Finite Group Databasebeta

← #13274217

Subgroups — #13274217

Order 1792.

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

normal subgroups only

This group has 304 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×#3223C211yesmaximal, characteristic
2#11729627C211yesmaximal, characteristic
3#11729626C211yesmaximal, characteristic
4#11725601C212yesmaximal, complemented
5#11725592C212yesmaximal, complemented
6C7×#541C411yescharacteristic
7C7×#536C2211yescharacteristic
8C7×#497C411yescharacteristic
9C7×#494C2211yescharacteristic
10C7×#491C2211yescharacteristic
11C7×#488C411yescharacteristic
12C7×#464—21no—
13C7×#462—21no—
14C7×#452—21no—
15C7×#451—21no—
16C7×#433C411yescharacteristic
17#90009—21no—
18C2×#6410—21no—
19#89343—21no—
20#89343C2212yes—
21#89343C2212yes—
22#89337—21no—
23#13612—71nomaximal, Sylow
24C2×C7×#41D411yescharacteristic
25C22×C7×Q8D411yescharacteristic
26C2×C7×Q16—21no—
27C2×C7×Q16—21no—
28C2×C7×Q16—21no—
29C2×C7×SD16—21no—
30C7×C4⋊Q8C2×C411yescharacteristic
31C7×C4⋊Q8D411yescharacteristic
32C7×C4⋊Q8C2311yescharacteristic
33C7×#126—41no—
34C7×#125—41no—
35C7×#125C2×C411yescharacteristic
36C7×#124C2×C411yescharacteristic
37C7×#124D411yescharacteristic
38C7×#123—41no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—41no—
41C4×C7×Q8—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#3223D711yescharacteristic
58#2135—71no—
59#2133—71no—
60#1413—72no—
61#1411—72no—
62C7×#41—81no—
63C2×C7×Q8—21no—
64C2×C7×Q8—21no—
65C2×C7×Q8—81no—
66C2×C7×Q8C22⋊C411yescharacteristic
67C2×C7×Q8C2×D411yescharacteristic
68C2×C7×Q8C2×D411yescharacteristic
69C2×C7×D4—21no—
70C2×C7×D4C22⋊C411yescharacteristic
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C22⋊C411yescharacteristic
74C7×Q16—81no—
75C7×Q16—81no—
76C7×Q16—81no—
77C7×SD16—81no—
78C2×C7×C8—21no—
79C2×C7×C8—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
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—
92C42×C7—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#541Dic711yescharacteristic
102#536C2×D711yescharacteristic
103#497Dic711yescharacteristic
104#494C2×D711yescharacteristic
105#491C2×D711yescharacteristic
106#488Dic711yescharacteristic
107#464—21no—
108#462—21no—
109#452—21no—
110#451—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#9911yescharacteristic
120C7×Q8—41no—
121C7×Q8—41no—
122C7×Q8—41no—
123C7×Q8—41no—
124C7×Q8—41no—
125C7×Q8—81no—
126C7×D4—41no—
127C7×D4—81no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7#9911yescharacteristic
141C2×C4×C7C2×C22⋊C411yescharacteristic
142C7×C8—41no—
143C7×C8—41no—
144C7⋊C8—82no—
145C7⋊C8—82no—
146C2×#41C22⋊D711yescharacteristic
147C22×Q8C22⋊D711yescharacteristic
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×SD16—21no—
152C2×M16—281no—
153C2×M16—281no—
154C4⋊Q8C2×Dic711yescharacteristic
155C4⋊Q8C22⋊D711yescharacteristic
156C4⋊Q8C22×D711yescharacteristic
157#126—41no—
158#125—41no—
159#125C2×Dic711yescharacteristic
160#124C2×Dic711yescharacteristic
161#124C22⋊D711yescharacteristic
162#123—41no—
163C22⋊Q8—21no—
164C22⋊Q8—41no—
165C4×Q8—21no—
166#117C2×Dic711yescharacteristic
167C4⋊C8—21no—
168#103—21no—
169#103—21no—
170#103—21no—
171#102—21no—
172#101—142no—
173#101—142no—
174#101—281no—
175#100—281no—
176C22⋊C8—21no—
177#97C2×Dic711yescharacteristic
178C4×C8C2×Dic711yescharacteristic
179C22×C7—21no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—81no—
183C22×C7C2×#9911yescharacteristic
184C4×C7—21no—
185C4×C7—21no—
186C4×C7—41no—
187C4×C7—41no—
188C4×C7—41no—
189C4×C7—41no—
190C4×C7—41no—
191C4×C7—81no—
192C4×C7—81no—
193#41—81no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—81no—
197C2×Q8C22⋊Dic711yescharacteristic
198C2×Q8C2×C22⋊D711yescharacteristic
199C2×Q8C2×C22⋊D711yescharacteristic
200C2×D4—21no—
201C2×D4C22⋊Dic711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4C22⋊Dic711yescharacteristic
205Q16—81no—
206Q16—81no—
207Q16—81no—
208SD16—81no—
209M16—282no—
210M16—282no—
211M16—561no—
212M16—561no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—561no—
216C2×C8—561no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—21no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—41no—
230C42C22⋊Dic711yescharacteristic
231C42C22×Dic711yescharacteristic
232C2×C7—21no—
233C2×C7—41no—
234C2×C7—81no—
235C2×C7#212711yescharacteristic
236C23—21no—
237C23#640811yescharacteristic
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—81no—
244D4—41no—
245D4—81no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4#640811yescharacteristic
259C2×C4C2×C22⋊Dic711yescharacteristic
260C8—41no—
261C8—41no—
262C8—562no—
263C8—562no—
264C7#1361211yescharacteristic, complemented, Sylow
265C22—21no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22C2×#640811yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—81no—
278C4—81no—
279C2—21no—
280C2—41no—
281C2—81no—
282C2#1172961411yescharacteristic
subgroupG/H
centerC2#11729614
commutatorC42×C7C22×C4
FrattiniC4⋊Q8C22×D7
FittingC7×#3223C2
socleC2×C7#2127
prime radicalC2×C7×#41D4
cosocleC7×C4⋊Q8C23
hypercenter#3223D7
hypocenterC7#13612
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3223

Complements

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

NH ≅ G/Ntype
4. #11725601C2semidirect
5. #11725592C2semidirect
264. C7#13612semidirect

Maximal subgroups

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

subgroupindex
C7×#32232
#117296272
#117296262
#117256012
#117255922
#136127