Finite Group Databasebeta

← #13273486

Subgroups — #13273486

Order 1792.

384 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11734047C211yesmaximal, characteristic
2C7×#1857C211yesmaximal, characteristic
3#11729412C212yesmaximal
4#11725505C212yesmaximal
5#11725450C211yesmaximal, characteristic
6C7×#519C2211yescharacteristic
7C7×#428—21no—
8C2×C7×#101C2211yescharacteristic
9C7×#391C2211yescharacteristic
10#90002C2212yes—
11C2×#6410C2212yes—
12#89767—22no—
13#89758—21no—
14#89700—21no—
15#89433—22no—
16#89427—22no—
17C7×#339—21no—
18#89327—22no—
19#13465—71nomaximal, Sylow
20C22×C7×Q8C2311yescharacteristic
21C2×C7×M16—21no—
22C4×C7×Q8—41no—
23C4×C7×Q8—41no—
24C7×#117—21no—
25C7×#117—22no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C42×C7D411yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7D412yes—
32C2×Dic7⋊C4—22no—
33C2×C4×Dic7D412yes—
34C2×#1136—22no—
35C7×#107—41no—
36C7×#106—41no—
37C7×#101—21no—
38C7×#101Q812yes—
39C7×#95—22no—
40#6410—22no—
41#6410Q814yes—
42C7⋊#95—21no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46#6382—42no—
47#6381—42no—
48#2809—71no—
49#1857—72no—
50#1857D711yescharacteristic
51#1394—72no—
52#1394—71no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8C2×Q811yescharacteristic
56C2×C7×Q8C2×Q812yes—
57C2×C7×Q8#4111yescharacteristic
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7C2×D412yes—
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—22no—
63C22×Dic7—22no—
64C7×M16—22no—
65C7×M16—41no—
66C2×C7×C8—41no—
67C7×C4⋊C4—21no—
68C7×C4⋊C4—21no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—41no—
72C7×C22⋊C4—41no—
73C42×C7—41no—
74C42×C7—42no—
75C42×C7—41no—
76C4⋊Dic7—42no—
77C4⋊Dic7—42no—
78C4⋊Dic7—81no—
79C4⋊Dic7—81no—
80Dic7⋊C4—44no—
81C4×Dic7—42no—
82#1136—24no—
83#1136—42no—
84C2×C7⋊C8—42no—
85#519C2×D711yescharacteristic
86#428—21no—
87#428—142no—
88C2×#101—72no—
89C2×#101C2×D711yescharacteristic
90#400—141no—
91#398—142no—
92#391—72no—
93#391C2×D711yescharacteristic
94#389—141no—
95#387—142no—
96#339—21no—
97#339—142no—
98C23×C7#12011yescharacteristic
99C7×Q8—41no—
100C7×Q8—42no—
101C7×Q8—41no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—41no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—41no—
113C2×C4×C7C22⋊Q812yes—
114C2×C4×C7C22⋊Q811yescharacteristic
115C2×C4×C7C4⋊Q812yes—
116C2×C4×C7C4⋊Q811yescharacteristic
117C2×Dic7—42no—
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—42no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123C7×C8—42no—
124C7⋊C8—44no—
125C22×Q8C22×D711yescharacteristic
126C2×M16—21no—
127C2×M16—142no—
128C4×Q8—41no—
129C4×Q8—41no—
130#117—22no—
131#117—21no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—142no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4—141no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4C22⋊D711yescharacteristic
138C2×C42—72no—
139C2×C42C22⋊D711yescharacteristic
140#107—41no—
141#107—282no—
142#106—41no—
143#106—282no—
144#101—21no—
145#101—74no—
146#101—142no—
147#101Dic1412yes—
148#95—22no—
149#95—141no—
150#95—142no—
151#95—142no—
152#95—142no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#39511yescharacteristic
156C22×C7B(4,𝔽2)11yescharacteristic
157C22×C7#45811yescharacteristic
158C4×C7—22no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—42no—
163C4×C7—81no—
164C4×C7—81no—
165C4×C7—81no—
166Dic7—82no—
167Dic7—82no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8C2×Dic1411yescharacteristic
171C2×Q8#115711yescharacteristic
172C2×Q8Q8×D712yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—142no—
176C22×C4—142no—
177C22×C4D4×D712yes—
178C22×C4C2×C22⋊D711yescharacteristic
179M16—22no—
180M16—41no—
181M16—144no—
182M16—282no—
183C2×C8—41no—
184C2×C8—282no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—282no—
191C4⋊C4—284no—
192C4⋊C4—282no—
193C4⋊C4—561no—
194C4⋊C4—561no—
195C22⋊C4—41no—
196C42—41no—
197C42—41no—
198C42—42no—
199C42—282no—
200C2×C7—21no—
201C2×C7—21no—
202C2×C7#206211yescharacteristic
203C2×C7#220311yescharacteristic
204C2×C7#220411yescharacteristic
205C23#650011yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—42no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—282no—
221C2×C4—282no—
222C2×C4—282no—
223C2×C4—282no—
224C2×C4—282no—
225C2×C4—282no—
226C2×C4#645112yes—
227C2×C4#645912yes—
228C2×C4#648711yescharacteristic
229C2×C4#650711yescharacteristic
230C8—42no—
231C8—284no—
232C7#1346511yescharacteristic, complemented, Sylow
233C22—21no—
234C22—21no—
235C22#8975111yescharacteristic
236C22#8980411yescharacteristic
237C22#8982511yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4—562no—
247C4—562no—
248C2—21no—
249C2—21no—
250C2#1172734811yescharacteristic
251C2#1172874311yescharacteristic
252C2#1172874811yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC22×C7×Q8C23
FrattiniC22×Q8C22×D7
FittingC7×#1857C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#1857D7
hypocenterC7#13465
LCSG ▷ C22×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
232. C7#13465semidirect

Maximal subgroups

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

subgroupindex
#117340472
C7×#18572
#117294122
#117255052
#117254502
#134657