Finite Group Databasebeta

← #13269832

Subgroups — #13269832

Order 1792.

412 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 89 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#11736103C211yesmaximal, characteristic
2#11727902C212yesmaximal
3#11727888C212yesmaximal, complemented
4C7×#1392C211yesmaximal, characteristic, complemented
5#11725499C211yesmaximal, characteristic, complemented
6#90523C2211yescharacteristic
7C2×#6511C2211yescharacteristic
8C2×#6470C411yescharacteristic
9C7×#413C2212yes—
10C7×#408C2211yescharacteristic
11#90009C2212yes—
12C2×#6396—22no—
13#89890C411yescharacteristic
14#89882C411yescharacteristic
15C2×#6389C411yescharacteristic
16#89502—22no—
17#89500—22no—
18C7⋊#338—22no—
19#89273—22no—
20#89272—22no—
21#12810—71nomaximal, Sylow
22C2×C7×#41C2311yescharacteristic
23C2×#1157C2×C411yescharacteristic
24C2×C7×M16—22no—
25C2×C7×M16C2×C411yescharacteristic
26C22×C7×C8C2×C411yescharacteristic
27#6511—42no—
28#6510D411yescharacteristic
29#6507D411yescharacteristic
30#6503Q811yescharacteristic
31#6502—22no—
32#6499Q811yescharacteristic
33C4×C22⋊D7—22no—
34C2×C4×Dic7C2×C411yescharacteristic
35C2×#1136—22no—
36C2×#1136C2×C411yescharacteristic
37C2×#1136D412yes—
38C22×C7⋊C8C2×C411yescharacteristic
39C22×C7⋊C8D412yes—
40#6470—42no—
41C2×#1131—21no—
42C2×C8×D7—21no—
43C7×#101—22no—
44C7×#100—22no—
45C7×C22⋊C8—22no—
46#6410—22no—
47#6407—22no—
48#6396—44no—
49(C2×D7)⋊C8—22no—
50#6389—21no—
51#6389—21no—
52Dic7⋊C8—22no—
53#2932—71no—
54#2132—72no—
55#1813—72no—
56#1392—71no—
57#1392D711yescharacteristic, complemented
58C7×#41—41no—
59C7×#41—41no—
60C2×C7×Q8C2×D411yescharacteristic
61C2×C7×D4C2×Q811yescharacteristic
62C2×C7×D4#4112yes—
63C22×C4×C7C22×C411yescharacteristic
64C22×C4×C7C2×D412yes—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—22no—
67C22×Dic7—21no—
68#1157—41no—
69#1157—41no—
70C2×D28C4⋊C411yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14C4⋊C411yescharacteristic
73C7×M16—41no—
74C7×M16—42no—
75C7×M16—42no—
76C2×C7×C8—21no—
77C2×C7×C8—21no—
78C2×C7×C8—41no—
79C2×C7×C8—42no—
80C22⋊Dic7—22no—
81(C2×D7)⋊C4—22no—
82Dic7⋊C4—22no—
83C4×Dic7—21no—
84C4×Dic7C4⋊C411yescharacteristic
85C4×Dic7C4⋊C411yescharacteristic
86#1136—41no—
87#1136—42no—
88#1136—42no—
89#1136—42no—
90C2×C7⋊C8—22no—
91C2×C7⋊C8—24no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8C22⋊C411yescharacteristic
94C2×C7⋊C8C22⋊C412yes—
95C2×C7⋊C8C22⋊C411yescharacteristic
96#1131—42no—
97C8×D7—42no—
98C2×#131—71no—
99C2×#131—71no—
100#532—71no—
101#435—142no—
102#435—142no—
103C2×#108—142no—
104#423—71no—
105#413—72no—
106#413C2×D712yes—
107#408—71no—
108#408C2×D711yescharacteristic
109C2×#97—71no—
110#356—142no—
111#355—142no—
112#338—142no—
113C23×C7#11711yescharacteristic
114C23×C7C22⋊Q812yes—
115C22×D7—21no—
116C7×Q8—41no—
117C7×D4—41no—
118C7×D4—42no—
119C2×C4×C7—22no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7C2×C22⋊C411yescharacteristic
124C2×C4×C7C2×C4⋊C411yescharacteristic
125C2×C4×C7#12112yes—
126C22⋊D7—42no—
127C22⋊D7—82no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—81no—
132D28—41no—
133C4×D7—41no—
134C4×D7—41no—
135Dic14—41no—
136C7×C8—41no—
137C7×C8—41no—
138C7×C8—82no—
139C7⋊C8—21no—
140C7⋊C8—22no—
141C7⋊C8—82no—
142C7⋊C8#9912yes—
143C2×#41—71no—
144C2×#41C22×D711yescharacteristic
145#131—282no—
146#131—282no—
147C2×M16—22no—
148C2×M16—71no—
149C2×M16—72no—
150C2×M16—141no—
151C2×M16—142no—
152C2×M16C4×D711yescharacteristic
153C22×C8—71no—
154C22×C8—72no—
155C22×C8—141no—
156C22×C8C4×D711yescharacteristic
157C4⋊Q8—71no—
158#127—71no—
159#124—71no—
160#124—71no—
161#121—142no—
162C4×D4—142no—
163C2×C42—71no—
164#108—284no—
165C4⋊C8—142no—
166#101—22no—
167#101—142no—
168#100—22no—
169#100—142no—
170C22⋊C8—22no—
171C22⋊C8—142no—
172#97—141no—
173#97—141no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—41no—
177C22×C7—42no—
178C22×C7#38011yescharacteristic
179C2×D7—42no—
180C2×D7—41no—
181C4×C7—22no—
182C4×C7—21no—
183C4×C7C2×#9911yescharacteristic
184C4×C7#41011yescharacteristic
185Dic7—41no—
186Dic7—41no—
187Dic7—41no—
188#41—41no—
189#41—41no—
190#41—281no—
191#41—281no—
192C2×Q8—71no—
193C2×Q8D4×D711yescharacteristic
194C2×D4—71no—
195C2×D4—141no—
196C2×D4—142no—
197C2×D4#115712yes—
198C2×D4Q8×D711yescharacteristic
199C22×C4—141no—
200C22×C4—141no—
201C22×C4C2×C4×D711yescharacteristic
202C22×C4D4×D712yes—
203M16—41no—
204M16—42no—
205M16—42no—
206M16—282no—
207M16—282no—
208M16—281no—
209M16—282no—
210M16—282no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—71no—
216C2×C8—71no—
217C2×C8—72no—
218C2×C8—144no—
219C2×C8—142no—
220C2×C8—282no—
221C2×C8—282no—
222C4⋊C4—142no—
223C22⋊C4—142no—
224C22⋊C4—142no—
225C42—71no—
226C42—71no—
227C42—141no—
228C2×C7—21no—
229C2×C7—41no—
230C2×C7—42no—
231C2×C7#179911yescharacteristic
232D7—81no—
233C23—141no—
234C23#643511yescharacteristic
235C23#645912yes—
236Q8—41no—
237Q8—281no—
238D4—42no—
239D4—41no—
240D4—281no—
241D4—282no—
242D4—562no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—141no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—281no—
251C2×C4—281no—
252C2×C4—561no—
253C2×C4D7×C22⋊C411yescharacteristic
254C2×C4#644712yes—
255C2×C4D7×C4⋊C411yescharacteristic
256C8—41no—
257C8—41no—
258C8—72no—
259C8—82no—
260C8—141no—
261C8—142no—
262C8—562no—
263C7#1281011yescharacteristic, complemented, Sylow
264C22—21no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—282no—
269C22—281no—
270C22#8943911yescharacteristic
271C4—22no—
272C4—21no—
273C4—281no—
274C4—281no—
275C4—281no—
276C4#8951211yescharacteristic
277C4D7×#9911yescharacteristic
278C2—21no—
279C2—41no—
280C2—42no—
281C2—561no—
282C2#1172671311yescharacteristic
subgroupG/H
centerC4#89512
commutatorC22×C4×C7C22×C4
FrattiniC2×#41C22×D7
FittingC7×#1392C2
socleC2×C7#1799
prime radical#90523C22
cosocleC2×C7×#41C23
hypercenter#1392D7
hypocenterC7#12810
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
3. #11727888C2semidirect
4. C7×#1392C2semidirect
5. #11725499C2semidirect
57. #1392D7semidirect
263. C7#12810semidirect

Maximal subgroups

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

subgroupindex
#117361032
#117279022
#117278882
C7×#13922
#117254992
#128107