Finite Group Databasebeta

← #13272117

Subgroups — #13272117

Order 1792.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#11739921C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic, complemented
3C4×#6410C211yesmaximal, characteristic, complemented
4C2×#89272C214yesmaximal, complemented
5C2×C4×C7×Q8C2211yescharacteristic
6C2×#6510C2212yes—
7C2×#6437C411yescharacteristic
8C7×#423—21no—
9C2×C7×#101C2212yes—
10C2×#6410C2212yes—
11#89763—21no—
12#89763—21no—
13#89760C412yes—
14#89711—21no—
15#89711—21no—
16C4×C4⋊Dic7C411yescharacteristic
17C4×#1136—21no—
18C7×#343—21no—
19#89325—21no—
20#89272—28no—
21#13226—71nomaximal, Sylow
22C22×C7×Q8C2311yescharacteristic
23C22×D28—21no—
24C22×D28—21no—
25C22×D28C2×C412yes—
26C2×C7×M16—22no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C2×C7×C4⋊C4C2×C411yescharacteristic
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C42×C7C2×C411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35#6510—24no—
36#6510—24no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—22no—
40C2×(C2×D7)⋊C4—22no—
41C2×C4⋊Dic7—21no—
42C2×C4×Dic7C2×C412yes—
43C2×#1136—22no—
44#6437—41no—
45#6437—41no—
46C7×C4⋊C8—41no—
47C7×C4⋊C8—41no—
48C7×#101—24no—
49#6410—22no—
50#6410D414yescomplemented
51C7⋊#95—21no—
52#6377—41no—
53C4×C7⋊C8—41no—
54#2548—71no—
55C2×#355—74no—
56#1801D711yescharacteristic, complemented
57C4×#101—71no—
58C23×D7—22no—
59C2×C7×Q8—41no—
60C2×C7×Q8—41no—
61C2×C7×Q8C2×D412yes—
62C2×C7×Q8#4112yes—
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—22no—
71C2×D28—24no—
72C2×D28—42no—
73C2×D28—42no—
74C2×D28—42no—
75C2×D28—42no—
76C2×D28—42no—
77C2×D28—41no—
78C2×D28—41no—
79C2×D28—82no—
80C7×M16—42no—
81C7×M16—42no—
82C2×C7×C8—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92(C2×D7)⋊C4—44no—
93(C2×D7)⋊C4—42no—
94(C2×D7)⋊C4—42no—
95(C2×D7)⋊C4—82no—
96(C2×D7)⋊C4—82no—
97C4⋊Dic7—42no—
98C4×Dic7—24no—
99#1136—24no—
100#1136—42no—
101C2×C7⋊C8—42no—
102C2×#127—71no—
103C2×#124—72no—
104C2×C4×Q8C2×D711yescharacteristic
105#423—21no—
106C2×#101—72no—
107C2×#101C2×D712yes—
108C4×M16—141no—
109#394—141no—
110#394—141no—
111#390—72no—
112#390—141no—
113#390—141no—
114C4×C4⋊C4—71no—
115#355—148no—
116#343—21no—
117#343—141no—
118C23×C7C2×C22⋊C411yescharacteristic
119C22×D7—42no—
120C22×D7—42no—
121C22×D7—42no—
122C22×D7—42no—
123C22×D7—42no—
124C22×D7—42no—
125C22×D7—42no—
126C7×Q8—42no—
127C7×Q8—42no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7C4×D412yes—
144C2×C4×C7#12112yes—
145C2×C4×C7#12411yescharacteristic
146C2×C4×C7#12711yescharacteristic
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150D28—84no—
151D28—82no—
152D28—84no—
153D28—82no—
154C7×C8—82no—
155C7⋊C8—44no—
156C22×Q8C22×D711yescharacteristic
157C22×D4—72no—
158C22×D4—141no—
159C22×D4—141no—
160C2×M16—22no—
161C2×M16—142no—
162#127—281no—
163#127—281no—
164#124—144no—
165#124—144no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4C4×D711yescharacteristic
170C2×C4⋊C4D2811yescharacteristic
171C2×C4⋊C4C22⋊D711yescharacteristic
172C2×C22⋊C4—142no—
173C2×C22⋊C4—142no—
174C2×C22⋊C4—142no—
175C2×C22⋊C4—142no—
176C2×C42—72no—
177C2×C42C4×D711yescharacteristic
178C2×C42D2811yescharacteristic
179C2×C42C22⋊D711yescharacteristic
180C4⋊C8—41no—
181C4⋊C8—41no—
182#101—24no—
183#101—74no—
184#101—142no—
185#97—281no—
186C4×C8—281no—
187#95—141no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7#39011yescharacteristic
191C22×C7C2×#9911yescharacteristic
192C22×C7#41011yescharacteristic
193C2×D7—82no—
194C2×D7—82no—
195C2×D7—82no—
196C2×D7—82no—
197C2×D7—82no—
198C2×D7—82no—
199C2×D7—82no—
200C4×C7—21no—
201C4×C7—22no—
202C4×C7—21no—
203C4×C7—22no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7#35512yescomplemented
209Dic7—82no—
210C24—142no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8D4×D712yes—
214C2×Q8#116112yes—
215C2×D4—144no—
216C2×D4—282no—
217C2×D4—282no—
218C2×D4—282no—
219C2×D4—282no—
220C2×D4—282no—
221C2×D4—281no—
222C2×D4—281no—
223C2×D4—562no—
224C22×C4—142no—
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4(C2×D7)⋊C411yescharacteristic
227C22×C4(C2×D7)⋊C411yescharacteristic
228C22×C4(C2×D7)⋊C411yescharacteristic
229C22×C4C2×C4×D711yescharacteristic
230C22×C4C2×D2811yescharacteristic
231C22×C4C2×C22⋊D711yescharacteristic
232M16—42no—
233M16—42no—
234M16—144no—
235M16—282no—
236C2×C8—42no—
237C2×C8—282no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—282no—
242C22⋊C4—284no—
243C22⋊C4—282no—
244C22⋊C4—282no—
245C22⋊C4—562no—
246C22⋊C4—562no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—144no—
254C2×C7—21no—
255C2×C7—21no—
256C2×C7#196411yescharacteristic
257C2×C7C2×#35511yescharacteristic
258C2×C7#213411yescharacteristic
259D7—162no—
260C23—282no—
261C23—282no—
262C23—282no—
263C23—282no—
264C23—282no—
265C23—282no—
266C23—282no—
267C23C2×(C2×D7)⋊C411yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—564no—
271D4—562no—
272D4—564no—
273D4—562no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—282no—
288C2×C4—282no—
289C2×C4—282no—
290C2×C4#638011yescharacteristic
291C2×C4#638011yescharacteristic
292C2×C4#645612yes—
293C2×C4#645812yes—
294C2×C4#650311yescharacteristic
295C2×C4#651011yescharacteristic
296C8—82no—
297C8—284no—
298C7#1322611yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—562no—
302C22—562no—
303C22—562no—
304C22—562no—
305C22—562no—
306C22—562no—
307C22—562no—
308C22C2×#638011yescharacteristic
309C22#8976011yescharacteristic
310C22#8977811yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—562no—
320C4#8927212yescomplemented
321C2—21no—
322C2—21no—
323C2—1122no—
324C2#1172728211yescharacteristic
325C2C2×#8927211yescharacteristic
326C2#1172790311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC7×#1801C2
socleC22×C7C2×#99
prime radicalC2×#6437C4
cosocleC22×C7×Q8C23
hypercenter#1801D7
hypocenterC7#13226
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C7×#1801C2semidirect
3. C4×#6410C2semidirect
4. C2×#89272C2semidirect
50. #6410D4semidirect
56. #1801D7semidirect
208. C4×C7#355semidirect
298. C7#13226semidirect
320. C4#89272semidirect

Maximal subgroups

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

subgroupindex
#117399212
C7×#18012
C4×#64102
C2×#892722
#132267