Finite Group Databasebeta

← #13256826

Subgroups — #13256826

Order 1792.

416 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 123 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#11731787C211yesmaximal, characteristic
2C7×#1956C211yesmaximal, characteristic
3#11729533C212yesmaximal
4#11725461C212yesmaximal
5#11725440C211yesmaximal, characteristic
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#103—22no—
8C2×C4×C7×C8C2211yescharacteristic
9C7×#384C2211yescharacteristic
10#89758—22no—
11#89757—22no—
12#89749—22no—
13C4×C4⋊Dic7C2212yes—
14C2×C4⋊(C7⋊C8)C2212yes—
15#89324—22no—
16#89324—22no—
17#10352—71nomaximal, Sylow
18C22×C7×Q8—21no—
19C22×C7×C8—21no—
20C7×C4⋊Q8—22no—
21C7×C4⋊Q8—21no—
22C7×C4⋊Q8Q812yes—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4—22no—
25C2×C7×C4⋊C4D412yes—
26C2×C42×C7C2311yescharacteristic
27C2×C4⋊Dic7—22no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×C4×Dic7—22no—
31C22×C7⋊C8—22no—
32C7×#103—44no—
33C4×C7×C8—22no—
34C7⋊#95—22no—
35C7⋊#95—24no—
36C7⋊#95—22no—
37C4⋊(C7⋊C8)—22no—
38C4⋊(C7⋊C8)—22no—
39#2571—71no—
40#1987—72no—
41#1956D711yescharacteristic
42#1296—71no—
43#1279—72no—
44C2×C7×Q8—22no—
45C2×C7×Q8—42no—
46C2×C7×Q8—42no—
47C2×C7×Q8—41no—
48C22×C4×C7—22no—
49C22×C4×C7—41no—
50C22×C4×C7C2×D411yescharacteristic
51C22×C4×C7C2×D412yes—
52C22×Dic7—22no—
53C22×Dic7—22no—
54C2×C7×C8—22no—
55C2×C7×C8—41no—
56C2×C7×C8—41no—
57C7×C4⋊C4—24no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—42no—
60C7×C4⋊C4—42no—
61C7×C4⋊C4—42no—
62C42×C7C2×Q811yescharacteristic
63C42×C7#4111yescharacteristic
64C42×C7#4112yes—
65C4⋊Dic7—22no—
66C4⋊Dic7—22no—
67C4⋊Dic7—22no—
68C4⋊Dic7—22no—
69C4⋊Dic7—82no—
70C4×Dic7—42no—
71C4×Dic7—42no—
72C2×C7⋊C8—42no—
73C2×C7⋊C8—42no—
74C2×C7⋊C8—42no—
75C2×C4⋊Q8C2×D711yescharacteristic
76C2×C4⋊C8—72no—
77C2×#103—22no—
78C2×C4×C8C2×D711yescharacteristic
79#400—142no—
80#389—142no—
81#384C2×D711yescharacteristic
82#382—142no—
83C4×C4⋊C4—72no—
84#340—142no—
85#340—142no—
86C23×C7#12011yescharacteristic
87C7×Q8—44no—
88C7×Q8—82no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—42no—
92C2×C4×C7—44no—
93C2×C4×C7—42no—
94C2×C4×C7—81no—
95C2×C4×C7—81no—
96C2×C4×C7C22⋊Q812yes—
97C2×C4×C7C22⋊Q811yescharacteristic
98C2×C4×C7#12412yes—
99C2×C4×C7#12411yescharacteristic
100C2×C4×C7#13512yes—
101C2×C4×C7#13512yes—
102C2×C4×C7#13712yes—
103C2×C4×C7#13712yes—
104C2×Dic7—42no—
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C7×C8—42no—
111C7⋊C8—82no—
112C22×Q8—21no—
113C22×C8—21no—
114C22×C8—142no—
115C4⋊Q8—21no—
116C4⋊Q8—22no—
117C4⋊Q8Dic1412yes—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—142no—
123C2×C4⋊C4C22⋊D712yes—
124C2×C42—142no—
125C2×C42C22×D711yescharacteristic
126C4⋊C8—142no—
127C4⋊C8—142no—
128#103—44no—
129C4×C8—22no—
130#95—142no—
131#95—144no—
132#95—142no—
133C22×C7#39311yescharacteristic
134C22×C7#45212yes—
135C22×C7#45212yes—
136C22×C7#45311yescharacteristic
137C22×C7#45411yescharacteristic
138C4×C7—21no—
139C4×C7—21no—
140C4×C7—82no—
141C4×C7—82no—
142C4×C7#47912yes—
143C4×C7#47912yes—
144C4×C7#48812yes—
145C4×C7#49112yes—
146Dic7—82no—
147Dic7—82no—
148C2×Q8—22no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—41no—
152C22×C4—22no—
153C22×C4—41no—
154C22×C4—142no—
155C22×C4—142no—
156C22×C4D4×D712yes—
157C22×C4C2×C22⋊D711yescharacteristic
158C2×C8—22no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—282no—
162C2×C8—282no—
163C2×C8—282no—
164C4⋊C4—24no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—562no—
174C42—282no—
175C42—282no—
176C42C2×Dic1411yescharacteristic
177C42#115711yescharacteristic
178C42#115912yes—
179C2×C7#163011yescharacteristic
180C2×C7#163411yescharacteristic
181C2×C7#163812yes—
182C2×C7#202612yes—
183C2×C7#202711yescharacteristic
184C23#650011yescharacteristic
185Q8—44no—
186Q8—82no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—282no—
195C2×C4—282no—
196C2×C4—282no—
197C2×C4—282no—
198C2×C4—282no—
199C2×C4—282no—
200C2×C4#644112yes—
201C2×C4#644812yes—
202C2×C4#647812yes—
203C2×C4#647912yes—
204C2×C4#648112yes—
205C2×C4#648212yes—
206C2×C4#648711yescharacteristic
207C2×C4#649911yescharacteristic
208C8—42no—
209C8—562no—
210C7#1035211yescharacteristic, complemented, Sylow
211C22#8971911yescharacteristic
212C22#8983411yescharacteristic
213C22#8984911yescharacteristic
214C22#8994512yes—
215C22#8995912yes—
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—562no—
221C4—562no—
222C4#8956812yes—
223C4#8956912yes—
224C4#8957112yes—
225C4#8957212yes—
226C2#1172739711yescharacteristic
227C2#1172798612yes—
228C2#1172898312yes—
229C2#1172902411yescharacteristic
230C2#1172905811yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1956C2
socle, prime radicalC23×C7#120
hypercenter#1956D7
hypocenterC7#10352
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
210. C7#10352semidirect

Maximal subgroups

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

subgroupindex
#117317872
C7×#19562
#117295332
#117254612
#117254402
#103527