Finite Group Databasebeta

← #13243106

Subgroups — #13243106

Order 1792.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2809C211yesmaximal, characteristic
2#11727438C211yesmaximal, characteristic
3#11727366C212yesmaximal
4#11725448C211yesmaximal, characteristic
5#11725444C212yesmaximal
6C7×#400—21no—
7C7×#398—22no—
8C7×#391—21no—
9C7×#391C2211yescharacteristic
10C7×#389C2212yes—
11C7×#387—22no—
12C2×#6409—21no—
13C2×#6381—21no—
14#89707C2211yescharacteristic
15#89324—22no—
16#89324—22no—
17#89319C2212yes—
18#89319C2211yescharacteristic
19#7831—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4D412yes—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C4⋊C4D412yes—
27C2×C42×C7—21no—
28C2×C42×C7C2311yescharacteristic
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—22no—
31C22×C7⋊C8—21no—
32C22×C7⋊C8—22no—
33C7×#95—21no—
34C7×#95—22no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38#6409—42no—
39#6381—42no—
40#2809D711yescharacteristic
41#2100—72no—
42#2095—71no—
43#1272—71no—
44#1271—72no—
45C2×C7×Q8—21no—
46C2×C7×Q8—41no—
47C2×C7×Q8—41no—
48C2×C7×Q8Q1612yes—
49C22×C4×C7—21no—
50C22×C4×C7—21no—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D412yes—
55C22×Dic7—41no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4—41no—
60C7×C4⋊C4—42no—
61C7×C4⋊C4—42no—
62C7×C4⋊C4—81no—
63C7×C4⋊C4—81no—
64C42×C7—41no—
65C42×C7—81no—
66C4⋊Dic7—41no—
67C4⋊Dic7—44no—
68C4⋊Dic7—41no—
69C2×C7⋊C8—22no—
70C2×C7⋊C8—41no—
71C2×C7⋊C8—41no—
72C2×C7⋊C8—42no—
73C2×C7⋊C8—42no—
74C2×C7⋊C8—42no—
75C2×#107—141no—
76C2×#103—141no—
77#400—21no—
78#398—22no—
79#391—21no—
80#391C2×D711yescharacteristic
81#389C2×D712yes—
82#387—22no—
83#384—71no—
84#340—142no—
85#340—142no—
86#336—71no—
87#336—72no—
88C23×C7#12011yescharacteristic
89C7×Q8—42no—
90C7×Q8—81no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—41no—
97C2×C4×C7—41no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—41no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7C2×Q1611yescharacteristic
107C2×C4×C7#13512yes—
108C2×C4×C7#13711yescharacteristic
109C2×C4×C7#13712yes—
110C2×Dic7—81no—
111C2×Dic7—82no—
112C7⋊C8—42no—
113C7⋊C8—82no—
114C22×Q8C22⋊D711yescharacteristic
115C22×C8—141no—
116C22×C8—142no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—142no—
121C2×C4⋊C4—141no—
122C2×C4⋊C4C22⋊D712yes—
123C2×C4⋊C4C22⋊D712yes—
124C2×C4⋊C4C22⋊D711yescharacteristic
125C2×C42—21no—
126C2×C42C22×D711yescharacteristic
127#107—282no—
128#103—282no—
129#95—21no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134C22×C7#45111yescharacteristic
135C22×C7#45212yes—
136C22×C7#45311yescharacteristic
137C22×C7#45412yes—
138C22×C7#45811yescharacteristic
139C4×C7—21no—
140C4×C7—21no—
141C4×C7—22no—
142C4×C7—22no—
143C4×C7—42no—
144C4×C7—82no—
145C4×C7—82no—
146C4×C7—81no—
147Dic7—161no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8#114412yes—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—281no—
157C22×C4C2×C22⋊D711yescharacteristic
158C22×C4C2×C22⋊D712yes—
159C2×C8—142no—
160C2×C8—281no—
161C2×C8—281no—
162C2×C8—282no—
163C2×C8—282no—
164C2×C8—282no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—281no—
174C4⋊C4—281no—
175C4⋊C4—284no—
176C42—41no—
177C42—81no—
178C2×C7#161411yescharacteristic
179C2×C7#161512yes—
180C2×C7#163411yescharacteristic
181C2×C7#163612yes—
182C2×C7#165711yescharacteristic
183C23#651611yescharacteristic
184Q8—42no—
185Q8—81no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—561no—
202C2×C4—562no—
203C2×C4#650511yescharacteristic
204C2×C4C2×#114411yescharacteristic
205C2×C4#651312yes—
206C2×C4#651412yes—
207C8—282no—
208C8—562no—
209C7#783111yescharacteristic, complemented, Sylow
210C22#9000011yescharacteristic
211C22#9001112yes—
212C22#9001311yescharacteristic
213C22#9001511yescharacteristic
214C22#9001712yes—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—82no—
221C4—81no—
222C4—82no—
223C4—1121no—
224C2#1172964112yes—
225C2#1172964511yescharacteristic
226C2#1172968211yescharacteristic
227C2#1172969012yes—
228C2#1172970911yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2809C2
socle, prime radicalC23×C7#120
hypercenter#2809D7
hypocenterC7#7831
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
209. C7#7831semidirect

Maximal subgroups

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

subgroupindex
C7×#28092
#117274382
#117273662
#117254482
#117254442
#78317