Finite Group Databasebeta

← #13254738

Subgroups — #13254738

Order 1792.

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

normal subgroups only

This group has 512 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×#3245C211yesmaximal, characteristic
2#11740761C212yesmaximal, complemented
3#11729552C212yesmaximal, complemented
4#11728227C211yesmaximal, characteristic, complemented
5#11725514C211yesmaximal, characteristic, complemented
6C7×#532C2212yes—
7C2×C4×Dic14C2211yescharacteristic
8C7×#474—22no—
9C7×#466—22no—
10C7×#459—22no—
11C2×C7×C4⋊C8C2211yescharacteristic
12#89863—22no—
13#89847—22no—
14#89812—22no—
15#89805—22no—
16#89790—24no—
17#89698C2212yes—
18C2×#6377C2211yescharacteristic
19#89345—22no—
20#89335—24no—
21#10028—71nomaximal, Sylow
22C2×C7×#41—22no—
23C22×Dic14D411yescharacteristic
24C2×C7×D8—22no—
25C22×C7×C8D412yes—
26C7×C4⋊Q8D412yes—
27C7×#127D412yes—
28C7×#124—22no—
29C7×#121—22no—
30C7×#121—22no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C2×C42×C7C2311yescharacteristic
34C2×#1142—24no—
35C2×C4⋊Dic7D411yescharacteristic
36C2×Dic7⋊C4—21no—
37C2×C4×Dic7—21no—
38C22×C7⋊C8—21no—
39C4×Dic14—22no—
40C4×Dic14—22no—
41C7×#107—22no—
42C7×C4⋊C8—22no—
43C7×#102—24no—
44#6406—24no—
45C22⋊(C7⋊C8)—22no—
46#6384—24no—
47#6381—24no—
48C4⋊(C7⋊C8)—22no—
49#6377—22no—
50#6377D414yes—
51#3245D711yescharacteristic
52#3112—72no—
53#1985—71no—
54#1530—72no—
55#1290—71no—
56C7×#41—84no—
57C2×C7×Q8—22no—
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—42no—
62C22×C4×C7—42no—
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7#4111yescharacteristic
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×Dic14—22no—
69C2×Dic14—42no—
70C7×D8—82no—
71C2×C7×C8—22no—
72C2×C7×C8—42no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C22⋊C4—42no—
76C7×C22⋊C4—42no—
77C42×C7C2×D412yes—
78C42×C7C2×D412yes—
79#1142—48no—
80C4⋊Dic7—22no—
81Dic7⋊C4—42no—
82C4×Dic7—41no—
83C4×Dic7—41no—
84C2×C7⋊C8—22no—
85C2×C7⋊C8—24no—
86C2×C7⋊C8—41no—
87C2×C7⋊C8—41no—
88#532C2×D712yes—
89C2×C4×Q8—71no—
90#480—142no—
91#477—142no—
92#474—22no—
93#466—22no—
94#460—142no—
95#459—22no—
96#448—142no—
97#440—144no—
98#432—72no—
99C2×C4⋊C8C2×D711yescharacteristic
100C2×#97—71no—
101#336—142no—
102#329—144no—
103C23×C7—42no—
104C23×C7#12111yescharacteristic
105C7×Q8—82no—
106C7×D4—44no—
107C7×D4—42no—
108C7×D4—42no—
109C7×D4—82no—
110C7×D4—84no—
111C2×C4×C7—22no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7—84no—
117C2×C4×C7#12012yes—
118C2×C4×C7#12011yescharacteristic
119C2×C4×C7#12112yes—
120C2×C4×C7#12711yescharacteristic
121C2×Dic7—22no—
122C2×Dic7—41no—
123C2×Dic7—41no—
124C2×Dic7—41no—
125C2×Dic7—42no—
126Dic14—24no—
127Dic14—42no—
128C7×C8—42no—
129C7⋊C8—42no—
130C7⋊C8—44no—
131C2×#41—22no—
132C22×Q8—71no—
133C2×SD16—144no—
134C2×D8—22no—
135C22×C8—141no—
136C22×C8C22⋊D712yes—
137C4⋊Q8C22⋊D712yes—
138#127C22⋊D712yes—
139#124—22no—
140#121—22no—
141#121—22no—
142C4×Q8—142no—
143C4×Q8—142no—
144C4×D4—22no—
145C4×D4—22no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4—141no—
148C2×C42—141no—
149C2×C42C22×D711yescharacteristic
150#107—22no—
151#107—144no—
152C4⋊C8—22no—
153C4⋊C8—142no—
154#103—144no—
155#102—24no—
156#102—144no—
157C22⋊C8—142no—
158#97—74no—
159#97—142no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7—82no—
163C22×C7—82no—
164C22×C7—82no—
165C22×C7#39211yescharacteristic
166C22×C7#47212yes—
167C4×C7—21no—
168C4×C7—22no—
169C4×C7—22no—
170C4×C7—82no—
171C4×C7#45511yescharacteristic
172C4×C7#45611yescharacteristic
173Dic7—42no—
174Dic7—81no—
175#41—84no—
176C2×Q8—22no—
177C2×Q8—142no—
178C2×Q8—282no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—42no—
183C22×C4—42no—
184C22×C4—141no—
185C22×C4—141no—
186C22×C4D4×D711yescharacteristic
187C22×C4#115911yescharacteristic
188C22×C4C2×C22⋊D711yescharacteristic
189SD16—288no—
190D8—82no—
191C2×C8—22no—
192C2×C8—42no—
193C2×C8—142no—
194C2×C8—144no—
195C2×C8—281no—
196C2×C8—281no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—142no—
200C4⋊C4—282no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—281no—
204C42—281no—
205C42D4×D712yes—
206C42C2×C22⋊D712yes—
207C2×C7—22no—
208C2×C7—162no—
209C2×C7#200711yescharacteristic
210C2×C7#201011yescharacteristic
211C2×C7#201811yescharacteristic
212C23—42no—
213C23#650111yescharacteristic
214Q8—82no—
215Q8—144no—
216Q8—282no—
217D4—44no—
218D4—42no—
219D4—42no—
220D4—82no—
221D4—84no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4—142no—
229C2×C4—281no—
230C2×C4—281no—
231C2×C4—282no—
232C2×C4—281no—
233C2×C4#650012yes—
234C2×C4#650212yes—
235C2×C4#650311yescharacteristic
236C2×C4#651611yescharacteristic
237C8—42no—
238C8—282no—
239C8—284no—
240C7#1002811yescharacteristic, complemented, Sylow
241C22—21no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#8993212yes—
247C22#8999511yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—82no—
252C4—282no—
253C4—561no—
254C4#9001411yescharacteristic
255C4#9001811yescharacteristic
256C2—22no—
257C2—162no—
258C2#1172963511yescharacteristic
259C2#1172980011yescharacteristic
260C2#1172987211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#3245C2
socleC22×C7#392
hypercenter#3245D7
hypocenterC7#10028
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
2. #11740761C2semidirect
3. #11729552C2semidirect
4. #11728227C2semidirect
5. #11725514C2semidirect
240. C7#10028semidirect

Maximal subgroups

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

subgroupindex
C7×#32452
#117407612
#117295522
#117282272
#117255142
#100287