Finite Group Databasebeta

← #13256442

Subgroups — #13256442

Order 1792.

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

normal subgroups only

This group has 626 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
1C7×#3206C211yesmaximal, characteristic
2#11740716C211yesmaximal, characteristic
3#11740698C211yesmaximal, characteristic, complemented
4#11729569C211yesmaximal, characteristic
5#11729528C211yesmaximal, characteristic, complemented
6#11728534C211yesmaximal, characteristic, complemented
7#11725542C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×#6507—22no—
11C2×#6432C2211yescharacteristic
12C7×#492—21no—
13C7×#487—21no—
14C7×#461—22no—
15C7×#448—22no—
16C7×#431C2211yescharacteristic
17C2×#6411—22no—
18#89851—21no—
19#89842—21no—
20#89806—22no—
21#89780—22no—
22#89748—22no—
23#89748—22no—
24C42×Dic7C2211yescharacteristic
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27#89341—22no—
28#89277—22no—
29#10283—71nomaximal, Sylow
30C2×C7×#41—21no—
31C22×C7×Q8D412yes—
32C22×Dic14—21no—
33C2×C7×SD16—24no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8D411yescharacteristic
39C7×#127D411yescharacteristic
40C7×#124—21no—
41C7×#121—21no—
42C7×#121—21no—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C42×C7C2311yescharacteristic
48#6507—48no—
49C2×C4⋊Dic7—21no—
50C2×C4⋊Dic7D412yes—
51C2×Dic7⋊C4—28no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—21no—
54C2×C4×Dic7D412yes—
55C2×#1136—21no—
56#6432—21no—
57#6432—21no—
58#6432—21no—
59#6432—21no—
60C7×C4⋊C8D412yes—
61C7×#103—24no—
62C7×C22⋊C8—21no—
63C4×C7×C8—21no—
64#6411—44no—
65#6406—24no—
66C22⋊(C7⋊C8)—21no—
67#6382—24no—
68C4⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)D412yes—
71C4×C7⋊C8—21no—
72#3227—71no—
73#3206D711yescharacteristic
74#2868—71no—
75#1962—71no—
76#1549—71no—
77#1548—71no—
78#1276—71no—
79C7×#41—42no—
80C7×#41—42no—
81C2×C7×Q8—22no—
82C2×C7×Q8—21no—
83C2×C7×Q8—22no—
84C2×C7×Q8—44no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—41no—
89C22×C4×C7—22no—
90C22×C4×C7—41no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×Dic7—22no—
95C22×Dic7—22no—
96C2×Dic14—44no—
97C2×Dic14—41no—
98C2×Dic14—41no—
99C7×SD16—84no—
100C2×C7×C8—22no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C42×C7C2×D411yescharacteristic
110C42×C7C2×D411yescharacteristic
111C42×C7C2×D411yescharacteristic
112C42×C7#4111yescharacteristic
113C4⋊Dic7—22no—
114C4⋊Dic7—22no—
115C4⋊Dic7—41no—
116C4⋊Dic7—41no—
117Dic7⋊C4—416no—
118C4×Dic7—24no—
119C4×Dic7—24no—
120C4×Dic7—22no—
121C4×Dic7SD1614yescomplemented
122#1136—42no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—41no—
125#532C2×D711yescharacteristic
126C2×C4⋊Q8—71no—
127C2×C4⋊Q8—142no—
128C2×C4⋊Q8C2×D711yescharacteristic
129#498—141no—
130#492—21no—
131#487—21no—
132#486—141no—
133#481—142no—
134#476—142no—
135#461—22no—
136#448—22no—
137#431—71no—
138#431C2×D711yescharacteristic
139#423—71no—
140C2×#104—142no—
141#384—142no—
142#384—142no—
143C43—71no—
144#333—142no—
145#332—142no—
146C23×C7—41no—
147C23×C7#12011yescharacteristic
148C7×Q8—44no—
149C7×Q8—42no—
150C7×Q8—82no—
151C7×D4—42no—
152C7×D4—42no—
153C7×D4—42no—
154C7×D4—82no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—44no—
158C2×C4×C7—41no—
159C2×C4×C7—42no—
160C2×C4×C7—81no—
161C2×C4×C7—81no—
162C2×C4×C7#12011yescharacteristic
163C2×C4×C7#12011yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#12711yescharacteristic
168C2×C4×C7C2×SD1612yes—
169C2×C4×C7#13712yes—
170C2×Dic7—24no—
171C2×Dic7—24no—
172C2×Dic7—24no—
173C2×Dic7—44no—
174C2×Dic7—42no—
175Dic14—82no—
176Dic14—82no—
177C7×C8—42no—
178C7⋊C8—42no—
179C7⋊C8—81no—
180C2×#41—21no—
181C22×Q8—141no—
182C22×Q8C22⋊D712yes—
183C2×SD16—24no—
184C2×M16—141no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—141no—
190C4⋊Q8—141no—
191C4⋊Q8—141no—
192C4⋊Q8—141no—
193C4⋊Q8—288no—
194C4⋊Q8C22⋊D711yescharacteristic
195#127C22⋊D711yescharacteristic
196#124—21no—
197#121—21no—
198#121—21no—
199C4×D4—21no—
200C4×D4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—72no—
204C2×C4⋊C4—148no—
205C2×C4⋊C4—141no—
206C2×C42—72no—
207C2×C42—141no—
208C2×C42—141no—
209C2×C42C22×D711yescharacteristic
210#106—144no—
211C4⋊C8—72no—
212C4⋊C8—141no—
213C4⋊C8—141no—
214C4⋊C8C22⋊D712yes—
215#104—284no—
216#103—24no—
217#102—144no—
218C22⋊C8—21no—
219C22⋊C8—141no—
220C4×C8—21no—
221C4×C8—141no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7#39211yescharacteristic
228C22×C7#44812yes—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—82no—
232C4×C7—81no—
233C4×C7#45312yes—
234C4×C7#45612yes—
235C4×C7#46112yes—
236C4×C7#49211yescharacteristic
237C4×C7#49711yescharacteristic
238Dic7—28no—
239Dic7—82no—
240#41—42no—
241#41—42no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—44no—
246C2×Q8—284no—
247C2×Q8—281no—
248C2×Q8—281no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—142no—
256C22×C4—142no—
257C22×C4D4×D711yescharacteristic
258C22×C4D4×D711yescharacteristic
259C22×C4C2×C22⋊D711yescharacteristic
260SD16—84no—
261M16—282no—
262C2×C8—22no—
263C2×C8—142no—
264C2×C8—281no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—142no—
272C4⋊C4—142no—
273C4⋊C4—2816no—
274C4⋊C4—281no—
275C4⋊C4—281no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42—74no—
279C42—144no—
280C42—144no—
281C42—142no—
282C42D4×D711yescharacteristic
283C42#115911yescharacteristic
284C42C2×C22⋊D711yescharacteristic
285C42C2×C22⋊D711yescharacteristic
286C2×C7—22no—
287C2×C7—161no—
288C2×C7#200211yescharacteristic
289C2×C7#200511yescharacteristic
290C2×C7#201311yescharacteristic
291C23—41no—
292C23#650011yescharacteristic
293Q8—42no—
294Q8—44no—
295Q8—82no—
296Q8—562no—
297Q8—562no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—144no—
310C2×C4—144no—
311C2×C4—144no—
312C2×C4—284no—
313C2×C4—282no—
314C2×C4D7×SD1612yes—
315C2×C4#647812yes—
316C2×C4#650011yescharacteristic
317C2×C4#650111yescharacteristic
318C2×C4#650211yescharacteristic
319C2×C4#650211yescharacteristic
320C2×C4#650311yescharacteristic
321C2×C4#651611yescharacteristic
322C8—42no—
323C8—282no—
324C8—561no—
325C7#1028311yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#8994212yes—
332C22#8999511yescharacteristic
333C4—21no—
334C4—21no—
335C4—81no—
336C4—82no—
337C4—148no—
338C4—562no—
339C4#8993412yes—
340C4#8993911yescharacteristic
341C4#8994711yescharacteristic
342C4#8997712yes—
343C4#9001312yes—
344C2—22no—
345C2—161no—
346C2#1172961011yescharacteristic
347C2#1172981111yescharacteristic
348C2#1172986211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#3206C2
socleC22×C7#392
prime radicalC7×#532C22
hypercenter#3206D7
hypocenterC7#10283
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
3. #11740698C2semidirect
5. #11729528C2semidirect
6. #11728534C2semidirect
7. #11725542C2semidirect
121. C4×Dic7SD16semidirect
325. C7#10283semidirect

Maximal subgroups

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

subgroupindex
C7×#32062
#117407162
#117406982
#117295692
#117295282
#117285342
#117255422
#102837