Finite Group Databasebeta

← #13251176

Subgroups — #13251176

Order 1792.

470 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2982C211yesmaximal, characteristic
2#11739924C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic
4#11728534C212yesmaximal
5#11725552C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×C8×Q8—24no—
9C7×#431—22no—
10C7×#431C2212yes—
11#90002—21no—
12C2×#6410—21no—
13#89758C414yes—
14#89749—21no—
15C4×C4⋊Dic7C2211yescharacteristic
16#89701—22no—
17#89696C2212yes—
18#89325—21no—
19#89277—24no—
20#9396—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×Dic14—21no—
23C4×C7×Q8—22no—
24C4×C7×Q8—22no—
25C2×C7×C4⋊C4C2×C412yes—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C42×C7D412yes—
28C2×C42×C7C2311yescharacteristic
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7C2×C412yes—
32C2×C4×Dic7—21no—
33C2×#1136—22no—
34#6432—22no—
35#6432—22no—
36C7×C4⋊C8—24no—
37C7×C4⋊C8D414yes—
38C7×C22⋊C8—22no—
39C4×C7×C8—24no—
40C4×C7×C8—22no—
41#6410—42no—
42C7⋊#95—24no—
43C7⋊#95—24no—
44C7⋊#95C2×C412yes—
45C4⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)—22no—
47#2982D711yescharacteristic
48#2568—71no—
49#1801—71no—
50#1549—72no—
51#1303—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—22no—
54C2×C7×Q8SD1612yes—
55C2×C7×Q8Q1612yes—
56C22×C4×C7C22⋊C412yes—
57C22×C4×C7C22⋊C412yes—
58C22×C4×C7C22×C411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D411yescharacteristic
61C22×Dic7—22no—
62C22×Dic7—22no—
63C2×Dic14—44no—
64C2×Dic14—41no—
65C2×Dic14—41no—
66C2×C7×C8—24no—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—22no—
69C42×C7—22no—
70C42×C7—22no—
71C42×C7C2×D412yes—
72C42×C7#4112yes—
73C4⋊Dic7—24no—
74C4⋊Dic7—42no—
75C4⋊Dic7—42no—
76C4×Dic7—42no—
77#1136—42no—
78#1136—42no—
79C2×C7⋊C8—42no—
80C2×C4⋊Q8—71no—
81C2×C4×Q8C2×D711yescharacteristic
82C8×Q8—24no—
83#431—22no—
84#431C2×D712yes—
85#423—72no—
86C2×#101—141no—
87#400—74no—
88#391—141no—
89#391—142no—
90#382—141no—
91C4×C4⋊C4—71no—
92#343—141no—
93#332—144no—
94C23×C7C2×C22⋊C411yescharacteristic
95C7×Q8—24no—
96C7×Q8—24no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—41no—
102C2×C4×C7—42no—
103C2×C4×C7#10314yes—
104C2×C4×C7C4×D412yes—
105C2×C4×C7#12011yescharacteristic
106C2×C4×C7#12112yes—
107C2×C4×C7#12311yescharacteristic
108C2×C4×C7C2×SD1611yescharacteristic
109C2×C4×C7C2×Q1611yescharacteristic
110C2×C4×C7#13512yes—
111C2×Dic7—42no—
112C2×Dic7—44no—
113C2×Dic7—42no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116Dic14—82no—
117Dic14—82no—
118C7×C8—44no—
119C7⋊C8—82no—
120C22×Q8—141no—
121C22×Q8D2811yescharacteristic
122C2×M16—142no—
123C4⋊Q8—142no—
124C4⋊Q8—142no—
125C4×Q8—22no—
126C4×Q8—22no—
127C2×C4⋊C4—72no—
128C2×C4⋊C4—141no—
129C2×C4⋊C4—141no—
130C2×C4⋊C4C4×D712yes—
131C2×C4⋊C4D2811yescharacteristic
132C2×C42—141no—
133C2×C42C22⋊D712yes—
134C2×C42C22×D711yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—142no—
137C4⋊C8—142no—
138C4⋊C8C22⋊D714yes—
139#101—282no—
140C22⋊C8—22no—
141C4×C8—24no—
142C4×C8—22no—
143#95—72no—
144#95—144no—
145#95—144no—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#38611yescharacteristic
149C22×C7C2×#10311yescharacteristic
150C22×C7#41611yescharacteristic
151C4×C7—21no—
152C4×C7—24no—
153C4×C7—22no—
154C4×C7—42no—
155C4×C7#45412yes—
156C4×C7#46512yes—
157C4×C7#46712yes—
158Dic7—82no—
159Dic7—82no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—284no—
163C2×Q8—281no—
164C2×Q8—281no—
165C2×Q8#113212yes—
166C2×Q8Dic2812yes—
167C22×C4—142no—
168C22×C4—142no—
169C22×C4(C2×D7)⋊C412yes—
170C22×C4(C2×D7)⋊C412yes—
171C22×C4C2×C4×D711yescharacteristic
172C22×C4C2×D2811yescharacteristic
173C22×C4C2×C22⋊D711yescharacteristic
174M16—282no—
175M16—282no—
176C2×C8—24no—
177C2×C8—282no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—144no—
181C4⋊C4—282no—
182C4⋊C4—282no—
183C42—22no—
184C42—22no—
185C42—282no—
186C42#115712yes—
187C42C2×C22⋊D712yes—
188C2×C7—21no—
189C2×C7—21no—
190C2×C7#188811yescharacteristic
191C2×C7#189211yescharacteristic
192C2×C7#191511yescharacteristic
193C23C2×(C2×D7)⋊C411yescharacteristic
194Q8—24no—
195Q8—24no—
196Q8—562no—
197Q8—562no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4—284no—
207C2×C4—282no—
208C2×C4—282no—
209C2×C4#639014yes—
210C2×C4C2×#113211yescharacteristic
211C2×C4#646712yes—
212C2×C4C2×Dic2811yescharacteristic
213C2×C4C4×C22⋊D712yes—
214C2×C4#649211yescharacteristic
215C2×C4#649312yes—
216C2×C4#651611yescharacteristic
217C8—44no—
218C8—562no—
219C7#939611yescharacteristic, complemented, Sylow
220C22—21no—
221C22—21no—
222C22C2×#639011yescharacteristic
223C22#8988511yescharacteristic
224C22#8998511yescharacteristic
225C4—21no—
226C4—24no—
227C4—22no—
228C4—42no—
229C4—562no—
230C4—562no—
231C4#8988612yes—
232C4#8988812yes—
233C4#9001712yes—
234C2—21no—
235C2—21no—
236C2#1172941511yescharacteristic
237C2#1172972611yescharacteristic
238C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2982C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2982D7
hypocenterC7#9396
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
219. C7#9396semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117399242
#117293912
#117285342
#117255522
#93967