Finite Group Databasebeta

← #13251188

Subgroups — #13251188

Order 1792.

514 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 109 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#11740705C211yesmaximal, characteristic
2#11736733C211yesmaximal, characteristic
3C7×#1548C211yesmaximal, characteristic, complemented
4#11729528C211yesmaximal, characteristic, complemented
5#11728281C211yesmaximal, characteristic
6#11725552C211yesmaximal, characteristic, complemented
7#11725521C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×D28C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12#90003C414yes—
13C2×#6397—21no—
14#89760—21no—
15#89759—22no—
16#89749—21no—
17C4×C4⋊Dic7C2211yescharacteristic
18#89697C2211yescharacteristic
19#89696C2211yescharacteristic
20#89610—22no—
21#89498—22no—
22#89460—22no—
23C7×#333—22no—
24#89349—21no—
25#89341—22no—
26#9399—71nomaximal, Sylow
27C22×C7×Q8C2×C412yes—
28C22×D28D411yescharacteristic
29C22×C4×D7D412yes—
30C2×C7×M16—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C42×C7C2311yescharacteristic
38C2×(C2×D7)⋊C4—24no—
39C2×(C2×D7)⋊C4C2×C412yes—
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7D411yescharacteristic
42C2×C4×Dic7—21no—
43C2×#1136—21no—
44C2×#1131—24no—
45C4×D28—21no—
46C4×D28—21no—
47C4×D28—21no—
48C4×D28—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8D412yes—
52C7×C22⋊C8—21no—
53C4×C7×C8—21no—
54C7⋊#95—24no—
55C7⋊#95C2×C412yes—
56C22⋊(C7⋊C8)—21no—
57#6397—42no—
58(C2×D7)⋊C8—24no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)D412yes—
62C4×C7⋊C8—21no—
63#2975—71no—
64#2563—71no—
65#1800—71no—
66#1548—71no—
67#1548D711yescharacteristic, complemented
68#1303—71no—
69#1303—71no—
70C23×D7C22⋊C412yes—
71C2×C7×Q8—22no—
72C2×C7×Q8—22no—
73C2×C7×Q8—82no—
74C22×C4×C7—22no—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×Dic7—22no—
79C22×Dic7C22⋊C412yes—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28—21no—
83C2×D28—21no—
84C2×D28—21no—
85C2×D28—21no—
86C2×C4×D7—22no—
87C2×C4×D7—22no—
88C2×C4×D7—44no—
89C7×M16—41no—
90C7×M16—41no—
91C2×C7×C8—22no—
92C2×C7×C8—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C42×C7C2×D411yescharacteristic
98C42×C7C2×D411yescharacteristic
99C42×C7#4111yescharacteristic
100C42×C7#4111yescharacteristic
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—84no—
104C4⋊Dic7—21no—
105C4⋊Dic7—21no—
106C4⋊Dic7—42no—
107C4×Dic7—41no—
108C4×Dic7—41no—
109#1136—41no—
110#1136—41no—
111C2×C7⋊C8—22no—
112C2×C7⋊C8—41no—
113#1131—84no—
114C2×C4⋊Q8C2×D711yescharacteristic
115C2×C4×D4—71no—
116#436—142no—
117#436—142no—
118#431—71no—
119#431C2×D711yescharacteristic
120#423—71no—
121#423C2×D711yescharacteristic
122C2×#100—141no—
123#407—142no—
124#393—74no—
125#390—141no—
126#382—141no—
127#380—142no—
128C4×C4⋊C4—71no—
129#343—141no—
130#333—22no—
131#333—142no—
132C23×C7C2×C22⋊C411yescharacteristic
133C22×D7—22no—
134C22×D7—22no—
135C22×D7—22no—
136C22×D7—42no—
137C22×D7—42no—
138C7×Q8—82no—
139C7×Q8—82no—
140C2×C4×C7—22no—
141C2×C4×C7—41no—
142C2×C4×C7—44no—
143C2×C4×C7—42no—
144C2×C4×C7C4×D411yescharacteristic
145C2×C4×C7C4×D411yescharacteristic
146C2×C4×C7#12011yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12311yescharacteristic
150C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C7#13711yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×C4×C7#13711yescharacteristic
154C2×Dic7—22no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159D28—41no—
160D28—41no—
161D28—41no—
162D28—41no—
163C4×D7—44no—
164C4×D7—82no—
165C7×C8—42no—
166C7×C8—81no—
167C7⋊C8—42no—
168C7⋊C8—81no—
169C22×Q8C4×D712yes—
170C22×D4—71no—
171C23×C4—72no—
172C2×M16—21no—
173C2×M16—144no—
174C2×M16—141no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4×D4—141no—
180C4×D4—141no—
181C4×D4—141no—
182C4×D4—141no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—141no—
187C2×C22⋊C4—72no—
188C2×C22⋊C4—144no—
189C2×C42—141no—
190C2×C42C22×D711yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—72no—
194C4⋊C8—141no—
195C4⋊C8—141no—
196C4⋊C8C22⋊D712yes—
197#100—282no—
198C22⋊C8—21no—
199C22⋊C8—144no—
200C22⋊C8—141no—
201C4×C8—21no—
202C4×C8—141no—
203#95—72no—
204#95—144no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7#38611yescharacteristic
208C22×C7#41711yescharacteristic
209C22×C7#41911yescharacteristic
210C2×D7—44no—
211C2×D7—42no—
212C2×D7—42no—
213C2×D7—42no—
214C2×D7—82no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7—82no—
219C4×C7#45512yes—
220C4×C7#46811yescharacteristic
221C4×C7#46811yescharacteristic
222C4×C7#47011yescharacteristic
223C4×C7#47011yescharacteristic
224Dic7—42no—
225Dic7—82no—
226C24—72no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—82no—
230C2×D4—141no—
231C2×D4—141no—
232C2×D4—141no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4—141no—
236C22×C4—22no—
237C22×C4—72no—
238C22×C4—142no—
239C22×C4—142no—
240C22×C4—142no—
241C22×C4—284no—
242C22×C4C2×C4×D711yescharacteristic
243C22×C4D4×D711yescharacteristic
244C22×C4D4×D711yescharacteristic
245M16—41no—
246M16—41no—
247M16—281no—
248M16—281no—
249M16—564no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—142no—
253C2×C8—281no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—282no—
261C22⋊C4—142no—
262C22⋊C4—142no—
263C22⋊C4—564no—
264C42—281no—
265C42—281no—
266C42#115711yescharacteristic
267C42D4×D711yescharacteristic
268C42#115911yescharacteristic
269C42C2×C22⋊D711yescharacteristic
270C2×C7—21no—
271C2×C7—21no—
272C2×C7#188511yescharacteristic
273C2×C7#189411yescharacteristic
274C2×C7#191411yescharacteristic
275D7—82no—
276C23—142no—
277C23—142no—
278C23—142no—
279C23—282no—
280C23—282no—
281C23D7×C22⋊C411yescharacteristic
282Q8—82no—
283Q8—82no—
284D4—281no—
285D4—281no—
286D4—281no—
287D4—281no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—142no—
293C2×C4—284no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—562no—
299C2×C4(C22⋊D7)⋊C411yescharacteristic
300C2×C4#644611yescharacteristic
301C2×C4#644711yescharacteristic
302C2×C4#647711yescharacteristic
303C2×C4#647811yescharacteristic
304C2×C4#648111yescharacteristic
305C2×C4#648111yescharacteristic
306C2×C4C4×C22⋊D711yescharacteristic
307C2×C4#650011yescharacteristic
308C2×C4#650111yescharacteristic
309C8—42no—
310C8—81no—
311C8—282no—
312C8—561no—
313C7#939911yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22—284no—
317C22—282no—
318C22—282no—
319C22—282no—
320C22—562no—
321C22#8957411yescharacteristic
322C22#8957511yescharacteristic
323C22#8972811yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—282no—
329C4—562no—
330C4#8958411yescharacteristic
331C4#8958811yescharacteristic
332C4#8994611yescharacteristic
333C4#8996111yescharacteristic
334C4#8997412yes—
335C2—21no—
336C2—21no—
337C2—562no—
338C2#1172784311yescharacteristic
339C2#1172802511yescharacteristic
340C2#1172809711yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1548C2
socleC22×C7#386
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1548D7
hypocenterC7#9399
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C7×#1548C2semidirect
4. #11729528C2semidirect
6. #11725552C2semidirect
7. #11725521C2semidirect
67. #1548D7semidirect
313. C7#9399semidirect

Maximal subgroups

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

subgroupindex
#117407052
#117367332
C7×#15482
#117295282
#117282812
#117255522
#117255212
#93997