Finite Group Databasebeta

← #13256445

Subgroups — #13256445

Order 1792.

626 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C7×#2868C211yesmaximal, characteristic, complemented
2#11740661C211yesmaximal, characteristic
3#11740658C211yesmaximal, characteristic
4#11729529C211yesmaximal, characteristic, complemented
5#11729528C211yesmaximal, characteristic, complemented
6#11727149C211yesmaximal, characteristic
7#11725417C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8—22no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11#90168C2211yescharacteristic
12C7×#384—22no—
13C7×#384—22no—
14C43×C7C2211yescharacteristic
15#89999—22no—
16#89860—21no—
17#89857—21no—
18#89856—22no—
19#89855—22no—
20#89853—21no—
21#89851—21no—
22#89740—22no—
23C2×#6379—22no—
24#89697C2211yescharacteristic
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27#89342—22no—
28#89341—22no—
29#10283—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D412yes—
32C2×#1157—21no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—48no—
42C2×C7×C4⋊C4—28no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4D412yes—
47C2×C42×C7—21no—
48C2×C42×C7—21no—
49C2×C42×C7D412yes—
50C2×C42×C7C2311yescharacteristic
51C2×#1143—24no—
52#6493—21no—
53#6493—21no—
54C2×#1136—21no—
55#6438—21no—
56#6437D411yescharacteristic
57C4×D28—21no—
58C4×D28—21no—
59#6432D411yescharacteristic
60#6409—24no—
61C22⋊(C7⋊C8)—21no—
62C22⋊(C7⋊C8)—21no—
63#6383—24no—
64#6382—24no—
65#6379—44no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)D412yes—
69C4⋊(C7⋊C8)D412yes—
70C4×C7⋊C8—21no—
71C4×C7⋊C8—21no—
72#3227—71no—
73#3206—71no—
74#2868D711yescharacteristic, complemented
75#1962—71no—
76#1549—71no—
77#1548—71no—
78#1276—71no—
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—44no—
82C2×C7×Q8—44no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C2×C22⋊D7—41no—
92#1157—42no—
93#1157—42no—
94C2×D28—21no—
95C2×D28—21no—
96C2×D28—21no—
97C2×C4×D7—41no—
98C2×Dic14—21no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—416no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C42×C7—24no—
109C42×C7—24no—
110C42×C7—22no—
111C42×C7SD1614yescomplemented
112C42×C7C2×D411yescharacteristic
113C42×C7C2×D411yescharacteristic
114C42×C7C2×D411yescharacteristic
115C42×C7#4111yescharacteristic
116#1143—84no—
117(C2×D7)⋊C4—41no—
118(C2×D7)⋊C4—41no—
119C4⋊Dic7—21no—
120C4⋊Dic7—21no—
121#1136—42no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—41no—
125#532—71no—
126C2×C4⋊Q8—22no—
127C2×C4⋊Q8C2×D711yescharacteristic
128C2×C4⋊Q8C2×D711yescharacteristic
129#498—141no—
130#492—141no—
131#487—141no—
132#486—141no—
133#481—142no—
134#476—142no—
135#461—142no—
136#448—142no—
137#431—71no—
138#431—71no—
139#423—71no—
140C2×#104—142no—
141#384—22no—
142#384—22no—
143C43C2×D711yescharacteristic
144#333—142no—
145#332—142no—
146C23×C7#12011yescharacteristic
147C22×D7—41no—
148C7×Q8—44no—
149C7×Q8—82no—
150C7×Q8—82no—
151C7×Q8—82no—
152C2×C4×C7—24no—
153C2×C4×C7—24no—
154C2×C4×C7—24no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—44no—
158C2×C4×C7—44no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7#12011yescharacteristic
162C2×C4×C7#12011yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12711yescharacteristic
167C2×C4×C7C2×SD1612yes—
168C2×C4×C7#13712yes—
169C22⋊D7—82no—
170C2×Dic7—41no—
171D28—42no—
172D28—42no—
173D28—42no—
174C4×D7—81no—
175C4×D7—81no—
176Dic14—42no—
177C7⋊C8—42no—
178C7⋊C8—42no—
179C7⋊C8—81no—
180C2×#41—141no—
181C22×Q8—21no—
182C22×Q8C22⋊D712yes—
183C2×SD16—144no—
184C2×M16—141no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—48no—
194C4⋊Q8—71no—
195#127—71no—
196#124—141no—
197#121—141no—
198#121—141no—
199C4×D4—141no—
200C4×D4—141no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D712yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22⋊D712yes—
209C2×C42C22×D711yescharacteristic
210#106—144no—
211C4⋊C8—72no—
212C4⋊C8—72no—
213C4⋊C8—141no—
214C4⋊C8—141no—
215#104—284no—
216#103—144no—
217#102—144no—
218C22⋊C8—141no—
219C22⋊C8—141no—
220C4×C8—141no—
221C4×C8—141no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7#39211yescharacteristic
225C22×C7#44812yes—
226C2×D7—81no—
227C2×D7—81no—
228C2×D7—81no—
229C4×C7—28no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—82no—
233C4×C7—82no—
234C4×C7#45312yes—
235C4×C7#45612yes—
236C4×C7#46112yes—
237C4×C7#49211yescharacteristic
238C4×C7#49711yescharacteristic
239Dic7—81no—
240#41—282no—
241#41—282no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—141no—
249C2×D4—141no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—281no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—281no—
257C22×C4C2×C22⋊D711yescharacteristic
258C22×C4C2×C22⋊D711yescharacteristic
259C22×C4C2×C22⋊D711yescharacteristic
260SD16—564no—
261M16—282no—
262C2×C8—142no—
263C2×C8—142no—
264C2×C8—281no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—416no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C22⋊C4—281no—
277C22⋊C4—281no—
278C42—24no—
279C42—24no—
280C42—22no—
281C42#114314yescomplemented
282C42D4×D711yescharacteristic
283C42D4×D711yescharacteristic
284C42D4×D711yescharacteristic
285C42#115911yescharacteristic
286C2×C7—22no—
287C2×C7#200211yescharacteristic
288C2×C7#200511yescharacteristic
289C2×C7#201311yescharacteristic
290D7—161no—
291C23—281no—
292C23#651611yescharacteristic
293Q8—44no—
294Q8—82no—
295Q8—82no—
296Q8—82no—
297Q8—282no—
298D4—282no—
299D4—282no—
300D4—282no—
301D4—562no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—281no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4#650011yescharacteristic
315C2×C4#650011yescharacteristic
316C2×C4#650111yescharacteristic
317C2×C4#650211yescharacteristic
318C2×C4#650211yescharacteristic
319C2×C4#650311yescharacteristic
320C2×C4C2×#114312yes—
321C2×C4#650512yes—
322C8—282no—
323C8—282no—
324C8—561no—
325C7#1028311yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—561no—
329C22—561no—
330C22—561no—
331C22#8999511yescharacteristic
332C22#8999912yes—
333C4—28no—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—82no—
338C4—561no—
339C4#8984912yes—
340C4#8985512yes—
341C4#8985711yescharacteristic
342C4#8985811yescharacteristic
343C4#8986712yes—
344C2—22no—
345C2—1121no—
346C2#1172952311yescharacteristic
347C2#1172954511yescharacteristic
348C2#1172954911yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2868C2
socleC22×C7#392
prime radical#90168C22
hypercenter#2868D7
hypocenterC7#10283
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
1. C7×#2868C2semidirect
4. #11729529C2semidirect
5. #11729528C2semidirect
7. #11725417C2semidirect
74. #2868D7semidirect
111. C42×C7SD16semidirect
281. C42#1143semidirect
325. C7#10283semidirect

Maximal subgroups

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

subgroupindex
C7×#28682
#117406612
#117406582
#117295292
#117295282
#117271492
#117254172
#102837