Finite Group Databasebeta

← #13246709

Subgroups — #13246709

Order 1792.

460 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 105 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×#89783C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11727736C212yesmaximal, complemented
5#11725493C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C22×#1136C411yescharacteristic
8C23×C7⋊C8C2211yescharacteristic
9C7×#407C2212yes—
10#89979C2212yes—
11C2×C22⋊(C7⋊C8)C411yescharacteristic
12C2×C22⋊(C7⋊C8)C2211yescharacteristic
13C2×#6396—22no—
14#89783—22no—
15#89783—22no—
16C2×C4⋊(C7⋊C8)C411yescharacteristic
17C2×#6377C411yescharacteristic
18C7×#324—21no—
19C7×#323—21no—
20#89353—22no—
21#89323—21no—
22#89320—21no—
23#8613—71nomaximal, Sylow
24C22×C7×D4Q811yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C2×C7×M16—22no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4Q811yescharacteristic
33C2×C42×C7C2×C411yescharacteristic
34C2×#1136—21no—
35C2×#1136—21no—
36C2×#1136—22no—
37C2×#1136—42no—
38C22×C7⋊C8—42no—
39C22×C7⋊C8C2×C411yescharacteristic
40C22×C7⋊C8C2×C411yescharacteristic
41C22×C7⋊C8C2×C411yescharacteristic
42C22×C7⋊C8C2×C411yescharacteristic
43C22×C7⋊C8D412yes—
44C22×C7⋊C8D412yes—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51#6396—44no—
52C4⋊(C7⋊C8)—22no—
53#6377—21no—
54#6377—21no—
55C2×#435—71no—
56#1810—72no—
57#1465—71no—
58#1465D711yescharacteristic
59#1293—72no—
60C24×C7C4⋊C411yescharacteristic
61C24×C7C2×Q811yescharacteristic
62C2×C7×D4—22no—
63C2×C7×D4—42no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7C4⋊C411yescharacteristic
71C22×C4×C7C4⋊C411yescharacteristic
72C22×C4×C7C4⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7#4112yes—
77C7×M16—42no—
78C7×M16—42no—
79C2×C7×C8—42no—
80C7×C4⋊C4—22no—
81C7×C22⋊C4—22no—
82C7×C22⋊C4—22no—
83C42×C7—21no—
84C42×C7—21no—
85#1136—42no—
86#1136—42no—
87#1136—42no—
88#1136—42no—
89C2×C7⋊C8—22no—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—21no—
92C2×C7⋊C8—21no—
93C2×C7⋊C8—41no—
94C2×C7⋊C8—42no—
95C2×C7⋊C8—41no—
96C2×C7⋊C8—41no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8C22⋊C412yes—
99C2×C7⋊C8C22⋊C411yescharacteristic
100C2×C7⋊C8C22⋊C411yescharacteristic
101C22×M16—71no—
102C23×C8—71no—
103C2×C4×D4C2×D711yescharacteristic
104#435—142no—
105#435—142no—
106C2×#108—142no—
107C2×C4⋊C8—71no—
108#407—72no—
109#407C2×D712yes—
110C2×C22⋊C8—71no—
111C2×C22⋊C8—71no—
112C2×#97—71no—
113#341—142no—
114#324—21no—
115#324—141no—
116#323—21no—
117#323—141no—
118C23×C7—22no—
119C23×C7—21no—
120C23×C7—22no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7C2×C4⋊C411yescharacteristic
126C23×C7C22⋊Q812yes—
127C7×D4—44no—
128C2×C4×C7—21no—
129C2×C4×C7—22no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7C2×C22⋊C411yescharacteristic
144C2×C4×C7#11711yescharacteristic
145C2×C4×C7#12112yes—
146C7×C8—82no—
147C7⋊C8—22no—
148C7⋊C8—21no—
149C7⋊C8—41no—
150C7⋊C8—41no—
151C7⋊C8—82no—
152C7⋊C8#9912yes—
153C22×D4Dic1411yescharacteristic
154C23×C4C4×D711yescharacteristic
155C23×C4C22×D711yescharacteristic
156C2×M16—22no—
157C2×M16—141no—
158C2×M16—141no—
159C2×M16—142no—
160C2×M16—282no—
161C22×C8—71no—
162C22×C8—72no—
163C22×C8—71no—
164C22×C8—72no—
165C22×C8—71no—
166C22×C8—71no—
167C22×C8—282no—
168C4×D4—22no—
169C4×D4—22no—
170C2×C4⋊C4C22⋊D711yescharacteristic
171C2×C22⋊C4Dic1411yescharacteristic
172C2×C22⋊C4C22⋊D711yescharacteristic
173C2×C42C4×D711yescharacteristic
174#108—284no—
175C4⋊C8—142no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—142no—
179C22⋊C8—142no—
180C22⋊C8—142no—
181C22⋊C8—142no—
182#97—141no—
183#97—141no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—22no—
188C22×C7—22no—
189C22×C7—44no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—82no—
194C22×C7#38011yescharacteristic
195C22×C7#43012yes—
196C4×C7—22no—
197C4×C7—21no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7C2×#9911yescharacteristic
202C4×C7#41011yescharacteristic
203C24Dic7⋊C411yescharacteristic
204C24C2×Dic1411yescharacteristic
205C2×D4—22no—
206C2×D4—42no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4Dic7⋊C411yescharacteristic
214C22×C4Dic7⋊C411yescharacteristic
215C22×C4Dic7⋊C411yescharacteristic
216C22×C4C2×C4×D711yescharacteristic
217C22×C4D4×D712yes—
218C22×C4#115912yes—
219C22×C4C2×C22⋊D711yescharacteristic
220M16—42no—
221M16—42no—
222M16—282no—
223M16—282no—
224M16—282no—
225M16—282no—
226C2×C8—42no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8—72no—
230C2×C8—144no—
231C2×C8—142no—
232C2×C8—141no—
233C2×C8—141no—
234C2×C8—281no—
235C2×C8—282no—
236C2×C8—281no—
237C2×C8—281no—
238C2×C8—282no—
239C4⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C42—21no—
243C42—21no—
244C2×C7—22no—
245C2×C7—42no—
246C2×C7—81no—
247C2×C7#179911yescharacteristic
248C2×C7#181111yescharacteristic
249C2×C7#181311yescharacteristic
250C23—22no—
251C23—21no—
252C23—22no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#644112yes—
258C23C2×Dic7⋊C411yescharacteristic
259D4—44no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4#644011yescharacteristic
276C2×C4D7×C22⋊C411yescharacteristic
277C2×C4#650212yes—
278C8—72no—
279C8—82no—
280C8—142no—
281C8—141no—
282C8—281no—
283C8—281no—
284C8—562no—
285C7#861311yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—44no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—82no—
296C22#8971411yescharacteristic
297C22#8989612yes—
298C4—22no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4#8951211yescharacteristic
304C4D7×#9911yescharacteristic
305C2—22no—
306C2—42no—
307C2—81no—
308C2#1172773711yescharacteristic
309C2#1172778211yescharacteristic
310C2#1172788811yescharacteristic
subgroupG/H
centerC22#89714
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1465C2
socleC22×C7#380
prime radicalC22×C7×D4Q8
cosocleC23×C4×C7C23
hypercenter#1465D7
hypocenterC7#8613
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. #11727736C2semidirect
5. #11725493C2semidirect
285. C7#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#897832
C7×#14652
#117293562
#117277362
#117254932
#86137