Finite Group Databasebeta

← #13267674

Subgroups — #13267674

Order 1792.

288 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3395C211yesmaximal, characteristic
2#11729064C211yesmaximal, characteristic
3#11728698C211yesmaximal, characteristic
4#11728664C211yesmaximal, characteristic
5#11725599C211yesmaximal, characteristic
6Q8⋊(C7⋊C16)C211yesmaximal, characteristic
7#11725571C211yesmaximal, characteristic
8C7×#557—21no—
9C7×#494—21no—
10C7×#494C2211yescharacteristic
11C7×#462—22no—
12C7×#462—21no—
13C7×C8×Q8C2211yescharacteristic
14C4×C7×Q16—21no—
15C4×C7×Q16C2211yescharacteristic
16#89803—21no—
17#89454C2211yescharacteristic
18#89359—21no—
19#89359—22no—
20#89285—21no—
21#89281—21no—
22C4⋊(C7⋊C16)C2211yescharacteristic
23C4×C7⋊C16C2211yescharacteristic
24#89253C2211yescharacteristic
25#12382—71nomaximal, Sylow
26C2×C7×Q16—22no—
27C2×C7×Q16—21no—
28C2×C7×Q16D411yescharacteristic
29C2×C7×Q16D412yes—
30C7×C4⋊Q8—22no—
31C7×C4⋊Q8—21no—
32C7×#125—42no—
33C7×#125—41no—
34C4×C7×Q8—22no—
35C4×C7×Q8—21no—
36C4×C7×Q8—41no—
37C4×C7×Q8D411yescharacteristic
38C2×#1144—41no—
39#6432—21no—
40C7×#107D411yescharacteristic
41C7×C4⋊C8—21no—
42C7×C4⋊C8D411yescharacteristic
43C7×#103—22no—
44C7×#103—21no—
45C4×C7×C8—21no—
46C4×C7×C8C2311yescharacteristic
47#6392—21no—
48#6392—21no—
49C2×C7⋊C16—21no—
50C2×C7⋊C16—21no—
51#6384—41no—
52C4⋊(C7⋊C8)—21no—
53#3395D711yescharacteristic
54#2241—71no—
55#2225—71no—
56#1674—71no—
57#1362—71no—
58Q8⋊C16—71no—
59#1332—71no—
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8D811yescharacteristic
63C2×Dic14—41no—
64C7×Q16—21no—
65C7×Q16—21no—
66C7×Q16—42no—
67C7×Q16—41no—
68C7×Q16—41no—
69C7×Q16—82no—
70C2×C7×C8—21no—
71C2×C7×C8C2×D411yescharacteristic
72C2×C7×C8C2×D411yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4D811yescharacteristic
82C42×C7—21no—
83C42×C7—42no—
84C42×C7—41no—
85C42×C7C2×D411yescharacteristic
86#1144—82no—
87C4⋊Dic7—41no—
88C4⋊Dic7—41no—
89C2×C7⋊C8—41no—
90C7⋊C16—41no—
91C7⋊C16—41no—
92#557—21no—
93#500—71no—
94#494—21no—
95#494C2×D711yescharacteristic
96#462—22no—
97#462—21no—
98#462—141no—
99C8×Q8C2×D711yescharacteristic
100C4×Q16—21no—
101C4×Q16C2×D711yescharacteristic
102#367—141no—
103C4⋊C16—71no—
104#358—141no—
105#358—142no—
106C4×C16—71no—
107#335—71no—
108#332—141no—
109C7×Q8—21no—
110C7×Q8—41no—
111C7×Q8—42no—
112C7×Q8—42no—
113C7×Q8—41no—
114C7×Q8SD3212yes—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—41no—
118C2×C4×C7—42no—
119C2×C4×C7—41no—
120C2×C4×C7#12011yescharacteristic
121C2×C4×C7C2×D811yescharacteristic
122C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
123C2×Dic7—81no—
124Dic14—81no—
125Dic14—81no—
126C7×C8—21no—
127C7×C8—41no—
128C7×C8#13511yescharacteristic
129C7×C8#13711yescharacteristic
130C7⋊C8—81no—
131C2×Q16—22no—
132C2×Q16—21no—
133C2×Q16—281no—
134C2×Q16C22⋊D711yescharacteristic
135C2×Q16C22⋊D712yes—
136C4⋊Q8—22no—
137C4⋊Q8—21no—
138C4⋊Q8—141no—
139#125—42no—
140#125—41no—
141C4×Q8—22no—
142C4×Q8—21no—
143C4×Q8—41no—
144C4×Q8C22⋊D711yescharacteristic
145C2×C16—141no—
146C2×C16—141no—
147#107—141no—
148#107—141no—
149#107C22⋊D711yescharacteristic
150C4⋊C8—21no—
151C4⋊C8—141no—
152C4⋊C8C22⋊D711yescharacteristic
153#103—22no—
154#103—21no—
155#103—281no—
156C4×C8—21no—
157C4×C8C22×D711yescharacteristic
158C22×C7#44711yescharacteristic
159C4×C7—22no—
160C4×C7—21no—
161C4×C7—41no—
162C4×C7—81no—
163C4×C7—82no—
164C4×C7—81no—
165C4×C7#45211yescharacteristic
166C4×C7#45511yescharacteristic
167C4×C7C2×SD3211yescharacteristic
168C4×C7#51011yescharacteristic
169Dic7—161no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—281no—
173C2×Q8#114111yescharacteristic
174Q16—21no—
175Q16—21no—
176Q16—42no—
177Q16—41no—
178Q16—41no—
179Q16—82no—
180Q16—562no—
181C2×C8—21no—
182C2×C8—281no—
183C2×C8C2×C22⋊D711yescharacteristic
184C2×C8C2×C22⋊D711yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—281no—
194C4⋊C4—281no—
195C4⋊C4#114111yescharacteristic
196C42—21no—
197C42—42no—
198C42—41no—
199C42C2×C22⋊D711yescharacteristic
200C16—281no—
201C16—281no—
202C2×C7#164111yescharacteristic
203C2×C7#218511yescharacteristic
204C2×C7#219511yescharacteristic
205Q8—21no—
206Q8—41no—
207Q8—42no—
208Q8—42no—
209Q8—41no—
210Q8—561no—
211Q8—561no—
212Q8#640212yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—561no—
219C2×C4C2×#114111yescharacteristic
220C2×C4#649511yescharacteristic
221C2×C4#651611yescharacteristic
222C8—21no—
223C8—41no—
224C8—561no—
225C8#651311yescharacteristic
226C8#651411yescharacteristic
227C7#1238211yescharacteristic, complemented, Sylow
228C22#8998911yescharacteristic
229C4—22no—
230C4—21no—
231C4—41no—
232C4—81no—
233C4—82no—
234C4—81no—
235C4—1121no—
236C4C2×#640211yescharacteristic
237C4#8995111yescharacteristic
238C4#9001111yescharacteristic
239C4#9001411yescharacteristic
240C2#1172967411yescharacteristic
241C2#1172981711yescharacteristic
242C2#1172987911yescharacteristic
subgroupG/H
centerC22#89989
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#3395C2
socle, prime radicalC22×C7#447
hypercenter#3395D7
hypocenterC7#12382
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
227. C7#12382semidirect

Maximal subgroups

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

subgroupindex
C7×#33952
#117290642
#117286982
#117286642
#117255992
Q8⋊(C7⋊C16)2
#117255712
#123827