Finite Group Databasebeta

← #13263077

Subgroups — #13263077

Order 1792.

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

normal subgroups only

This group has 428 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#11731765C211yesmaximal, characteristic
2#11725879C212yesmaximal
3C7×#1325C211yesmaximal, characteristic
4#11725441C211yesmaximal, characteristic
5#11725261C212yesmaximal
6C2×#6450C411yescharacteristic
7C2×#6450C2211yescharacteristic
8C2×#6432C411yescharacteristic
9C7×#384C2211yescharacteristic
10C4⋊C4×Dic7—22no—
11#89746C411yescharacteristic
12#89746C2211yescharacteristic
13#89745—22no—
14C42×Dic7C411yescharacteristic
15#89490—22no—
16C7×#324C2212yes—
17#89323C2212yes—
18#89264—22no—
19#11478—71nomaximal, Sylow
20C22×Dic14C2×C411yescharacteristic
21C22×Dic14C2×C411yescharacteristic
22C2×C7×C4⋊C4—22no—
23C2×C7×C4⋊C4C2×C411yescharacteristic
24C2×C7×C4⋊C4C2311yescharacteristic
25C2×C42×C7C2×C411yescharacteristic
26C2×C4⋊Dic7—22no—
27C2×Dic7⋊C4—21no—
28C2×Dic7⋊C4D412yes—
29C2×C4×Dic7—21no—
30C2×C4×Dic7—21no—
31C2×C4×Dic7—21no—
32C2×C4×Dic7C2×C411yescharacteristic
33C2×C4×Dic7C2×C411yescharacteristic
34C2×C4×Dic7D412yes—
35#6450—41no—
36#6450—41no—
37#6450—41no—
38#6450—41no—
39#6432—22no—
40#6432C814yescomplemented
41C7×C22⋊C8—22no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C22⋊(C7⋊C8)—22no—
45Dic7⋊C8—42no—
46C8×Dic7—42no—
47C4×C4⋊Q8—71no—
48#1461—72no—
49#1325—71no—
50#1325D711yescharacteristic
51#1319—72no—
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7C22×C411yescharacteristic
55C22×C4×C7C2×D412yes—
56C22×Dic7—21no—
57C22×Dic7—21no—
58C22×Dic7C22⋊C412yes—
59C22×Dic7C22⋊C411yescharacteristic
60C22×Dic7C22⋊C411yescharacteristic
61C2×Dic14—21no—
62C2×Dic14—41no—
63C2×Dic14—41no—
64C2×Dic14—41no—
65C2×Dic14—41no—
66C2×Dic14C2×C812yes—
67C2×Dic14C2×C812yes—
68C2×Dic14M1612yes—
69C2×C7×C8—42no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—44no—
73C42×C7—21no—
74C42×C7C2×C812yes—
75C4⋊Dic7—28no—
76Dic7⋊C4—42no—
77Dic7⋊C4—42no—
78C4×Dic7—24no—
79C4×Dic7—21no—
80C4×Dic7—21no—
81C4×Dic7—22no—
82C4×Dic7—22no—
83C4×Dic7—42no—
84C4×Dic7—41no—
85C4×Dic7—41no—
86C4×Dic7—41no—
87C2×C7⋊C8—42no—
88C2×C4⋊Q8—71no—
89C2×C4×Q8—71no—
90C2×C4×Q8—71no—
91#431—142no—
92#391—71no—
93#391—71no—
94#389—142no—
95#384C2×D711yescharacteristic
96C4×C4⋊C4—142no—
97C43—71no—
98#324—72no—
99#324—142no—
100#324C2×D712yes—
101C23×C7C2×C22⋊C411yescharacteristic
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—41no—
111C2×C4×C7C22×C811yescharacteristic
112C2×C4×C7C2×M1611yescharacteristic
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—21no—
119C2×Dic7—41no—
120C2×Dic7—41no—
121C2×Dic7—41no—
122C2×Dic7—41no—
123C2×Dic7—41no—
124C2×Dic7C22⋊C814yes—
125C2×Dic7#9911yescharacteristic
126C2×Dic7#9911yescharacteristic
127Dic14—42no—
128Dic14—42no—
129Dic14—42no—
130Dic14—42no—
131C7×C8—82no—
132C7⋊C8—82no—
133C22×Q8—71no—
134C22×Q8—71no—
135C4⋊Q8—74no—
136C4⋊Q8—142no—
137C4×Q8—281no—
138C4×Q8—281no—
139C4×Q8—281no—
140C4×Q8—281no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—72no—
143C2×C4⋊C4—142no—
144C2×C4⋊C4—141no—
145C2×C4⋊C4C4×D711yescharacteristic
146C2×C4⋊C4C22×D711yescharacteristic
147C2×C42—71no—
148C2×C42—72no—
149C2×C42—71no—
150C2×C42—141no—
151C2×C42—141no—
152C2×C42—141no—
153C2×C42C4×D711yescharacteristic
154C4⋊C8—282no—
155C22⋊C8—22no—
156C22⋊C8—142no—
157C4×C8—282no—
158#95—142no—
159#95—142no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7C2×C22⋊C811yescharacteristic
163C22×C7C2×#9911yescharacteristic
164C22×C7#41311yescharacteristic
165C4×C7—24no—
166C4×C7—21no—
167C4×C7—21no—
168C4×C7—42no—
169C4×C7—81no—
170Dic7—21no—
171Dic7—24no—
172Dic7—42no—
173Dic7—41no—
174Dic7—42no—
175Dic7#35612yes—
176C2×Q8—72no—
177C2×Q8—72no—
178C2×Q8—72no—
179C2×Q8—141no—
180C2×Q8—281no—
181C2×Q8—281no—
182C2×Q8—281no—
183C2×Q8—281no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—72no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—141no—
190C22×C4—141no—
191C22×C4C2×C4×D711yescharacteristic
192C22×C4D4×D712yes—
193C2×C8—42no—
194C2×C8—282no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—148no—
199C4⋊C4—282no—
200C4⋊C4—282no—
201C42—21no—
202C42—142no—
203C42—142no—
204C42—141no—
205C42—141no—
206C42—144no—
207C42—281no—
208C42—282no—
209C42—281no—
210C42—281no—
211C42C8×D712yes—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#145911yescharacteristic
215C2×C7#212611yescharacteristic
216C2×C7C2×#35611yescharacteristic
217C23D7×C22⋊C411yescharacteristic
218Q8—282no—
219Q8—282no—
220Q8—282no—
221Q8—282no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—71no—
232C2×C4—74no—
233C2×C4—71no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4—141no—
240C2×C4—281no—
241C2×C4—281no—
242C2×C4—281no—
243C2×C4—281no—
244C2×C4—281no—
245C2×C4C2×C8×D711yescharacteristic
246C2×C4D7×M1611yescharacteristic
247C8—82no—
248C8—562no—
249C7#1147811yescharacteristic, complemented, Sylow
250C22—21no—
251C22—21no—
252C22D7×C22⋊C811yescharacteristic
253C22D7×#9911yescharacteristic
254C22#8952311yescharacteristic
255C4—24no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—72no—
260C4—81no—
261C4—144no—
262C4—141no—
263C4—282no—
264C4—281no—
265C4—282no—
266C2—21no—
267C2—21no—
268C2#1172589711yescharacteristic
269C2#1172690511yescharacteristic
270C2D7×#35611yescharacteristic
subgroupG/H
centerC22D7×#99
commutatorC2×C4×C7C22×C8
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1325C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1325D7
hypocenterC7#11478
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
40. #6432C8semidirect
249. C7#11478semidirect

Maximal subgroups

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

subgroupindex
#117317652
#117258792
C7×#13252
#117254412
#117252612
#114787