Finite Group Databasebeta

← #13272001

Subgroups — #13272001

Order 1792.

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

normal subgroups only

This group has 438 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#11735998C211yesmaximal, characteristic
2C7×#2110C211yesmaximal, characteristic, complemented
3#11729191C212yesmaximal, complemented
4#11729185C212yesmaximal, complemented
5#11727639C211yesmaximal, characteristic, complemented
6C2×#6464C2211yescharacteristic
7C22×C8×D7C2211yescharacteristic
8#90215C411yescharacteristic
9C2×C7×M32C2212yes—
10C7×#405C2211yescharacteristic
11#89901C411yescharacteristic
12#89898C411yescharacteristic
13C2×#6386C2212yes—
14#89771C411yescharacteristic
15#89478—22no—
16C8×D28—22no—
17C7×#347—22no—
18#89303—24no—
19(C2×D7)⋊C16—24no—
20#89258—22no—
21#13200—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×C8×D7—21no—
33C2×C8×D7—21no—
34C2×C8×D7—22no—
35C2×C8×D7—21no—
36C2×C8×D7—22no—
37#6460—21no—
38#6458—21no—
39#6449—21no—
40#6445—21no—
41C4×D28—22no—
42C7×M32—22no—
43C7×M32—22no—
44C2×C7×C16D414yescomplemented
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#2929—71no—
54#2153—71no—
55#2117—72no—
56#2112—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—
89C8×D7—42no—
90C8×D7—41no—
91C8×D7—41no—
92C8×D7—41no—
93C8×D7—41no—
94C7×C16—24no—
95C7⋊C16—42no—
96C2×#131—71no—
97C23×C8—71no—
98#525—71no—
99C2×M32—72no—
100C2×M32C2×D712yes—
101#435—142no—
102C8×D4—142no—
103#424—71no—
104#408—71no—
105#407—71no—
106#405C2×D711yescharacteristic
107#364—142no—
108#350—144no—
109C22⋊C16—144no—
110#347—22no—
111C23×C7C2×M1611yescharacteristic
112C22×D7—21no—
113C22×D7—22no—
114C22×D7—21no—
115C22×D7—42no—
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—44no—
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—141no—
145C2×M16C4×D711yescharacteristic
146C22×C8—71no—
147C22×C8—142no—
148C22×C8—141no—
149C22×C8—141no—
150C22×C8—142no—
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—22no—
161M32—142no—
162M32—142no—
163C2×C16—142no—
164C2×C16D2814yescomplemented
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—22no—
174C2×D7—24no—
175C2×D7—41no—
176C2×D7—42no—
177C2×D7—41no—
178C2×D7—42no—
179C2×D7—41no—
180C4×C7—42no—
181C4×C7#39011yescharacteristic
182C4×C7#40811yescharacteristic
183C4×C7#43512yes—
184Dic7—41no—
185Dic7—41no—
186C24—71no—
187#41—142no—
188#41—142no—
189C2×Q8—71no—
190C2×D4—71no—
191C2×D4—142no—
192C2×D4—141no—
193C22×C4—71no—
194C22×C4—142no—
195C22×C4—141no—
196C22×C4—142no—
197C22×C4—141no—
198C22×C4—141no—
199C22×C4C2×C4×D711yescharacteristic
200M16—22no—
201M16—142no—
202M16—282no—
203C2×C8—142no—
204C2×C8—141no—
205C2×C8—141no—
206C2×C8—281no—
207C2×C8—281no—
208C2×C8—281no—
209C2×C8—281no—
210C2×C8—282no—
211C2×C8(C2×D7)⋊C412yes—
212C2×C8C2×D2811yescharacteristic
213C2×C8C2×D2812yes—
214C2×C8#115712yes—
215C2×C8C2×C22⋊D711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—142no—
218C22⋊C4—21no—
219C22⋊C4—142no—
220C22⋊C4—142no—
221C42(C2×D7)⋊C412yes—
222C16—24no—
223C16—282no—
224C2×C7—21no—
225C2×C7—22no—
226C2×C7#195511yescharacteristic
227D7—42no—
228D7—81no—
229C23—141no—
230C23—142no—
231C23—141no—
232C23—282no—
233C23D7×M1611yescharacteristic
234Q8—142no—
235D4—142no—
236D4—284no—
237D4—282no—
238C2×C4—22no—
239C2×C4—141no—
240C2×C4—141no—
241C2×C4—142no—
242C2×C4—281no—
243C2×C4—281no—
244C2×C4—281no—
245C2×C4—281no—
246C2×C4—282no—
247C2×C4C4×D2812yes—
248C2×C4D7×M1611yescharacteristic
249C2×C4#647012yes—
250C2×C4C2×(C2×D7)⋊C411yescharacteristic
251C8—22no—
252C8—281no—
253C8—281no—
254C8#643711yescharacteristic
255C8#643811yescharacteristic
256C8#649312yes—
257C7#1320011yescharacteristic, complemented, Sylow
258C22—41no—
259C22—142no—
260C22—144no—
261C22—281no—
262C22—282no—
263C22—282no—
264C22—281no—
265C22—281no—
266C22#8947812yes—
267C22#8989811yescharacteristic
268C4—42no—
269C4—281no—
270C4—281no—
271C4#8947812yes—
272C4#8971111yescharacteristic
273C4#8990111yescharacteristic
274C2—21no—
275C2—22no—
276C2—282no—
277C2—561no—
278C2#1172759111yescharacteristic
subgroupG/H
centerC8#6437
commutatorC22×C4×C7C22×C4
FrattiniC22×C8C22×D7
FittingC7×#2110C2
socleC2×C7#1955
prime radical#90215C4
cosocleC22×C7×C8C23
hypercenter#2110D7
hypocenterC7#13200
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. C7×#2110C2semidirect
3. #11729191C2semidirect
4. #11729185C2semidirect
5. #11727639C2semidirect
44. C2×C7×C16D4semidirect
57. #2110D7semidirect
164. C2×C16D28semidirect
257. C7#13200semidirect

Maximal subgroups

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

subgroupindex
#117359982
C7×#21102
#117291912
#117291852
#117276392
#132007