Finite Group Databasebeta

← #13274212

Subgroups — #13274212

Order 1792.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3221C211yesmaximal, characteristic
2#11729627C211yesmaximal, characteristic, complemented
3#11729626C211yesmaximal, characteristic, complemented
4#11725595C212yesmaximal, complemented
5#11725592C212yesmaximal, complemented
6C7×#540C411yescharacteristic
7C7×#536C2211yescharacteristic
8C7×#496C411yescharacteristic
9C7×#492C2211yescharacteristic
10C7×#491C2211yescharacteristic
11C7×#488C411yescharacteristic
12C7×#463—21no—
13C7×#461—21no—
14C7×#449—21no—
15C7×#448—21no—
16C7×#433C411yescharacteristic
17#90009—21no—
18C2×#6410—21no—
19#89343—21no—
20#89343C2212yes—
21#89343C2212yes—
22#89337—21no—
23#13611—71nomaximal, Sylow
24C2×C7×#41D411yescharacteristic
25C22×C7×Q8D411yescharacteristic
26C2×C7×SD16—21no—
27C2×C7×SD16—21no—
28C2×C7×SD16—21no—
29C2×C7×D8—21no—
30C7×C4⋊Q8D411yescharacteristic
31C7×C4⋊Q8C2311yescharacteristic
32C7×#127C2×C411yescharacteristic
33C7×#125C2×C411yescharacteristic
34C7×#124C2×C411yescharacteristic
35C7×#124D411yescharacteristic
36C7×#123—41no—
37C7×#123—41no—
38C7×C22⋊Q8—41no—
39C7×#121—21no—
40C7×#121—41no—
41C4×C7×D4—21no—
42C7×#117C2×C411yescharacteristic
43C2×#1136—41no—
44C2×#1136—41no—
45C7×C4⋊C8—21no—
46C7×#103—21no—
47C7×#103—21no—
48C7×#103—21no—
49C7×#102—21no—
50C7×C22⋊C8—21no—
51C7×#97C2×C411yescharacteristic
52C4×C7×C8C2×C411yescharacteristic
53#6410—22no—
54#6410—22no—
55#6410—41no—
56#6407—41no—
57#3221D711yescharacteristic
58#2135—71no—
59#2133—71no—
60#1413—72no—
61#1410—72no—
62C7×#41—81no—
63C2×C7×Q8—21no—
64C2×C7×Q8—81no—
65C2×C7×Q8C22⋊C411yescharacteristic
66C2×C7×Q8C2×D411yescharacteristic
67C2×C7×Q8C2×D411yescharacteristic
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4C22⋊C411yescharacteristic
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—41no—
76C22×C4×C7C22⋊C411yescharacteristic
77C7×SD16—81no—
78C7×SD16—81no—
79C7×SD16—81no—
80C7×D8—81no—
81C2×C7×C8—21no—
82C2×C7×C8—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C22⋊C4—21no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C42×C7C22⋊C411yescharacteristic
94C42×C7C22×C411yescharacteristic
95#1136—42no—
96#1136—42no—
97#1136—81no—
98#1136—81no—
99C2×C7⋊C8—81no—
100C2×C7⋊C8—81no—
101#540Dic711yescharacteristic
102#536C2×D711yescharacteristic
103#496Dic711yescharacteristic
104#492C2×D711yescharacteristic
105#491C2×D711yescharacteristic
106#488Dic711yescharacteristic
107#463—21no—
108#461—21no—
109#449—21no—
110#448—21no—
111#433Dic711yescharacteristic
112#413—141no—
113C2×#101—141no—
114#356—72no—
115#356—72no—
116#356—141no—
117#355—141no—
118C23×C7—21no—
119C23×C7—41no—
120C23×C7#9911yescharacteristic
121C7×Q8—41no—
122C7×Q8—41no—
123C7×Q8—41no—
124C7×Q8—81no—
125C7×D4—41no—
126C7×D4—41no—
127C7×D4—41no—
128C7×D4—81no—
129C7×D4—161no—
130C7×D4—161no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—161no—
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7C2×C22⋊C411yescharacteristic
145C7×C8—41no—
146C7×C8—41no—
147C7⋊C8—82no—
148C7⋊C8—82no—
149C2×#41C22⋊D711yescharacteristic
150C22×Q8C22⋊D711yescharacteristic
151C2×SD16—21no—
152C2×SD16—21no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—281no—
156C2×M16—281no—
157C4⋊Q8C22⋊D711yescharacteristic
158C4⋊Q8C22×D711yescharacteristic
159#127C2×Dic711yescharacteristic
160#125C2×Dic711yescharacteristic
161#124C2×Dic711yescharacteristic
162#124C22⋊D711yescharacteristic
163#123—41no—
164#123—41no—
165C22⋊Q8—41no—
166#121—21no—
167#121—41no—
168C4×D4—21no—
169#117C2×Dic711yescharacteristic
170C4⋊C8—21no—
171#103—21no—
172#103—21no—
173#103—21no—
174#102—21no—
175#101—142no—
176#101—142no—
177#101—281no—
178#100—281no—
179C22⋊C8—21no—
180#97C2×Dic711yescharacteristic
181C4×C8C2×Dic711yescharacteristic
182C22×C7—21no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—81no—
186C22×C7—81no—
187C22×C7—81no—
188C22×C7—81no—
189C22×C7C2×#9911yescharacteristic
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—41no—
193C4×C7—41no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—81no—
198#41—81no—
199C2×Q8—21no—
200C2×Q8—81no—
201C2×Q8C22⋊Dic711yescharacteristic
202C2×Q8C2×C22⋊D711yescharacteristic
203C2×Q8C2×C22⋊D711yescharacteristic
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C22⋊Dic711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4C22⋊Dic711yescharacteristic
213SD16—81no—
214SD16—81no—
215SD16—81no—
216D8—81no—
217M16—282no—
218M16—282no—
219M16—561no—
220M16—561no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—561no—
224C2×C8—561no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—21no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42C22⋊Dic711yescharacteristic
236C42C22×Dic711yescharacteristic
237C2×C7—21no—
238C2×C7—41no—
239C2×C7—81no—
240C2×C7—161no—
241C2×C7#212711yescharacteristic
242C23—21no—
243C23—41no—
244C23#640811yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—81no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—81no—
253D4—161no—
254D4—161no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—161no—
267C2×C4#640811yescharacteristic
268C2×C4C2×C22⋊Dic711yescharacteristic
269C8—41no—
270C8—41no—
271C8—562no—
272C8—562no—
273C7#1361111yescharacteristic, complemented, Sylow
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22C2×#640811yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—161no—
294C2#1172961411yescharacteristic
subgroupG/H
centerC2#11729614
commutatorC42×C7C22×C4
FrattiniC4⋊Q8C22×D7
Fitting, prime radicalC7×#3221C2
socleC2×C7#2127
cosocleC7×C4⋊Q8C23
hypercenter#3221D7
hypocenterC7#13611
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3221

Complements

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

NH ≅ G/Ntype
2. #11729627C2semidirect
3. #11729626C2semidirect
4. #11725595C2semidirect
5. #11725592C2semidirect
273. C7#13611semidirect

Maximal subgroups

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

subgroupindex
C7×#32212
#117296272
#117296262
#117255952
#117255922
#136117