Finite Group Databasebeta

← #13272006

Subgroups — #13272006

Order 1792.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#11735999C211yesmaximal, characteristic
2C7×#2110C211yesmaximal, characteristic, complemented
3#11729192C212yesmaximal, complemented
4C2×#89309C212yesmaximal, complemented
5#11727639C211yesmaximal, characteristic, complemented
6C2×#6464C2211yescharacteristic
7C22×#1131C2211yescharacteristic
8#90215C411yescharacteristic
9C2×C7×M32C2212yes—
10C7×#405C2211yescharacteristic
11#89901C411yescharacteristic
12#89898C411yescharacteristic
13C2×#6386C2212yes—
14#89771C411yescharacteristic
15#89478—22no—
16#89460—22no—
17C7×#347—22no—
18#89310—24no—
19#89309—24no—
20#89258—22no—
21#13201—71nomaximal, Sylow
22C2×#1157C2×C411yescharacteristic
23C22×C4×D7C2×C411yescharacteristic
24C2×C7×M16C2×C411yescharacteristic
25C22×C7×C8C2311yescharacteristic
26C7×#117C2×C411yescharacteristic
27C2×#1136C2×C411yescharacteristic
28C22×C7⋊C8C2×C411yescharacteristic
29#6464—22no—
30#6464—22no—
31C2×#1131—21no—
32C2×#1131—21no—
33C2×#1131—22no—
34C2×#1131—21no—
35C2×#1131—22no—
36C2×C8×D7—21no—
37#6460—21no—
38#6458—21no—
39#6449—21no—
40#6445—21no—
41C4×D28—22no—
42C7×M32—22no—
43C7×M32D414yescomplemented
44C2×C7×C16—22no—
45C7×#97D412yes—
46C4×C7×C8D412yes—
47(C2×D7)⋊C8—22no—
48(C2×D7)⋊C8—22no—
49#6386—22no—
50#6386—22no—
51C2×C7⋊C16—22no—
52C4⋊(C7⋊C8)—22no—
53#2930—71no—
54#2153—71no—
55#2117—72no—
56C2×#349—72no—
57#2110D711yescharacteristic, complemented
58C23×D7M1611yescharacteristic
59C22×C4×C7C22×C411yescharacteristic
60C2×C22⋊D7—21no—
61C22×Dic7M1611yescharacteristic
62#1157—22no—
63#1157—22no—
64C2×D28—22no—
65C2×D28M1611yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—22no—
69C2×C4×D7—21no—
70C2×C4×D7—22no—
71C2×Dic14M1611yescharacteristic
72C7×M16—22no—
73C2×C7×C8C22⋊C412yes—
74C2×C7×C8C2×D411yescharacteristic
75C2×C7×C8C2×D412yes—
76C2×C7×C8C2×D411yescharacteristic
77C2×C7×C8#4112yes—
78C7×C4⋊C4—21no—
79C7×C22⋊C4—21no—
80C42×C7C22⋊C412yes—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—22no—
83C4⋊Dic7—22no—
84#1136—22no—
85C2×C7⋊C8—21no—
86C2×C7⋊C8—21no—
87C2×C7⋊C8—22no—
88#1131—42no—
89#1131—41no—
90#1131—41no—
91#1131—41no—
92#1131—41no—
93C8×D7—42no—
94C7×C16—24no—
95C7⋊C16—42no—
96C2×#131—71no—
97C22×M16—71no—
98#525—71no—
99C2×M32—72no—
100C2×M32C2×D712yes—
101#436—142no—
102#435—142no—
103#424—71no—
104#408—71no—
105#407—71no—
106#405C2×D711yescharacteristic
107#364—142no—
108#350—144no—
109#349—144no—
110#347—22no—
111C23×C7C2×M1611yescharacteristic
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—22no—
115C22×D7—81no—
116C2×C4×C7—22no—
117C2×C4×C7C2×C22⋊C411yescharacteristic
118C2×C4×C7C4×D412yes—
119C2×C4×C7C2×M1611yescharacteristic
120C2×C4×C7#13112yes—
121C22⋊D7—42no—
122C2×Dic7—21no—
123C2×Dic7—21no—
124C2×Dic7—22no—
125D28—22no—
126D28—82no—
127C4×D7—41no—
128C4×D7—41no—
129C4×D7—41no—
130C4×D7—41no—
131C4×D7—42no—
132Dic14—22no—
133C7×C8—22no—
134C7×C8#12112yes—
135C7×C8#12411yescharacteristic
136C7×C8#12711yescharacteristic
137C7⋊C8—41no—
138C7⋊C8—41no—
139C2×#41—71no—
140C23×C4—71no—
141#131—142no—
142#131—142no—
143C2×M16—71no—
144C2×M16—142no—
145C2×M16—141no—
146C2×M16—141no—
147C2×M16—142no—
148C2×M16—141no—
149C2×M16C4×D711yescharacteristic
150C22×C8—71no—
151C22×C8—141no—
152C22×C8C22×D711yescharacteristic
153#123—141no—
154C22⋊Q8—141no—
155#121—141no—
156#120—141no—
157C4×D4—142no—
158#117C4×D711yescharacteristic
159M32—22no—
160M32—142no—
161M32—142no—
162M32D2814yescomplemented
163C2×C16—22no—
164C2×C16—142no—
165C4⋊C8—142no—
166C22⋊C8—142no—
167C22⋊C8—142no—
168#97C22⋊D712yes—
169C4×C8D2812yes—
170C22×C7—41no—
171C22×C7#40711yescharacteristic
172C22×C7#43512yes—
173C2×D7—41no—
174C2×D7—42no—
175C2×D7—41no—
176C2×D7—41no—
177C2×D7—41no—
178C2×D7—82no—
179C4×C7—42no—
180C4×C7#39011yescharacteristic
181C4×C7#40811yescharacteristic
182C4×C7#43512yes—
183Dic7—41no—
184Dic7—41no—
185C24—71no—
186#41—142no—
187#41—142no—
188C2×Q8—71no—
189C2×D4—71no—
190C2×D4—142no—
191C2×D4—141no—
192C22×C4—71no—
193C22×C4—142no—
194C22×C4—141no—
195C22×C4—142no—
196C22×C4—141no—
197C22×C4—141no—
198C22×C4C2×C4×D711yescharacteristic
199M16—22no—
200M16—142no—
201M16—281no—
202M16—281no—
203M16—281no—
204M16—281no—
205M16—282no—
206C2×C8—142no—
207C2×C8—141no—
208C2×C8—141no—
209C2×C8—282no—
210C2×C8(C2×D7)⋊C412yes—
211C2×C8C2×D2811yescharacteristic
212C2×C8C2×D2812yes—
213C2×C8#115712yes—
214C2×C8C2×C22⋊D711yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—142no—
217C22⋊C4—21no—
218C22⋊C4—142no—
219C22⋊C4—142no—
220C42(C2×D7)⋊C412yes—
221C16—24no—
222C16—282no—
223C2×C7—21no—
224C2×C7—22no—
225C2×C7#195511yescharacteristic
226D7—81no—
227D7—81no—
228C23—141no—
229C23—141no—
230C23—142no—
231C23—561no—
232C23D7×M1611yescharacteristic
233Q8—142no—
234D4—142no—
235D4—282no—
236D4—562no—
237C2×C4—22no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—281no—
242C2×C4—281no—
243C2×C4—281no—
244C2×C4—281no—
245C2×C4—282no—
246C2×C4C4×D2812yes—
247C2×C4D7×M1611yescharacteristic
248C2×C4#647012yes—
249C2×C4C2×(C2×D7)⋊C411yescharacteristic
250C8—22no—
251C8—281no—
252C8—281no—
253C8#643711yescharacteristic
254C8#643811yescharacteristic
255C8#649312yes—
256C7#1320111yescharacteristic, complemented, Sylow
257C22—41no—
258C22—281no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22—281no—
263C22—562no—
264C22#8947812yes—
265C22#8989811yescharacteristic
266C4—42no—
267C4—281no—
268C4—281no—
269C4#8947812yes—
270C4#8971111yescharacteristic
271C4#8990111yescharacteristic
272C2—21no—
273C2—22no—
274C2—561no—
275C2—561no—
276C2#1172759111yescharacteristic
subgroupG/H
centerC4#89711
commutatorC22×C4×C7C22×C4
FrattiniC22×C8C22×D7
FittingC7×#2110C2
socleC2×C7#1955
prime radical#90215C4
cosocleC22×C7×C8C23
hypercenter#2110D7
hypocenterC7#13201
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. C7×#2110C2semidirect
3. #11729192C2semidirect
4. C2×#89309C2semidirect
5. #11727639C2semidirect
43. C7×M32D4semidirect
57. #2110D7semidirect
162. M32D28semidirect
256. C7#13201semidirect

Maximal subgroups

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

subgroupindex
#117359992
C7×#21102
#117291922
C2×#893092
#117276392
#132017