—
90C23×C7#12011yescharacteristic
91C7×Q8—44no—
92C7×Q8—82no—
93C7×Q8—82no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—41no—
99C2×C4×C7—41no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—41no—
110C2×C4×C7—42no—
111C2×C4×C7C2×Q1611yescharacteristic
112C2×C4×C7#13711yescharacteristic
113C7×C8—42no—
114C7×C8—82no—
115C22×Q8C7×D412yes—
116C2×Q16—42no—
117C2×M16—21no—
118C4⋊Q8—41no—
119#125—41no—
120C22⋊Q8—22no—
121#117—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C7×D411yescharacteristic
127C2×C4⋊C4C23×C711yescharacteristic
128C2×C42C7×D411yescharacteristic
129#107—41no—
130#106—41no—
131#103—42no—
132C22⋊C8—22no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—22no—
137#95—22no—
138#95C7×D412yes—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7#45111yescharacteristic
142C22×C7#45611yescharacteristic
143C22×C7B(4,𝔽2)11yescharacteristic
144C4×C7—21no—
145C4×C7—21no—
146C4×C7—42no—
147C4×C7—42no—
148C4×C7—82no—
149C4×C7—82no—
150C4×C7—81no—
151C4×C7—81no—
152C4×C7—81no—
153C2×Q8—22no—
154C2×Q8—42no—
155C2×Q8—82no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4C2×C7×D411yescharacteristic
161C22×C4C2×C7×D412yes—
162Q16—84no—
163M16—22no—
164M16—41no—
165C2×C8—41no—
166C2×C8—42no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—82no—
176C4⋊C4—81no—
177C22⋊C4—41no—
178C42—21no—
179C42—41no—
180C42C7×Q1612yescomplemented
181C2×C7—21no—
182C2×C7—21no—
183C2×C7#160511yescharacteristic
184C2×C7#219911yescharacteristic
185C2×C7#220311yescharacteristic
186C23C7×#12011yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4C2×C7×Q1611yescharacteristic
208C2×C4C7×#13711yescharacteristic
209C8—42no—
210C8—82no—
211C7#1226011yescharacteristic, direct product factor, Sylow
212C22—21no—
213C22—21no—
214C22C7×#45111yescharacteristic
215C22C7×#45611yescharacteristic
216C22C7×B(4,𝔽2)11yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—81no—
226C2—21no—
227C2—21no—
228C2C7×#160511yescharacteristic
229C2C7×#219911yescharacteristic
230C2C7×#220311yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12260C7direct (N × H)
76. C42×C7Q16semidirect
180. C42C7×Q16semidirect
211. C7#12260direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28442
C7×#20612
C7×#16052
C7×#13542
C7×#13242
#122607