Finite Group Databasebeta

← #13251193

Subgroups — #13251193

Order 1792.

466 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 109 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#11740688C211yesmaximal, characteristic
2#11736709C211yesmaximal, characteristic
3C7×#1548C211yesmaximal, characteristic
4#11729528C211yesmaximal, characteristic
5#11728282C211yesmaximal, characteristic
6#11725552C211yesmaximal, characteristic
7#11725521C211yesmaximal, characteristic
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12#90001C414yes—
13C2×#6398—21no—
14#89748—21no—
15#89746—21no—
16#89745—22no—
17C4×C4⋊Dic7C2211yescharacteristic
18#89697C2211yescharacteristic
19#89696C2211yescharacteristic
20#89603—22no—
21#89491—22no—
22#89451—22no—
23C7×#333—22no—
24#89349—21no—
25#89341—22no—
26#9400—71nomaximal, Sylow
27C22×C7×Q8C2×C412yes—
28C22×Dic14D411yescharacteristic
29C2×C7×M16—21no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C42×C7C2311yescharacteristic
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7D411yescharacteristic
39C2×Dic7⋊C4—24no—
40C2×Dic7⋊C4C2×C412yes—
41C2×C4×Dic7—21no—
42C2×C4×Dic7D412yes—
43C2×#1136—21no—
44C4×Dic14—21no—
45C4×Dic14—21no—
46C4×Dic14—21no—
47C4×Dic14—21no—
48C7×C4⋊C8—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8D412yes—
51C7×C22⋊C8—21no—
52C4×C7×C8—21no—
53C7⋊#95—24no—
54C7⋊#95C2×C412yes—
55C22⋊(C7⋊C8)—21no—
56#6398—42no—
57#6389—24no—
58Dic7⋊C8—24no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)D412yes—
62C4×C7⋊C8—21no—
63#2983—71no—
64#2567—71no—
65#1801—71no—
66#1548—71no—
67#1548D711yescharacteristic
68#1303—71no—
69#1303—71no—
70C2×C7×Q8—22no—
71C2×C7×Q8—22no—
72C2×C7×Q8—82no—
73C22×C4×C7—22no—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—22no—
78C22×Dic7C22⋊C412yes—
79C22×Dic7C22⋊C412yes—
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14—21no—
83C2×Dic14—21no—
84C2×Dic14—21no—
85C2×Dic14—21no—
86C7×M16—41no—
87C7×M16—41no—
88C2×C7×C8—22no—
89C2×C7×C8—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C42×C7C2×D411yescharacteristic
95C42×C7C2×D411yescharacteristic
96C42×C7#4111yescharacteristic
97C42×C7#4111yescharacteristic
98C4⋊Dic7—24no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—84no—
104C4×Dic7—22no—
105C4×Dic7—22no—
106C4×Dic7—41no—
107C4×Dic7—41no—
108#1136—41no—
109#1136—41no—
110C2×C7⋊C8—22no—
111C2×C7⋊C8—41no—
112C2×C4⋊Q8C2×D711yescharacteristic
113C2×C4×Q8—71no—
114#446—142no—
115#446—142no—
116#432—142no—
117#431—71no—
118#431C2×D711yescharacteristic
119#423—71no—
120#423C2×D711yescharacteristic
121C2×#101—141no—
122#395—74no—
123#391—141no—
124#389—142no—
125#384—141no—
126C4×C4⋊C4—71no—
127#343—141no—
128#333—22no—
129#333—142no—
130C23×C7C2×C22⋊C411yescharacteristic
131C7×Q8—82no—
132C7×Q8—82no—
133C2×C4×C7—22no—
134C2×C4×C7—41no—
135C2×C4×C7—44no—
136C2×C4×C7—42no—
137C2×C4×C7C4×D411yescharacteristic
138C2×C4×C7C4×D411yescharacteristic
139C2×C4×C7#12011yescharacteristic
140C2×C4×C7#12111yescharacteristic
141C2×C4×C7#12111yescharacteristic
142C2×C4×C7#12311yescharacteristic
143C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C7#13711yescharacteristic
145C2×C4×C7#13711yescharacteristic
146C2×C4×C7#13711yescharacteristic
147C2×Dic7—24no—
148C2×Dic7—22no—
149C2×Dic7—22no—
150C2×Dic7—22no—
151C2×Dic7—22no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155Dic14—41no—
156Dic14—41no—
157Dic14—41no—
158Dic14—41no—
159C7×C8—42no—
160C7×C8—81no—
161C7⋊C8—42no—
162C7⋊C8—81no—
163C22×Q8—71no—
164C22×Q8C4×D712yes—
165C2×M16—21no—
166C2×M16—141no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4×Q8—141no—
172C4×Q8—141no—
173C4×Q8—141no—
174C4×Q8—141no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—72no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4—144no—
181C2×C42—72no—
182C2×C42—141no—
183C2×C42C22×D711yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—72no—
187C4⋊C8—144no—
188C4⋊C8—141no—
189C4⋊C8—141no—
190C4⋊C8C22⋊D712yes—
191#101—282no—
192C22⋊C8—21no—
193C22⋊C8—141no—
194#97—144no—
195C4×C8—21no—
196C4×C8—141no—
197#95—72no—
198#95—144no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#38611yescharacteristic
202C22×C7#41711yescharacteristic
203C22×C7#41911yescharacteristic
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—82no—
208C4×C7#45612yes—
209C4×C7#46811yescharacteristic
210C4×C7#46811yescharacteristic
211C4×C7#47011yescharacteristic
212C4×C7#47011yescharacteristic
213Dic7—44no—
214Dic7—42no—
215Dic7—42no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—82no—
219C2×Q8—141no—
220C2×Q8—141no—
221C2×Q8—141no—
222C2×Q8—141no—
223C2×Q8—141no—
224C2×Q8—141no—
225C22×C4—22no—
226C22×C4—72no—
227C22×C4—72no—
228C22×C4—142no—
229C22×C4C2×C4×D711yescharacteristic
230C22×C4D4×D711yescharacteristic
231C22×C4D4×D711yescharacteristic
232M16—41no—
233M16—41no—
234M16—281no—
235M16—281no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—142no—
239C2×C8—281no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—144no—
245C4⋊C4—142no—
246C4⋊C4—141no—
247C4⋊C4—142no—
248C4⋊C4—141no—
249C4⋊C4—564no—
250C42—142no—
251C42—142no—
252C42—281no—
253C42—281no—
254C42#115711yescharacteristic
255C42D4×D711yescharacteristic
256C42#115911yescharacteristic
257C42C2×C22⋊D711yescharacteristic
258C2×C7—21no—
259C2×C7—21no—
260C2×C7#188611yescharacteristic
261C2×C7#189411yescharacteristic
262C2×C7#191411yescharacteristic
263C23D7×C22⋊C411yescharacteristic
264Q8—82no—
265Q8—82no—
266Q8—281no—
267Q8—281no—
268Q8—281no—
269Q8—281no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—144no—
275C2×C4—142no—
276C2×C4—142no—
277C2×C4—142no—
278C2×C4—142no—
279C2×C4—282no—
280C2×C4—282no—
281C2×C4—282no—
282C2×C4(C22⋊D7)⋊C411yescharacteristic
283C2×C4#644611yescharacteristic
284C2×C4#644711yescharacteristic
285C2×C4#647711yescharacteristic
286C2×C4#647811yescharacteristic
287C2×C4#648111yescharacteristic
288C2×C4#648111yescharacteristic
289C2×C4C4×C22⋊D711yescharacteristic
290C2×C4#650011yescharacteristic
291C2×C4#650111yescharacteristic
292C8—42no—
293C8—81no—
294C8—282no—
295C8—561no—
296C7#940011yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22#8957411yescharacteristic
300C22#8957511yescharacteristic
301C22#8972811yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—284no—
307C4—282no—
308C4—282no—
309C4#8958411yescharacteristic
310C4#8958811yescharacteristic
311C4#8994611yescharacteristic
312C4#8996111yescharacteristic
313C4#8997712yes—
314C2—21no—
315C2—21no—
316C2#1172784311yescharacteristic
317C2#1172802511yescharacteristic
318C2#1172809911yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1548C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1548D7
hypocenterC7#9400
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
296. C7#9400semidirect

Maximal subgroups

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

subgroupindex
#117406882
#117367092
C7×#15482
#117295282
#117282822
#117255522
#117255212
#94007