Finite Group Databasebeta

← #13257742

Subgroups — #13257742

Order 1792.

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

normal subgroups only

This group has 344 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#11736716C211yesmaximal, characteristic
2C7×#1574C211yesmaximal, characteristic
3#11728534C211yesmaximal, characteristic
4#11728523C211yesmaximal, characteristic
5#11728278C211yesmaximal, characteristic
6#11728265C211yesmaximal, characteristic
7#11725549C211yesmaximal, characteristic
8#90161C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13#89696C2211yescharacteristic
14C4×#1136C2211yescharacteristic
15#89623—21no—
16#89623—21no—
17#89493—22no—
18#89453—21no—
19#89452—21no—
20C7×#335—21no—
21C7×#334—21no—
22#89349—22no—
23#89281—21no—
24#89279—22no—
25#89277—21no—
26#89277—22no—
27#10494—71nomaximal, Sylow
28C22×Dic14—21no—
29C2×C7×M16—21no—
30C2×C42×C7C2311yescharacteristic
31#6489—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7D412yes—
34#6487—21no—
35#6487—21no—
36C2×#1136—21no—
37C2×#1136—21no—
38#6433—21no—
39#6432—21no—
40#6432—21no—
41#6432—21no—
42#6432—21no—
43#6432D411yescharacteristic
44#6432D411yescharacteristic
45C4×Dic14—21no—
46C4×Dic14—21no—
47C7×C4⋊C8—21no—
48C7×C4⋊C8—21no—
49C7×C4⋊C8D412yes—
50C7×C4⋊C8D411yescharacteristic
51C7×C4⋊C8D411yescharacteristic
52C7×C22⋊C8—21no—
53C7×C22⋊C8—21no—
54C4×C7×C8—21no—
55C4×C7×C8—21no—
56#6391—22no—
57#6391—22no—
58#6390—22no—
59#6390—22no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62#6377—21no—
63C4×C7⋊C8—21no—
64#3229—71no—
65#1997—71no—
66#1574D711yescharacteristic
67#1561—71no—
68#1549—71no—
69#1485—71no—
70#1299—71no—
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7#4111yescharacteristic
73C22×C4×C7#4111yescharacteristic
74C22×Dic7—22no—
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×Dic14—41no—
78C2×Dic14—41no—
79C2×Dic14—82no—
80C7×M16—42no—
81C2×C7×C8—21no—
82C2×C7×C8—21no—
83C2×C7×C8—22no—
84C2×C7×C8—41no—
85C42×C7C2×D411yescharacteristic
86C42×C7C2×D411yescharacteristic
87C42×C7C2×D411yescharacteristic
88C42×C7#4111yescharacteristic
89C22⋊Dic7—41no—
90C4⋊Dic7—21no—
91C4⋊Dic7—22no—
92C4⋊Dic7—21no—
93C4⋊Dic7—22no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—41no—
97C4⋊Dic7—41no—
98Dic7⋊C4—41no—
99Dic7⋊C4—41no—
100C4×Dic7—41no—
101#1136—22no—
102#1136—41no—
103#1136—81no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106#533—71no—
107C2×C4⋊Q8—71no—
108#498—141no—
109#487—141no—
110#483—142no—
111#475—141no—
112#475—141no—
113#431C2×D711yescharacteristic
114#431C2×D711yescharacteristic
115#423—71no—
116#423C2×D711yescharacteristic
117C4×M16—71no—
118#343—142no—
119#335—21no—
120#334—21no—
121#332—141no—
122#332—141no—
123#332—142no—
124Q8⋊C8—142no—
125C23×C7#12311yescharacteristic
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—22no—
130C2×C4×C7—21no—
131C2×C4×C7#12011yescharacteristic
132C2×C4×C7#12111yescharacteristic
133C2×C4×C7#12111yescharacteristic
134C2×C4×C7#12111yescharacteristic
135C2×C4×C7#12311yescharacteristic
136C2×C4×C7#12411yescharacteristic
137C2×Dic7—41no—
138C2×Dic7—41no—
139C2×Dic7—42no—
140C2×Dic7—82no—
141Dic14—41no—
142Dic14—41no—
143Dic14—41no—
144Dic14—41no—
145Dic14—82no—
146Dic14—82no—
147C7×C8—41no—
148C7×C8—41no—
149C7×C8—42no—
150C7×C8—81no—
151C7⋊C8—42no—
152C7⋊C8—81no—
153C22×Q8—141no—
154C2×M16—21no—
155C2×M16—141no—
156C2×M16—141no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—141no—
160C4⋊Q8—141no—
161C4⋊Q8—141no—
162C4⋊Q8—141no—
163#125—141no—
164C22⋊Q8—141no—
165C22⋊Q8—141no—
166C4×Q8—141no—
167C4×Q8—141no—
168#117—141no—
169C2×C4⋊C4—72no—
170C2×C4⋊C4—141no—
171C2×C42C22×D711yescharacteristic
172#106—142no—
173#106—142no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—141no—
177C4⋊C8—141no—
178C4⋊C8D2812yes—
179C4⋊C8C22⋊D711yescharacteristic
180C4⋊C8C22⋊D711yescharacteristic
181#103—142no—
182#103—142no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185#97—141no—
186C4×C8—21no—
187C4×C8—21no—
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#45411yescharacteristic
197C4×C7#45611yescharacteristic
198C4×C7#47112yes—
199C4×C7#47111yescharacteristic
200C4×C7#47311yescharacteristic
201Dic7—81no—
202Dic7—81no—
203Dic7—82no—
204C2×Q8—141no—
205C2×Q8—141no—
206C2×Q8—281no—
207C2×Q8—281no—
208C2×Q8—562no—
209C22×C4—142no—
210C22×C4#115711yescharacteristic
211C22×C4D4×D711yescharacteristic
212C22×C4#116111yescharacteristic
213M16—42no—
214M16—142no—
215M16—281no—
216M16—561no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—281no—
222C2×C8—281no—
223C4⋊C4—141no—
224C4⋊C4—141no—
225C4⋊C4—142no—
226C4⋊C4—141no—
227C4⋊C4—142no—
228C4⋊C4—141no—
229C4⋊C4—281no—
230C4⋊C4—281no—
231C4⋊C4—281no—
232C4⋊C4—281no—
233C22⋊C4—281no—
234C42—281no—
235C42C2×D2811yescharacteristic
236C42D4×D711yescharacteristic
237C42#116111yescharacteristic
238C42C2×C22⋊D711yescharacteristic
239C2×C7—41no—
240C2×C7#203811yescharacteristic
241C2×C7#204611yescharacteristic
242C2×C7#205211yescharacteristic
243C23#645711yescharacteristic
244Q8—281no—
245Q8—281no—
246Q8—281no—
247Q8—281no—
248Q8—562no—
249Q8—562no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—281no—
256C2×C4—281no—
257C2×C4—282no—
258C2×C4—562no—
259C2×C4#645711yescharacteristic
260C2×C4#645811yescharacteristic
261C2×C4#645811yescharacteristic
262C2×C4#649311yescharacteristic
263C2×C4#650011yescharacteristic
264C2×C4#651011yescharacteristic
265C8—42no—
266C8—41no—
267C8—41no—
268C8—81no—
269C8—282no—
270C8—561no—
271C7#1049411yescharacteristic, complemented, Sylow
272C22—21no—
273C22—21no—
274C22—21no—
275C22#8976311yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—561no—
280C4—561no—
281C4—562no—
282C4#8966811yescharacteristic
283C4#8967011yescharacteristic
284C4#8991312yes—
285C4#8997311yescharacteristic
286C4#8997711yescharacteristic
287C2—41no—
288C2#1172829511yescharacteristic
289C2#1172843211yescharacteristic
290C2#1172847011yescharacteristic
subgroupG/H
centerC22#89763
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1574C2
socleC22×C7#394
prime radicalC23×C7#123
hypercenter#1574D7
hypocenterC7#10494
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1574

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
#117367162
C7×#15742
#117285342
#117285232
#117282782
#117282652
#117255492
#104947