Finite Group Databasebeta

← #13257737

Subgroups — #13257737

Order 1792.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3229C211yesmaximal, characteristic
2#11729588C211yesmaximal, characteristic
3#11729579C211yesmaximal, characteristic
4#11729544C211yesmaximal, characteristic
5#11729529C211yesmaximal, characteristic
6#11728502C211yesmaximal, characteristic
7#11725549C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C7×#498—21no—
11C7×#487—21no—
12C7×#483—22no—
13C7×#475—21no—
14C7×#475—21no—
15C7×#431C2211yescharacteristic
16#89697C2211yescharacteristic
17#89696C2211yescharacteristic
18#89696C2211yescharacteristic
19C4×#1136C2211yescharacteristic
20#89342—22no—
21#89342—21no—
22#89342—21no—
23Q8⋊(C7⋊C8)—22no—
24#89325—22no—
25#89275—21no—
26#89274—21no—
27#10494—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C7×C4⋊Q8—21no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8D411yescharacteristic
34C7×C4⋊Q8D411yescharacteristic
35C7×#125—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C7×#117—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4D412yes—
43C2×C42×C7C2311yescharacteristic
44C2×#1136—21no—
45C2×#1136—21no—
46C2×#1136—21no—
47C7×#106—22no—
48C7×#106—22no—
49C7×C4⋊C8D411yescharacteristic
50C7×C4⋊C8D411yescharacteristic
51C7×#103—22no—
52C7×#103—22no—
53C7×C22⋊C8—21no—
54C4×C7×C8—21no—
55C22⋊(C7⋊C8)—21no—
56C4⋊(C7⋊C8)—21no—
57C4⋊(C7⋊C8)—21no—
58C4⋊(C7⋊C8)—21no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)D412yes—
61#6377—21no—
62C4×C7⋊C8—21no—
63C4×C7⋊C8—21no—
64#3229D711yescharacteristic
65#1997—71no—
66#1574—71no—
67#1561—71no—
68#1549—71no—
69#1485—71no—
70#1299—71no—
71C2×C7×Q8—21no—
72C2×C7×Q8—21no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C2×C7×Q8—82no—
76C22×C4×C7—22no—
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7#4111yescharacteristic
79C22×C4×C7#4111yescharacteristic
80C2×C7×C8—21no—
81C2×C7×C8—21no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C22⋊C4—41no—
93C42×C7—41no—
94C42×C7C2×D411yescharacteristic
95C42×C7C2×D411yescharacteristic
96C42×C7C2×D411yescharacteristic
97C42×C7#4111yescharacteristic
98#1136—22no—
99#1136—41no—
100#1136—42no—
101#1136—81no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106#533C2×D711yescharacteristic
107C2×C4⋊Q8C2×D711yescharacteristic
108#498—21no—
109#487—21no—
110#483—22no—
111#475—21no—
112#475—21no—
113#431—71no—
114#431C2×D711yescharacteristic
115#423—71no—
116#423—71no—
117C4×M16—71no—
118#343—142no—
119#335—141no—
120#334—141no—
121#332—142no—
122#332—141no—
123#332—141no—
124Q8⋊C8—142no—
125C23×C7#12311yescharacteristic
126C7×Q8—41no—
127C7×Q8—41no—
128C7×Q8—41no—
129C7×Q8—41no—
130C7×Q8—82no—
131C7×Q8—82no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
136C2×C4×C7—21no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—82no—
141C2×C4×C7#12011yescharacteristic
142C2×C4×C7#12111yescharacteristic
143C2×C4×C7#12111yescharacteristic
144C2×C4×C7#12111yescharacteristic
145C2×C4×C7#12311yescharacteristic
146C2×C4×C7#12411yescharacteristic
147C7×C8—41no—
148C7×C8—41no—
149C7⋊C8—42no—
150C7⋊C8—42no—
151C7⋊C8—81no—
152C7⋊C8—81no—
153C22×Q8—21no—
154C2×M16—141no—
155C2×M16—141no—
156C2×M16—141no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C22⋊D711yescharacteristic
162C4⋊Q8C22⋊D711yescharacteristic
163#125—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168#117—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C22⋊D712yes—
171C2×C42C22×D711yescharacteristic
172#106—22no—
173#106—22no—
174C4⋊C8—72no—
175C4⋊C8—141no—
176C4⋊C8—141no—
177C4⋊C8—141no—
178C4⋊C8—141no—
179C4⋊C8C22⋊D711yescharacteristic
180C4⋊C8C22⋊D711yescharacteristic
181#103—22no—
182#103—22no—
183C22⋊C8—21no—
184C22⋊C8—141no—
185#97—141no—
186C4×C8—21no—
187C4×C8—141no—
188C4×C8—141no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7#39411yescharacteristic
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—81no—
197C4×C7—81no—
198C4×C7—82no—
199C4×C7#45411yescharacteristic
200C4×C7#45611yescharacteristic
201C4×C7#47112yes—
202C4×C7#47111yescharacteristic
203C4×C7#47311yescharacteristic
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4#115911yescharacteristic
211C22×C4#115911yescharacteristic
212C22×C4C2×C22⋊D711yescharacteristic
213M16—142no—
214M16—281no—
215M16—282no—
216M16—561no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—142no—
220C2×C8—281no—
221C2×C8—281no—
222C2×C8—281no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—41no—
234C42—41no—
235C42D4×D711yescharacteristic
236C42#115911yescharacteristic
237C42C2×C22⋊D711yescharacteristic
238C42C2×C22⋊D711yescharacteristic
239C2×C7—41no—
240C2×C7#203811yescharacteristic
241C2×C7#204611yescharacteristic
242C2×C7#205211yescharacteristic
243C23#649811yescharacteristic
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—82no—
249Q8—82no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4#649811yescharacteristic
260C2×C4#649911yescharacteristic
261C2×C4#650111yescharacteristic
262C2×C4#650211yescharacteristic
263C2×C4#650211yescharacteristic
264C2×C4#651611yescharacteristic
265C8—41no—
266C8—41no—
267C8—282no—
268C8—282no—
269C8—561no—
270C8—561no—
271C7#1049411yescharacteristic, complemented, Sylow
272C22—21no—
273C22—21no—
274C22—21no—
275C22#8999411yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4#8994912yes—
283C4#8997511yescharacteristic
284C4#8997611yescharacteristic
285C4#9001711yescharacteristic
286C4#9001811yescharacteristic
287C2—41no—
288C2#1172961711yescharacteristic
289C2#1172980711yescharacteristic
290C2#1172985911yescharacteristic
subgroupG/H
centerC22#89994
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3229C2
socleC22×C7#394
prime radicalC23×C7#123
hypercenter#3229D7
hypocenterC7#10494
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
271. C7#10494semidirect

Maximal subgroups

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

subgroupindex
C7×#32292
#117295882
#117295792
#117295442
#117295292
#117285022
#117255492
#104947