Finite Group Databasebeta

← #13242374

Subgroups — #13242374

Order 1792.

500 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2857C211yesmaximal, characteristic
2#11729285C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725444C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C7×C22⋊Q8—21no—
7C22×C42×C7C2211yescharacteristic
8C7×#391—21no—
9C7×#389—21no—
10C7×#389C2212yes—
11C7×#384C411yescharacteristic
12C7×#380—21no—
13C7×#380C412yes—
14C2×C22⋊(C7⋊C8)—21no—
15C2×C22⋊(C7⋊C8)—21no—
16C2×#6377—21no—
17#89319—21no—
18#89319C2212yes—
19#89319C2212yes—
20#7708—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—42no—
26C7×C22⋊Q8—44no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4C2×C412yes—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C22⋊C4—22no—
34C2×C42×C7—42no—
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7C2311yescharacteristic
39C22×C7⋊C8—22no—
40C22×C7⋊C8—22no—
41C7×#95—22no—
42C7×#95—22no—
43C22⋊(C7⋊C8)—24no—
44C22⋊(C7⋊C8)—42no—
45#6377—42no—
46#2857D711yescharacteristic
47#1817—71no—
48#1749—71no—
49#1271—74no—
50C24×C7C22⋊C411yescharacteristic
51C2×C7×Q8—42no—
52C2×C7×Q8—42no—
53C2×C7×Q8—42no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C7×C4⋊C4—24no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—44no—
77C7×C22⋊C4—44no—
78C42×C7—22no—
79C42×C7—22no—
80C42×C7—41no—
81C42×C7—41no—
82C2×C7⋊C8—24no—
83C2×C7⋊C8—42no—
84C2×C7⋊C8—42no—
85C2×C7⋊C8—42no—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—42no—
88C2×C4⋊Q8Dic711yescharacteristic
89C2×C22⋊Q8—21no—
90C22×C42C2×D711yescharacteristic
91C2×C22⋊C8—141no—
92C2×C22⋊C8—141no—
93C2×#97—141no—
94#391—21no—
95#389—21no—
96#389C2×D712yes—
97#384Dic711yescharacteristic
98#380—21no—
99#380Dic712yes—
100#336—72no—
101#336—72no—
102#336—141no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7#10112yes—
110C23×C7C2×C22⋊C411yescharacteristic
111C7×Q8—84no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—44no—
126C2×C4×C7—44no—
127C2×C4×C7—44no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7#9912yes—
135C2×C4×C7#10412yes—
136C2×C4×C7#13112yes—
137C2×C4×C7AGL(1,ℤ/8)12yes—
138C2×C4×C7#13712yes—
139C7⋊C8—44no—
140C7⋊C8—82no—
141C22×Q8—21no—
142C23×C4C2×Dic711yescharacteristic
143C23×C4C22⋊D711yescharacteristic
144C23×C4C22⋊D711yescharacteristic
145C22×C8—142no—
146C22×C8—142no—
147C4⋊Q8—42no—
148C22⋊Q8—44no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C2×Dic712yes—
154C2×C4⋊C4C2×Dic712yes—
155C2×C22⋊C4—22no—
156C2×C42—42no—
157C2×C42C2×Dic711yescharacteristic
158C2×C42C22⋊D711yescharacteristic
159C2×C42C22⋊D711yescharacteristic
160C2×C42C22×D711yescharacteristic
161C22⋊C8—144no—
162C22⋊C8—282no—
163#97—282no—
164#95—22no—
165#95—22no—
166C22×C7—22no—
167C22×C7—22no—
168C22×C7—22no—
169C22×C7—41no—
170C22×C7—41no—
171C22×C7—41no—
172C22×C7—41no—
173C22×C7#40811yescharacteristic
174C22×C7C2×#9911yescharacteristic
175C22×C7C2×#10111yescharacteristic
176C22×C7#41811yescharacteristic
177C22×C7#41911yescharacteristic
178C22×C7C2×#10411yescharacteristic
179C22×C7#42111yescharacteristic
180C4×C7—22no—
181C4×C7—22no—
182C4×C7—22no—
183C4×C7—22no—
184C4×C7—22no—
185C4×C7—41no—
186C4×C7—82no—
187C4×C7—82no—
188C24C22⋊Dic711yescharacteristic
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4C22⋊Dic711yescharacteristic
204C22×C4C22⋊Dic711yescharacteristic
205C22×C4C22⋊Dic711yescharacteristic
206C22×C4C22×Dic711yescharacteristic
207C22×C4C2×C22⋊D711yescharacteristic
208C22×C4C2×C22⋊D711yescharacteristic
209C2×C8—144no—
210C2×C8—282no—
211C2×C8—282no—
212C2×C8—282no—
213C2×C8—282no—
214C2×C8—282no—
215C4⋊C4—24no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C22⋊C4—44no—
222C42—22no—
223C42—22no—
224C42—41no—
225C42—41no—
226C2×C7—42no—
227C2×C7#147211yescharacteristic
228C2×C7#148411yescharacteristic
229C2×C7#149711yescharacteristic
230C2×C7#150811yescharacteristic
231C2×C7#151011yescharacteristic
232C2×C7#153111yescharacteristic
233C2×C7#155611yescharacteristic
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#641012yes—
241C23C2×C22⋊Dic711yescharacteristic
242Q8—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4#640812yes—
266C2×C4#641112yes—
267C2×C4#649512yes—
268C2×C4#650512yes—
269C2×C4#651112yes—
270C8—284no—
271C8—562no—
272C7#770811yescharacteristic, complemented, Sylow
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#8998711yescharacteristic
281C22C2×#640811yescharacteristic
282C22#8999711yescharacteristic
283C22C2×#641011yescharacteristic
284C22#9000611yescharacteristic
285C22C2×#641111yescharacteristic
286C22#9000811yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—82no—
294C4—82no—
295C2—42no—
296C2#1172942811yescharacteristic
297C2#1172944211yescharacteristic
298C2#1172944511yescharacteristic
299C2#1172953011yescharacteristic
300C2#1172955511yescharacteristic
301C2#1172957611yescharacteristic
302C2#1172958111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2857C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2857D7
hypocenterC7#7708
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
4. #11725444C2semidirect
272. C7#7708semidirect

Maximal subgroups

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

subgroupindex
C7×#28572
#117292852
#117292832
#117254442
#77087