Finite Group Databasebeta

← #13245359

Subgroups — #13245359

Order 1792.

686 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 167 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×#3502C211yesmaximal, characteristic
2#11729584C212yesmaximal
3#11729580C212yesmaximal
4#11729391C211yesmaximal, characteristic
5#11729388C211yesmaximal, characteristic
6C7×Q82—22no—
7C7×#557—22no—
8C7×#557—24no—
9C7×#544—21no—
10C7×#544C414yes—
11C7×#533—22no—
12C7×#533C2212yes—
13C7×#519—21no—
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6410—21no—
16#89696C2212yes—
17C2×C4⋊(C7⋊C8)C2211yescharacteristic
18C2×C4×C7⋊C8C2211yescharacteristic
19#89342—22no—
20#89341—22no—
21Q8⋊(C7⋊C8)—24no—
22#89325—21no—
23#89319—22no—
24#8356—71nomaximal, Sylow
25C22×C7×Q8D411yescharacteristic
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—22no—
28C7×C4⋊Q8—24no—
29C7×C4⋊Q8—28no—
30C7×C4⋊Q8D412yes—
31C7×C4⋊Q8D412yes—
32C7×#125—24no—
33C7×#125—24no—
34C7×#125—28no—
35C7×#125—22no—
36C7×C22⋊Q8—24no—
37C7×C22⋊Q8—22no—
38C7×C22⋊Q8C2×C414yes—
39C4×C7×Q8—24no—
40C4×C7×Q8—24no—
41C4×C7×Q8—24no—
42C4×C7×Q8—24no—
43C4×C7×Q8—22no—
44C4×C7×Q8—22no—
45C4×C7×Q8—22no—
46C4×C7×Q8D414yes—
47C7×#117—24no—
48C7×#117—22no—
49C2×C7×C4⋊C4C2×C412yes—
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C42×C7D412yes—
52C2×C42×C7C2311yescharacteristic
53C2×#1136—22no—
54C22×C7⋊C8—21no—
55C22×C7⋊C8—21no—
56#6410—42no—
57C4⋊(C7⋊C8)—22no—
58C4⋊(C7⋊C8)—22no—
59C4⋊(C7⋊C8)—22no—
60C4×C7⋊C8—22no—
61#3502D711yescharacteristic
62#1984—71no—
63#1801—71no—
64#1550—72no—
65#1493—72no—
66C2×C7×Q8—24no—
67C2×C7×Q8—22no—
68C2×C7×Q8—22no—
69C2×C7×Q8—22no—
70C2×C7×Q8C22⋊C414yes—
71C2×C7×Q8SD1612yes—
72C2×C7×Q8Q1612yes—
73C22×C4×C7C22⋊C412yes—
74C22×C4×C7C22⋊C412yes—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C7×C4⋊C4—24no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—28no—
82C7×C4⋊C4—24no—
83C7×C4⋊C4—24no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—24no—
89C7×C4⋊C4C22⋊C414yes—
90C7×C22⋊C4—24no—
91C42×C7—28no—
92C42×C7—22no—
93C42×C7—22no—
94C42×C7—24no—
95C42×C7C2×D412yes—
96C42×C7C2×D412yes—
97#1136—42no—
98#1136—42no—
99C2×C7⋊C8—22no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8—41no—
102C2×C7⋊C8—42no—
103C2×C7⋊C8—41no—
104C2×C7⋊C8—42no—
105Q82—22no—
106#557—22no—
107#557—24no—
108#544—21no—
109#544Dic714yes—
110#533—22no—
111#533C2×D712yes—
112#519—21no—
113C2×C4×Q8C2×D711yescharacteristic
114#423—72no—
115C2×C4⋊C8—71no—
116C2×#101—141no—
117C2×C4×C8—71no—
118#343—141no—
119#336—142no—
120#333—142no—
121#332—142no—
122Q8⋊C8—144no—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—24no—
125C7×Q8—24no—
126C7×Q8—44no—
127C7×Q8—42no—
128C7×Q8—42no—
129C7×Q8—84no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—24no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—41no—
138C2×C4×C7—42no—
139C2×C4×C7#10314yes—
140C2×C4×C7C2×C22⋊C412yes—
141C2×C4×C7#12011yescharacteristic
142C2×C4×C7#12012yes—
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7C2×SD1611yescharacteristic
145C2×C4×C7C2×Q1611yescharacteristic
146C2×C4×C7#13712yes—
147C7⋊C8—42no—
148C7⋊C8—82no—
149C7⋊C8—81no—
150C22×Q8C22⋊D711yescharacteristic
151C2×M16—142no—
152C22×C8—141no—
153C22×C8—141no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—24no—
157C4⋊Q8—28no—
158C4⋊Q8C22⋊D712yes—
159C4⋊Q8C22⋊D712yes—
160#125—24no—
161#125—24no—
162#125—28no—
163#125—22no—
164C22⋊Q8—24no—
165C22⋊Q8—22no—
166C22⋊Q8C2×Dic714yes—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8C22⋊D714yes—
175#117—24no—
176#117—22no—
177C2×C4⋊C4C2×Dic712yes—
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C42C22⋊D712yes—
180C2×C42C22×D711yescharacteristic
181C4⋊C8—142no—
182C4⋊C8—142no—
183C4⋊C8—142no—
184#101—282no—
185C4×C8—142no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7#37911yescharacteristic
189C22×C7C2×#10311yescharacteristic
190C22×C7#41911yescharacteristic
191C4×C7—21no—
192C4×C7—24no—
193C4×C7—22no—
194C4×C7—44no—
195C4×C7—42no—
196C4×C7—44no—
197C4×C7#44812yes—
198C4×C7#45112yes—
199C4×C7#45411yescharacteristic
200C4×C7#45611yescharacteristic
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8#114312yes—
206C2×Q8#114412yes—
207C2×Q8C22⋊Dic714yes—
208C22×C4C22⋊Dic712yes—
209C22×C4C22⋊Dic712yes—
210C22×C4C22×Dic711yescharacteristic
211C22×C4C2×C22⋊D711yescharacteristic
212C22×C4C2×C22⋊D711yescharacteristic
213M16—282no—
214M16—282no—
215C2×C8—142no—
216C2×C8—281no—
217C2×C8—281no—
218C2×C8—282no—
219C2×C8—281no—
220C2×C8—282no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—24no—
232C4⋊C4C22⋊Dic714yes—
233C22⋊C4—24no—
234C42—28no—
235C42—22no—
236C42—22no—
237C42—24no—
238C42C2×C22⋊D712yes—
239C42C2×C22⋊D712yes—
240C2×C7—21no—
241C2×C7—21no—
242C2×C7#178711yescharacteristic
243C2×C7#179111yescharacteristic
244C2×C7#179311yescharacteristic
245C23C2×C22⋊Dic711yescharacteristic
246Q8—24no—
247Q8—24no—
248Q8—44no—
249Q8—42no—
250Q8—42no—
251Q8—84no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4#640914yes—
262C2×C4C2×#114311yescharacteristic
263C2×C4#650512yes—
264C2×C4C2×#114411yescharacteristic
265C2×C4C2×C22⋊Dic712yes—
266C2×C4#651611yescharacteristic
267C2×C4#651612yes—
268C2×C4#651611yescharacteristic
269C8—282no—
270C8—562no—
271C8—561no—
272C7#835611yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22C2×#640911yescharacteristic
276C22#8999711yescharacteristic
277C22#9001911yescharacteristic
278C4—24no—
279C4—21no—
280C4—22no—
281C4—44no—
282C4—42no—
283C4—44no—
284C4#8999912yes—
285C4#9000012yes—
286C4#9001711yescharacteristic
287C4#9001811yescharacteristic
288C2—21no—
289C2—21no—
290C2#1172988611yescharacteristic
291C2#1172989711yescharacteristic
292C2#1172989911yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3502C2
socleC22×C7#379
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#3502D7
hypocenterC7#8356
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
272. C7#8356semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117295842
#117295802
#117293912
#117293882
#83567