Finite Group Databasebeta

← #11737922

Subgroups — #11737922

Order 896.

480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#6481C212yesmaximal
2D7×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#1161C211yesmaximal, characteristic
4C7×#462C211yesmaximal, characteristic, complemented
5#89954C212yesmaximal, complemented
6#89855C211yesmaximal, characteristic, complemented
7#89803C211yesmaximal, characteristic, complemented
8#89613C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89590C212yesmaximal, complemented
11#89574C212yesmaximal, complemented
12C2×#1161C2211yescharacteristic
13C2×Q8×D7C2212yes—
14C2×C7×Q16C2212yes—
15C7×C4⋊Q8C2211yescharacteristic
16C4×C7×Q8C2211yescharacteristic
17Q8×Dic7C2211yescharacteristic
18#6507C2212yes—
19C2×#1144C2212yes—
20C2×#1143C2212yes—
21#6481—28no—
22C2×#1132C2212yes—
23C2×#1131C2212yes—
24#6456—21no—
25#6456C2211yescharacteristic
26#6455—21no—
27#6455C2211yescharacteristic
28D7×C4⋊C4—21no—
29D7×C4⋊C4—21no—
30#6451—21no—
31#6451—21no—
32C4×D28—21no—
33C4×D28C2211yescharacteristic
34C42×D7—21no—
35C42×D7C2211yescharacteristic
36#6432C2211yescharacteristic
37C7×C4⋊C8C2211yescharacteristic
38C7×#103C2212yes—
39#6409C2212yes—
40#6390C2212yes—
41Dic7⋊C8C2212yes—
42#6384C2212yes—
43C4⋊(C7⋊C8)C2211yescharacteristic
44#3039—71nomaximal, Sylow
45C2×C7×Q8C2312yes—
46C2×C7×Q8C2311yescharacteristic
47#1161—22no—
48#1161D414yes—
49Q8×D7—24no—
50Q8×D7—24no—
51C2×D28—21no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7—22no—
55C2×C4×D7—21no—
56C2×C4×D7—21no—
57C2×C4×D7D411yescharacteristic
58C2×C4×D7D411yescharacteristic
59C2×C4×D7C2311yescharacteristic
60C2×Dic14—22no—
61C2×Dic14C2312yes—
62C7×Q16—24no—
63C2×C7×C8C2312yes—
64C7×C4⋊C4—21no—
65C7×C4⋊C4—21no—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2311yescharacteristic
68C42×C7—21no—
69C42×C7C2311yescharacteristic
70#1144—24no—
71#1143—24no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—21no—
75C4⋊Dic7—21no—
76C4⋊Dic7—21no—
77C4⋊Dic7—21no—
78C4⋊Dic7C2311yescharacteristic
79Dic7⋊C4—22no—
80Dic7⋊C4—22no—
81C4×Dic7—21no—
82C4×Dic7—21no—
83C4×Dic7D411yescharacteristic
84C4×Dic7D411yescharacteristic
85C4×Dic7C2311yescharacteristic
86C2×C7⋊C8C2312yes—
87#1132—24no—
88#1131—24no—
89C2×#137—72no—
90C2×C4⋊Q8—71no—
91C4×#41—71no—
92#462—72no—
93#462—71no—
94#462D711yescharacteristic, complemented
95#461—71no—
96#461—71no—
97#461—72no—
98#423—71no—
99#419—72no—
100C22×D7—21no—
101C22×D7C2×D411yescharacteristic
102C7×Q8—21no—
103C7×Q8—22no—
104C7×Q8—22no—
105C7×Q8C2×D412yes—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2×D411yescharacteristic
111C2×C4×C7C2411yescharacteristic
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117D28—24no—
118D28—21no—
119D28C2×D412yes—
120C4×D7—22no—
121C4×D7—24no—
122C4×D7—22no—
123C4×D7—44no—
124C4×D7—42no—
125C4×D7—42no—
126C4×D7C2×D412yes—
127C4×D7#4112yes—
128Dic14—22no—
129Dic14—22no—
130Dic14—44no—
131C7×C8—22no—
132C7⋊C8—22no—
133C2×#41—71no—
134C22×Q8—72no—
135#137—148no—
136C2×Q16—72no—
137C2×Q16C2×D712yes—
138C2×SD16—72no—
139C2×SD16—72no—
140C2×M16—72no—
141C4⋊Q8—72no—
142C4⋊Q8—71no—
143C4⋊Q8—141no—
144C4⋊Q8—141no—
145C4⋊Q8C2×D711yescharacteristic
146C4×Q8—71no—
147C4×Q8C2×D711yescharacteristic
148C4×D4—71no—
149C4×D4—71no—
150C4×D4—141no—
151C4×D4—141no—
152#117—71no—
153#117—141no—
154C2×C4⋊C4—141no—
155C2×C4⋊C4—141no—
156C2×C42—71no—
157C2×C42—141no—
158C4⋊C8—72no—
159C4⋊C8—71no—
160C4⋊C8C2×D711yescharacteristic
161#103—72no—
162#103—72no—
163#103—72no—
164#103C2×D712yes—
165C22×C7C22×D411yescharacteristic
166C2×D7—21no—
167C2×D7—21no—
168C2×D7—22no—
169C2×D7—41no—
170C2×D7—41no—
171C2×D7#12112yes—
172C4×C7—22no—
173C4×C7—21no—
174C4×C7—42no—
175C4×C7—41no—
176C4×C7#13712yescomplemented
177C4×C7C22×D411yescharacteristic
178C4×C7C2×#4111yescharacteristic
179Dic7—21no—
180Dic7—42no—
181Dic7—41no—
182Dic7#12112yes—
183Dic7#13712yescomplemented
184#41—74no—
185#41—142no—
186C2×Q8—72no—
187C2×Q8—142no—
188C2×Q8—144no—
189C2×Q8—144no—
190C2×Q8C22×D712yes—
191C2×Q8C22×D711yescharacteristic
192C2×D4—71no—
193C2×D4—141no—
194C22×C4—71no—
195C22×C4—71no—
196C22×C4—71no—
197C22×C4—141no—
198C22×C4—141no—
199C22×C4—142no—
200C22×C4—141no—
201Q16—24no—
202Q16—144no—
203SD16—144no—
204SD16—144no—
205M16—144no—
206C2×C8—72no—
207C2×C8C22×D712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—71no—
212C4⋊C4—142no—
213C4⋊C4—142no—
214C4⋊C4—141no—
215C4⋊C4—141no—
216C4⋊C4—141no—
217C4⋊C4C22×D711yescharacteristic
218C22⋊C4—141no—
219C22⋊C4—141no—
220C22⋊C4—141no—
221C42—21no—
222C42—71no—
223C42—71no—
224C42—71no—
225C42—141no—
226C42—141no—
227C42C22×D711yescharacteristic
228C2×C7C2×#12111yescharacteristic
229C2×C7C2×#13711yescharacteristic
230C2×C7C2×#13711yescharacteristic
231D7—22no—
232D7—42no—
233C23—71no—
234C23—141no—
235Q8—21no—
236Q8—22no—
237Q8—22no—
238Q8—142no—
239Q8—142no—
240Q8—284no—
241Q8D4×D712yes—
242D4—72no—
243D4—144no—
244D4—141no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—71no—
249C2×C4—71no—
250C2×C4—72no—
251C2×C4—72no—
252C2×C4—71no—
253C2×C4—142no—
254C2×C4—144no—
255C2×C4—142no—
256C2×C4—141no—
257C2×C4—142no—
258C2×C4—282no—
259C2×C4—284no—
260C2×C4—282no—
261C2×C4D4×D711yescharacteristic
262C2×C4D4×D711yescharacteristic
263C2×C4C23×D711yescharacteristic
264C8—22no—
265C8—142no—
266C7#303911yescharacteristic, complemented, Sylow
267C22—72no—
268C22—141no—
269C22—141no—
270C22—142no—
271C22—281no—
272C22—281no—
273C22C2×D4×D711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—42no—
278C4—72no—
279C4—72no—
280C4—141no—
281C4—281no—
282C4—282no—
283C4#648112yescomplemented
284C4C2×D4×D711yescharacteristic
285C4D7×#4111yescharacteristic
286C2—142no—
287C2—282no—
288C2D7×#12111yescharacteristic
289C2C2×#648111yescharacteristic
290C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#462D7
hypocenterC7#3039
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊Q8C2semidirect
4. C7×#462C2semidirect
5. #89954C2semidirect
6. #89855C2semidirect
7. #89803C2semidirect
8. #89613C2semidirect
9. #89609C2semidirect
10. #89590C2semidirect
11. #89574C2semidirect
94. #462D7semidirect
176. C4×C7#137semidirect
183. Dic7#137semidirect
266. C7#3039semidirect
283. C4#6481semidirect

Maximal subgroups

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

subgroupindex
C2×#64812
D7×C4⋊Q82
C4×#11612
C7×#4622
#899542
#898552
#898032
#896132
#896092
#895902
#895742
#30397