Finite Group Databasebeta

← #13251190

Subgroups — #13251190

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#11736733C211yesmaximal, characteristic
2#11731252C211yesmaximal, characteristic
3#11728533C211yesmaximal, characteristic, complemented
4#11728281C211yesmaximal, characteristic
5#11728261C211yesmaximal, characteristic, complemented
6C7×#1303C211yesmaximal, characteristic, complemented
7#11725426C211yesmaximal, characteristic, complemented
8C2×C4×D28C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6397—21no—
14#89759—22no—
15#89711—21no—
16#89708—21no—
17#89697C2211yescharacteristic
18#89696C2211yescharacteristic
19#89610—22no—
20#89498—22no—
21#89460—22no—
22#89445C414yes—
23#89349—21no—
24#89276—22no—
25#89276—22no—
26#9399—71nomaximal, Sylow
27C22×D28D411yescharacteristic
28C22×C4×D7D412yes—
29C22×Dic14C2×C412yes—
30C2×C7×M16—21no—
31C2×C7×C4⋊C4—21no—
32C2×C42×C7—21no—
33C2×C42×C7C2311yescharacteristic
34C2×(C2×D7)⋊C4—24no—
35C2×(C2×D7)⋊C4C2×C412yes—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7D411yescharacteristic
39C2×#1136—21no—
40C2×#1131—24no—
41C4×D28—21no—
42C4×D28—21no—
43C4×D28—21no—
44C4×D28—21no—
45#6432—21no—
46#6432—21no—
47#6432—21no—
48#6432—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8D412yes—
52C7×C22⋊C8—21no—
53C4×C7×C8—21no—
54C7×#95C2×C412yes—
55C7⋊#95—24no—
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#1548—71no—
68#1303—71no—
69#1303D711yescharacteristic, complemented
70C23×D7C22⋊C412yes—
71C22×C4×C7—22no—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—22no—
76C22×Dic7C22⋊C412yes—
77C2×D28—21no—
78C2×D28—21no—
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28—21no—
83C2×C4×D7—22no—
84C2×C4×D7—22no—
85C2×C4×D7—44no—
86C2×Dic14—22no—
87C2×Dic14—22no—
88C2×Dic14—82no—
89C7×M16—41no—
90C7×M16—41no—
91C2×C7×C8—22no—
92C2×C7×C8—41no—
93C7×C4⋊C4—42no—
94C42×C7—41no—
95C42×C7—41no—
96C42×C7C2×D411yescharacteristic
97C42×C7C2×D411yescharacteristic
98C42×C7#4111yescharacteristic
99C42×C7#4111yescharacteristic
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—84no—
103C4⋊Dic7—21no—
104C4⋊Dic7—21no—
105C4⋊Dic7—41no—
106C4⋊Dic7—41no—
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⋊Q8—71no—
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⋊C4C2×D711yescharacteristic
129#343—141no—
130#333—142no—
131#333—142no—
132C23×C7C2×C22⋊C411yescharacteristic
133C22×D7—22no—
134C22×D7—22no—
135C22×D7—22no—
136C22×D7—42no—
137C22×D7—42no—
138C2×C4×C7—22no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7C4×D411yescharacteristic
144C2×C4×C7C4×D411yescharacteristic
145C2×C4×C7#12011yescharacteristic
146C2×C4×C7#12111yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12311yescharacteristic
149C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C7#13711yescharacteristic
151C2×C4×C7#13711yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×Dic7—22no—
154C2×Dic7—42no—
155C2×Dic7—44no—
156C2×Dic7—42no—
157D28—41no—
158D28—41no—
159D28—41no—
160D28—41no—
161C4×D7—44no—
162C4×D7—82no—
163Dic14—82no—
164Dic14—82no—
165C7×C8—42no—
166C7×C8—81no—
167C7⋊C8—42no—
168C7⋊C8—81no—
169C22×Q8—72no—
170C22×D4—71no—
171C23×C4—72no—
172C2×M16—21no—
173C2×M16—144no—
174C2×M16—141no—
175C4⋊Q8—141no—
176C4⋊Q8—141no—
177C4⋊Q8—141no—
178C4⋊Q8—141no—
179C4×D4—141no—
180C4×D4—141no—
181C4×D4—141no—
182C4×D4—141no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—141no—
186C2×C4⋊C4—141no—
187C2×C22⋊C4—72no—
188C2×C22⋊C4—144no—
189C2×C42—21no—
190C2×C42C22×D711yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—72no—
194C4⋊C8—141no—
195C4⋊C8—141no—
196C4⋊C8D2812yes—
197#100—282no—
198C22⋊C8—21no—
199C22⋊C8—144no—
200C22⋊C8—141no—
201C4×C8—21no—
202C4×C8—141no—
203#95—144no—
204#95C4×D712yes—
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—142no—
228C2×Q8—142no—
229C2×Q8—562no—
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—42no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—281no—
258C4⋊C4—281no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C22⋊C4—142no—
262C22⋊C4—142no—
263C22⋊C4—564no—
264C42—41no—
265C42—41no—
266C42C2×D2811yescharacteristic
267C42#115711yescharacteristic
268C42D4×D711yescharacteristic
269C42#116111yescharacteristic
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—562no—
283Q8—562no—
284D4—281no—
285D4—281no—
286D4—281no—
287D4—281no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—142no—
294C2×C4—284no—
295C2×C4—282no—
296C2×C4—284no—
297C2×C4—282no—
298C2×C4—562no—
299C2×C4C4×D2811yescharacteristic
300C2×C4#644511yescharacteristic
301C2×C4#644711yescharacteristic
302C2×C4#645611yescharacteristic
303C2×C4#645711yescharacteristic
304C2×C4#645811yescharacteristic
305C2×C4#647411yescharacteristic
306C2×C4#647811yescharacteristic
307C2×C4#647811yescharacteristic
308C2×C4#648111yescharacteristic
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#8943711yescharacteristic
322C22#8954311yescharacteristic
323C22#8957411yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—282no—
329C4—562no—
330C4#8955211yescharacteristic
331C4#8958811yescharacteristic
332C4#8959712yes—
333C4#8964011yescharacteristic
334C4#8965811yescharacteristic
335C2—21no—
336C2—21no—
337C2—562no—
338C2#1172651311yescharacteristic
339C2#1172663811yescharacteristic
340C2#1172679411yescharacteristic
subgroupG/H
centerC22#89437
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1303C2
socleC22×C7#386
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1303D7
hypocenterC7#9399
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #11728533C2semidirect
5. #11728261C2semidirect
6. C7×#1303C2semidirect
7. #11725426C2semidirect
69. #1303D7semidirect
313. C7#9399semidirect

Maximal subgroups

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

subgroupindex
#117367332
#117312522
#117285332
#117282812
#117282612
C7×#13032
#117254262
#93997