Finite Group Databasebeta

← #13247380

Subgroups — #13247380

Order 1792.

378 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 99 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#11731604C211yesmaximal, characteristic
2#11728540C211yesmaximal, characteristic
3#11728282C211yesmaximal, characteristic
4C7×#1286C211yesmaximal, characteristic
5#11725552C211yesmaximal, characteristic
6#11725521C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C2×C4×Dic14C2211yescharacteristic
9C7×#431C2211yescharacteristic
10C7×#423C2211yescharacteristic
11C7×#382C2211yescharacteristic
12C2×#6398—21no—
13C4⋊C4×Dic7—21no—
14#89746—21no—
15C4×C4⋊Dic7C2211yescharacteristic
16#89697C2211yescharacteristic
17#89696C2211yescharacteristic
18#89427—22no—
19#89423C412yes—
20#89420C412yes—
21#89349—21no—
22#89264—22no—
23#8768—71nomaximal, Sylow
24C22×Dic14D411yescharacteristic
25C2×C7×M16—21no—
26C2×C7×C4⋊C4—21no—
27C2×C42×C7C2311yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7D411yescharacteristic
30C2×Dic7⋊C4D412yes—
31C2×C4×Dic7—21no—
32C2×C4×Dic7—22no—
33C2×C4×Dic7C2×C412yes—
34C2×#1136—21no—
35C4×Dic14—21no—
36C4×Dic14—21no—
37C4×Dic14—21no—
38C4×Dic14—21no—
39C7×C4⋊C8—21no—
40C7×C4⋊C8—21no—
41C7×C4⋊C8—21no—
42C7×C22⋊C8—21no—
43C4×C7×C8—21no—
44C7×#95—21no—
45C7×#95C2×C412yes—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95C2×C411yescharacteristic
50C7⋊#95C2×C411yescharacteristic
51C22⋊(C7⋊C8)—21no—
52#6398—42no—
53C4⋊(C7⋊C8)—21no—
54C4⋊(C7⋊C8)—21no—
55C4⋊(C7⋊C8)—21no—
56C4×C7⋊C8—21no—
57#2356—71no—
58#1801—71no—
59#1461—71no—
60#1303—71no—
61#1303—71no—
62#1286—71no—
63#1286D711yescharacteristic
64C22×C4×C7—22no—
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—21no—
70C22×Dic7C22⋊C412yes—
71C22×Dic7C22⋊C412yes—
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×Dic14—21no—
78C7×M16—41no—
79C7×M16—41no—
80C2×C7×C8—41no—
81C2×C7×C8—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C42×C7#4111yescharacteristic
85C42×C7#4111yescharacteristic
86C42×C7#4111yescharacteristic
87C42×C7#4111yescharacteristic
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—42no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93C4×Dic7—22no—
94C4×Dic7—22no—
95C4×Dic7—44no—
96C4×Dic7—41no—
97C4×Dic7—41no—
98#1136—41no—
99#1136—41no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8—41no—
102C2×C4×Q8—71no—
103#431—71no—
104#431C2×D711yescharacteristic
105#423—71no—
106#423C2×D711yescharacteristic
107C2×#101—141no—
108#391—141no—
109#387—142no—
110#383—72no—
111#382C2×D711yescharacteristic
112C4×C4⋊C4—71no—
113C4×C4⋊C4—141no—
114#376—72no—
115#343—141no—
116#324—142no—
117C23×C7C2×C22⋊C411yescharacteristic
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7#11711yescharacteristic
124C2×C4×C7#11711yescharacteristic
125C2×C4×C7#12311yescharacteristic
126C2×C4×C7#12311yescharacteristic
127C2×C4×C7#12311yescharacteristic
128C2×C4×C7#12311yescharacteristic
129C2×C4×C7#13511yescharacteristic
130C2×C4×C7#13511yescharacteristic
131C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C7#13711yescharacteristic
133C2×Dic7—22no—
134C2×Dic7—22no—
135C2×Dic7—22no—
136C2×Dic7—22no—
137C2×Dic7—22no—
138C2×Dic7—41no—
139C2×Dic7—41no—
140C2×Dic7—42no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143C2×Dic7—41no—
144Dic14—41no—
145Dic14—41no—
146Dic14—41no—
147Dic14—41no—
148C7×C8—81no—
149C7×C8—81no—
150C7⋊C8—81no—
151C7⋊C8—81no—
152C22×Q8—71no—
153C2×M16—21no—
154C2×M16—141no—
155C4×Q8—141no—
156C4×Q8—141no—
157C4×Q8—141no—
158C4×Q8—141no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—72no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—141no—
163C2×C42—72no—
164C2×C42—142no—
165C2×C42—141no—
166C2×C42C22×D711yescharacteristic
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—141no—
171C4⋊C8—141no—
172C4⋊C8—141no—
173#101—282no—
174C22⋊C8—21no—
175C22⋊C8—141no—
176C4×C8—21no—
177C4×C8—141no—
178#95—21no—
179#95—71no—
180#95—71no—
181#95—142no—
182#95—142no—
183#95—142no—
184#95C4×D712yes—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7#38111yescharacteristic
188C22×C7#41611yescharacteristic
189C22×C7#41711yescharacteristic
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—21no—
193C4×C7—21no—
194C4×C7—82no—
195C4×C7#48211yescharacteristic
196C4×C7#48211yescharacteristic
197C4×C7#48511yescharacteristic
198C4×C7#48511yescharacteristic
199Dic7—42no—
200Dic7—42no—
201Dic7—42no—
202Dic7—81no—
203C2×Q8—141no—
204C2×Q8—141no—
205C2×Q8—141no—
206C2×Q8—141no—
207C2×Q8—141no—
208C2×Q8—141no—
209C22×C4—22no—
210C22×C4—72no—
211C22×C4—72no—
212C22×C4—141no—
213C22×C4—141no—
214C22×C4C2×C4×D711yescharacteristic
215C22×C4D4×D711yescharacteristic
216C22×C4D4×D711yescharacteristic
217M16—41no—
218M16—41no—
219M16—281no—
220M16—281no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—281no—
224C2×C8—281no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—142no—
228C4⋊C4—141no—
229C4⋊C4—142no—
230C4⋊C4—141no—
231C4⋊C4—282no—
232C42—142no—
233C42—142no—
234C42—284no—
235C42—281no—
236C42—281no—
237C42#115711yescharacteristic
238C42#115711yescharacteristic
239C42#115911yescharacteristic
240C42#116111yescharacteristic
241C2×C7—21no—
242C2×C7—21no—
243C2×C7#182511yescharacteristic
244C2×C7#182711yescharacteristic
245C2×C7#183011yescharacteristic
246C23D7×C22⋊C411yescharacteristic
247Q8—281no—
248Q8—281no—
249Q8—281no—
250Q8—281no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—281no—
262C2×C4—281no—
263C2×C4—282no—
264C2×C4—281no—
265C2×C4—281no—
266C2×C4—281no—
267C2×C4#643511yescharacteristic
268C2×C4#644611yescharacteristic
269C2×C4#644611yescharacteristic
270C2×C4#645511yescharacteristic
271C2×C4#645711yescharacteristic
272C2×C4#645711yescharacteristic
273C2×C4#647511yescharacteristic
274C2×C4#647711yescharacteristic
275C2×C4#647911yescharacteristic
276C2×C4#648111yescharacteristic
277C8—81no—
278C8—81no—
279C8—561no—
280C8—561no—
281C7#876811yescharacteristic, complemented, Sylow
282C22—21no—
283C22—21no—
284C22#8943611yescharacteristic
285C22#8954411yescharacteristic
286C22#8957511yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—82no—
292C4—282no—
293C4—282no—
294C4—282no—
295C4—561no—
296C4#8955611yescharacteristic
297C4#8958711yescharacteristic
298C4#8963811yescharacteristic
299C4#8966011yescharacteristic
300C2—21no—
301C2—21no—
302C2#1172646511yescharacteristic
303C2#1172662711yescharacteristic
304C2#1172678311yescharacteristic
subgroupG/H
centerC22#89436
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1286C2
socleC22×C7#381
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1286D7
hypocenterC7#8768
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
281. C7#8768semidirect

Maximal subgroups

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

subgroupindex
#117316042
#117285402
#117282822
C7×#12862
#117255522
#117255212
#117254372
#87687