Finite Group Databasebeta

← C7×#12292 #13189508

Subgroups — C7×#12292 #13189508

Order 1792.

310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3249C211yesmaximal, characteristic
2C7×#2236C212yesmaximal
3C7×#2087C211yesmaximal, characteristic
4C7×#1348C212yesmaximal
5C7×#1322C211yesmaximal, characteristic
6C7×#533—21no—
7C7×#477—22no—
8C7×#467—21no—
9C7×#467C2212yes—
10C7×#462—22no—
11C2×C7×#107C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C7×#389—21no—
14C7×#366—22no—
15C7×#358—22no—
16C7×C22⋊C16C2212yes—
17C7×#340—21no—
18C7×#340C2211yescharacteristic
19#12292C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×Q16—21no—
21C2×C7×Q16D412yes—
22C22×C7×C8D411yescharacteristic
23C22×C7×C8C2311yescharacteristic
24C7×C4⋊Q8—21no—
25C7×C4⋊Q8—21no—
26C7×#125—41no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—41no—
29C4×C7×Q8—22no—
30C4×C7×Q8—41no—
31C7×#117—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—41no—
35C2×C42×C7D411yescharacteristic
36C2×C7×C16—22no—
37C7×#107—21no—
38C7×#107—22no—
39C7×#107D412yes—
40C7×C4⋊C8—22no—
41C7×#103—22no—
42C7×#103—22no—
43C7×#95—41no—
44#3249C2×C711yescharacteristic
45#2236C2×C712yes—
46#2087C2×C711yescharacteristic
47#1348C2×C712yes—
48#1322C2×C711yescharacteristic
49C2×C7×Q8—22no—
50C2×C7×Q8—41no—
51C22×C4×C7—41no—
52C22×C4×C7—41no—
53C22×C4×C7D811yescharacteristic
54C22×C4×C7D811yescharacteristic
55C22×C4×C7C2×D411yescharacteristic
56C7×Q16—42no—
57C7×Q16—81no—
58C2×C7×C8—21no—
59C2×C7×C8—41no—
60C2×C7×C8—41no—
61C2×C7×C8C2×D412yes—
62C7×C4⋊C4—22no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—41no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—82no—
73C7×C22⋊C4—42no—
74C42×C7—21no—
75C42×C7—21no—
76C42×C7—42no—
77C7×C16—42no—
78#533—21no—
79#477—22no—
80#467—21no—
81#467C22×C712yes—
82#462—22no—
83C2×#107C22×C711yescharacteristic
84C2×C4⋊C8C22×C711yescharacteristic
85#389—21no—
86#366—22no—
87#358—22no—
88C22⋊C16C22×C712yes—
89#340—21no—
90#340C22×C711yescharacteristic
91C23×C7C2×D811yescharacteristic
92C7×Q8—42no—
93C7×Q8—42no—
94C7×Q8—82no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—41no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—81no—
105C2×C4×C7—81no—
106C2×C4×C7—81no—
107C2×C4×C7—81no—
108C2×C4×C7—81no—
109C2×C4×C7Q3212yes—
110C2×C4×C7#12011yescharacteristic
111C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
112C7×C8—22no—
113C7×C8—41no—
114C2×Q16—21no—
115C2×Q16C7×D412yes—
116C22×C8C7×D411yescharacteristic
117C22×C8C23×C711yescharacteristic
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C4×Q8—22no—
124C4×Q8—41no—
125#117—41no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C42C7×D411yescharacteristic
130C2×C16—22no—
131#107—21no—
132#107—22no—
133#107C7×D412yes—
134C4⋊C8—22no—
135#103—22no—
136#103—22no—
137#95—41no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7#44711yescharacteristic
141C22×C7C2×Q3211yescharacteristic
142C22×C7#51011yescharacteristic
143C4×C7—21no—
144C4×C7—22no—
145C4×C7—41no—
146C4×C7—82no—
147C4×C7—82no—
148C4×C7—82no—
149C4×C7—161no—
150C4×C7#45611yescharacteristic
151C4×C7#45811yescharacteristic
152C2×Q8—22no—
153C2×Q8—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4C7×D811yescharacteristic
157C22×C4C7×D811yescharacteristic
158C22×C4C2×C7×D411yescharacteristic
159Q16—42no—
160Q16—81no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8C2×C7×D412yes—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C22⋊C4—42no—
177C42—21no—
178C42—21no—
179C42—42no—
180C16—42no—
181C2×C7—21no—
182C2×C7—21no—
183C2×C7#161011yescharacteristic
184C2×C7#218811yescharacteristic
185C2×C7#219011yescharacteristic
186C23C2×C7×D811yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4C7×Q3212yes—
205C2×C4C7×#12011yescharacteristic
206C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
207C8—22no—
208C8—41no—
209C7#1229211yescharacteristic, direct product factor, Sylow
210C22—21no—
211C22—21no—
212C22C7×#44711yescharacteristic
213C22C2×C7×Q3211yescharacteristic
214C22C7×#51011yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—161no—
222C4C7×#45611yescharacteristic
223C4C7×#45811yescharacteristic
224C2—21no—
225C2—21no—
226C2C7×#161011yescharacteristic
227C2C7×#218811yescharacteristic
228C2C7×#219011yescharacteristic
subgroupG/H
center, socleC22×C7#447
commutator, Frattini, cosocleC22×C8C23×C7
prime radicalC23×C7C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12292C7direct (N × H)
209. C7#12292direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32492
C7×#22362
C7×#20872
C7×#13482
C7×#13222
#122927