Finite Group Databasebeta

← #13243116

Subgroups — #13243116

Order 1792.

438 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 83 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#11733866C211yesmaximal, characteristic
2#11727410C211yesmaximal, characteristic
3#11726445C212yesmaximal, complemented
4C7×#1270C211yesmaximal, characteristic
5#11725286C212yesmaximal, complemented
6C7×#390C2211yescharacteristic
7#89992—21no—
8C2×#6406—21no—
9C2×#6392—22no—
10C2×#6390—22no—
11#89758—21no—
12C2×#6382—21no—
13#89722—22no—
14#89719—22no—
15#89707C2211yescharacteristic
16#89701C2212yes—
17C7×#336C2212yes—
18#89319C2211yescharacteristic
19#7833—71nomaximal, Sylow
20C22×C7×D4D411yescharacteristic
21C23×Dic7—21no—
22C22×Dic14D412yes—
23C22×C7×C8—22no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C7×C22⋊C4—22no—
26C2×C42×C7C2311yescharacteristic
27C2×C22⋊Dic7—21no—
28C2×C22⋊Dic7—22no—
29C2×C4⋊Dic7—22no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7D412yes—
33C22×C7⋊C8—21no—
34#6406—42no—
35C7⋊#95—21no—
36C7⋊#95—22no—
37C7⋊#95—22no—
38C7⋊#95—22no—
39#6392—44no—
40#6390—44no—
41#6382—42no—
42#2804—71no—
43#2095—72no—
44#2094—71no—
45#1272—72no—
46#1270D711yescharacteristic
47C24×C7—21no—
48C2×C7×D4—21no—
49C2×C7×D4—41no—
50C2×C7×D4—41no—
51C2×C7×D4Q1612yes—
52C22×C4×C7—21no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D412yes—
55C22×Dic7—22no—
56C22×Dic7—22no—
57C22×Dic7—41no—
58C22×Dic7—82no—
59C22×Dic7—81no—
60C2×Dic14—22no—
61C2×Dic14—22no—
62C2×Dic14—42no—
63C2×Dic14—42no—
64C2×C7×C8—24no—
65C2×C7×C8—42no—
66C2×C7×C8—42no—
67C7×C4⋊C4—41no—
68C7×C22⋊C4—42no—
69C7×C22⋊C4—42no—
70C42×C7—41no—
71C22⋊Dic7—82no—
72C22⋊Dic7—81no—
73C4⋊Dic7—42no—
74C4⋊Dic7—41no—
75C4⋊Dic7—41no—
76C4⋊Dic7—42no—
77C4⋊Dic7—42no—
78C4⋊Dic7—81no—
79C2×C7⋊C8—22no—
80C2×C7⋊C8—41no—
81C2×C7⋊C8—41no—
82C2×#107—142no—
83C2×#106—141no—
84C2×#103—142no—
85C2×#102—141no—
86#400—141no—
87#393—142no—
88#391—72no—
89#390C2×D711yescharacteristic
90#386—141no—
91#385—142no—
92#384—71no—
93#336—71no—
94#336C2×D712yes—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—42no—
98C23×C7—41no—
99C23×C7—41no—
100C23×C7#12011yescharacteristic
101C7×D4—42no—
102C7×D4—81no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—42no—
108C2×C4×C7—41no—
109C2×C4×C7C2×Q1611yescharacteristic
110C2×C4×C7#13512yes—
111C2×C4×C7#13711yescharacteristic
112C2×C4×C7#13712yes—
113C2×Dic7—42no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—42no—
117C2×Dic7—42no—
118C2×Dic7—42no—
119C2×Dic7—84no—
120C2×Dic7—81no—
121C2×Dic7—82no—
122Dic14—44no—
123Dic14—82no—
124C7×C8—44no—
125C7⋊C8—42no—
126C22×Q8—72no—
127C22×D4D2811yescharacteristic
128C23×C4—141no—
129C22×C8—22no—
130C22×C8—141no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4—141no—
133C2×C4⋊C4—142no—
134C2×C4⋊C4—141no—
135C2×C4⋊C4D2811yescharacteristic
136C2×C22⋊C4—22no—
137C2×C22⋊C4—142no—
138C2×C22⋊C4—141no—
139C2×C42C22×D711yescharacteristic
140#107—284no—
141#106—282no—
142#103—284no—
143#102—282no—
144#95—142no—
145#95—141no—
146#95—142no—
147#95—142no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7—82no—
151C22×C7—81no—
152C22×C7—81no—
153C22×C7#45111yescharacteristic
154C22×C7#45212yes—
155C22×C7#45311yescharacteristic
156C22×C7#45312yes—
157C22×C7#45811yescharacteristic
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—81no—
163Dic7—82no—
164Dic7—82no—
165Dic7—161no—
166C24—21no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—282no—
170C2×Q8—282no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4Dic2812yes—
175C22×C4—21no—
176C22×C4—142no—
177C22×C4—142no—
178C22×C4—281no—
179C22×C4—561no—
180C22×C4—562no—
181C22×C4C2×D2811yescharacteristic
182C22×C4D4×D712yes—
183C2×C8—24no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—142no—
187C2×C8—281no—
188C2×C8—281no—
189C4⋊C4—41no—
190C4⋊C4—282no—
191C4⋊C4—281no—
192C4⋊C4—282no—
193C4⋊C4—281no—
194C4⋊C4—282no—
195C4⋊C4—561no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—561no—
199C22⋊C4—562no—
200C42—41no—
201C2×C7—82no—
202C2×C7#161411yescharacteristic
203C2×C7#161812yes—
204C2×C7#163011yescharacteristic
205C2×C7#163212yes—
206C2×C7#165511yescharacteristic
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23#644511yescharacteristic
213Q8—284no—
214Q8—562no—
215D4—42no—
216D4—81no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—282no—
224C2×C4—282no—
225C2×C4—282no—
226C2×C4—282no—
227C2×C4—282no—
228C2×C4—282no—
229C2×C4—564no—
230C2×C4—561no—
231C2×C4—562no—
232C2×C4C2×Dic2811yescharacteristic
233C2×C4#647211yescharacteristic
234C2×C4#647512yes—
235C2×C4#647812yes—
236C8—44no—
237C8—282no—
238C7#783311yescharacteristic, complemented, Sylow
239C22—42no—
240C22—44no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#8951111yescharacteristic
245C22#8951411yescharacteristic
246C22#8951911yescharacteristic
247C22#8955512yes—
248C22#8959412yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—562no—
255C4—562no—
256C4—1121no—
257C2—82no—
258C2#1172587111yescharacteristic
259C2#1172595612yes—
260C2#1172604512yes—
261C2#1172610411yescharacteristic
262C2#1172615811yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1270C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenter#1270D7
hypocenterC7#7833
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #11726445C2semidirect
5. #11725286C2semidirect
238. C7#7833semidirect

Maximal subgroups

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

subgroupindex
#117338662
#117274102
#117264452
C7×#12702
#117252862
#78337