Finite Group Databasebeta

← #13264185

Subgroups — #13264185

Order 1792.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2632C211yesmaximal, characteristic
2C4×#6410C211yesmaximal, characteristic, complemented
3#11729378C211yesmaximal, characteristic, complemented
4#11725454C212yesmaximal, complemented
5#11725453C212yesmaximal
6C2×C7×#124C411yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#399—21no—
9C7×#395C411yescharacteristic
10C7×#395C2211yescharacteristic
11C7×#394C411yescharacteristic
12C7×#394C2211yescharacteristic
13C7×#393—21no—
14C7×#388—21no—
15C7×#388—21no—
16C7×#383C411yescharacteristic
17C2×#6410—21no—
18#89697—21no—
19C4×#1136—21no—
20#89325C2212yes—
21#89323—21no—
22#89323C2212yes—
23#11738—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×Q8D411yescharacteristic
26C22×C7×D4C2×C411yescharacteristic
27C7×#124—41no—
28C7×#124—41no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×#1136—22no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95C2×C411yescharacteristic
51C7×#95C2×C411yescharacteristic
52#6410—24no—
53C22⋊(C7⋊C8)—22no—
54C4⋊(C7⋊C8)—41no—
55#6377—41no—
56C4×C7⋊C8—41no—
57C4×C7⋊C8—41no—
58#2632D711yescharacteristic
59C4×#101—71no—
60#1461—71no—
61#1354—72no—
62#1345—72no—
63C24×C7—21no—
64C2×C7×Q8—22no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×Q8—81no—
69C2×C7×D4—41no—
70C2×C7×D4—81no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—81no—
85C7×C4⋊C4—81no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—81no—
92C7×C22⋊C4—81no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—21no—
99#1136—24no—
100#1136—42no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—42no—
103C2×#124Dic711yescharacteristic
104C2×C4×Q8C2×D711yescharacteristic
105#431—141no—
106C2×#101—141no—
107C4×M16—141no—
108#399—21no—
109#395Dic711yescharacteristic
110#395C2×D711yescharacteristic
111#394Dic711yescharacteristic
112#394C2×D711yescharacteristic
113#393—21no—
114#388—21no—
115#388—21no—
116#383Dic711yescharacteristic
117#343—72no—
118#324—72no—
119#324—141no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7C2×C22⋊C411yescharacteristic
128C7×Q8—44no—
129C7×Q8—82no—
130C7×D4—82no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7#9911yescharacteristic
155C2×C4×C7#9911yescharacteristic
156C2×C4×C7#10412yes—
157C2×C4×C7#13512yes—
158C7⋊C8—44no—
159C7⋊C8—82no—
160C22×Q8C2×Dic711yescharacteristic
161C22×Q8C22⋊D711yescharacteristic
162C22×D4C2×Dic711yescharacteristic
163C2×M16—142no—
164#124—41no—
165#124—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×Dic711yescharacteristic
171C2×C4⋊C4C22⋊D711yescharacteristic
172C2×C4⋊C4C22×D711yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42C2×Dic711yescharacteristic
180C2×C42C22⋊D711yescharacteristic
181C2×C42C22⋊D711yescharacteristic
182C4⋊C8—281no—
183#101—144no—
184C22⋊C8—142no—
185#97—281no—
186C4×C8—281no—
187C4×C8—281no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95C2×Dic711yescharacteristic
193#95C2×Dic711yescharacteristic
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—81no—
197C22×C7—81no—
198C22×C7—81no—
199C22×C7—81no—
200C22×C7—81no—
201C22×C7—81no—
202C22×C7—81no—
203C22×C7C2×#9911yescharacteristic
204C22×C7#41611yescharacteristic
205C22×C7C2×#10411yescharacteristic
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—81no—
215C4×C7—81no—
216C4×C7—81no—
217C24—21no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—81no—
223C2×D4—41no—
224C2×D4—81no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4C22⋊Dic711yescharacteristic
229C22×C4C22⋊Dic711yescharacteristic
230C22×C4C22⋊Dic711yescharacteristic
231C22×C4C22⋊Dic711yescharacteristic
232C22×C4C22×Dic711yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235M16—144no—
236M16—282no—
237C2×C8—282no—
238C2×C8—282no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C22⋊C4—42no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C2×C7—21no—
258C2×C7—21no—
259C2×C7—161no—
260C2×C7#150111yescharacteristic
261C2×C7#212211yescharacteristic
262C2×C7#213311yescharacteristic
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23C2×C22⋊Dic711yescharacteristic
271Q8—44no—
272Q8—82no—
273D4—82no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4#640811yescharacteristic
298C2×C4#640811yescharacteristic
299C2×C4#641112yes—
300C2×C4#651312yes—
301C8—284no—
302C8—562no—
303C7#1173811yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#640811yescharacteristic
314C22#9000511yescharacteristic
315C22C2×#641111yescharacteristic
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—21no—
328C2—21no—
329C2—161no—
330C2#1172946111yescharacteristic
331C2#1172961311yescharacteristic
332C2#1172962611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2632C2
socleC22×C7C2×#99
prime radicalC22×C7×D4C2×C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2632D7
hypocenterC7#11738
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2632

Complements

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

NH ≅ G/Ntype
2. C4×#6410C2semidirect
3. #11729378C2semidirect
4. #11725454C2semidirect
303. C7#11738semidirect

Maximal subgroups

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

subgroupindex
C7×#26322
C4×#64102
#117293782
#117254542
#117254532
#117387