Finite Group Databasebeta

← #13251195

Subgroups — #13251195

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#11736709C211yesmaximal, characteristic
2#11731166C211yesmaximal, characteristic
3#11728533C211yesmaximal, characteristic
4#11728282C211yesmaximal, characteristic
5#11728261C211yesmaximal, characteristic
6C7×#1303C211yesmaximal, characteristic
7#11725426C211yesmaximal, characteristic
8C2×#6432C2211yescharacteristic
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6398—21no—
14#89757—22no—
15#89707—21no—
16#89701—21no—
17#89697C2211yescharacteristic
18#89696C2211yescharacteristic
19#89603—22no—
20#89491—22no—
21#89451—22no—
22#89428C414yes—
23#89349—21no—
24#89276—22no—
25#89276—22no—
26#9400—71nomaximal, Sylow
27C22×Dic14C2×C412yes—
28C22×Dic14D411yescharacteristic
29C2×C7×M16—21no—
30C2×C7×C4⋊C4—21no—
31C2×C42×C7—21no—
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—24no—
36C2×C4⋊Dic7D411yescharacteristic
37C2×Dic7⋊C4C2×C412yes—
38C2×C4×Dic7D412yes—
39C2×#1136—21no—
40#6432—21no—
41#6432—21no—
42#6432—21no—
43#6432—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×#95C2×C412yes—
54C7⋊#95—24no—
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#1548—71no—
68#1303—71no—
69#1303D711yescharacteristic
70C22×C4×C7—22no—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—22no—
75C22×Dic7C22⋊C412yes—
76C22×Dic7C22⋊C412yes—
77C2×Dic14—21no—
78C2×Dic14—21no—
79C2×Dic14—21no—
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14—21no—
83C2×Dic14—22no—
84C2×Dic14—22no—
85C2×Dic14—82no—
86C7×M16—41no—
87C7×M16—41no—
88C2×C7×C8—22no—
89C2×C7×C8—41no—
90C7×C4⋊C4—24no—
91C42×C7—41no—
92C42×C7—41no—
93C42×C7C2×D411yescharacteristic
94C42×C7C2×D411yescharacteristic
95C42×C7#4111yescharacteristic
96C42×C7#4111yescharacteristic
97C4⋊Dic7—21no—
98C4⋊Dic7—21no—
99C4⋊Dic7—41no—
100C4⋊Dic7—41no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103C4⋊Dic7—84no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106C4×Dic7—22no—
107C4×Dic7—22no—
108#1136—41no—
109#1136—41no—
110C2×C7⋊C8—22no—
111C2×C7⋊C8—41no—
112C2×C4⋊Q8—71no—
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⋊C4C2×D711yescharacteristic
127#343—141no—
128#333—142no—
129#333—142no—
130C23×C7C2×C22⋊C411yescharacteristic
131C2×C4×C7—22no—
132C2×C4×C7—24no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7C4×D411yescharacteristic
137C2×C4×C7C4×D411yescharacteristic
138C2×C4×C7#12011yescharacteristic
139C2×C4×C7#12111yescharacteristic
140C2×C4×C7#12111yescharacteristic
141C2×C4×C7#12311yescharacteristic
142C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C7#13711yescharacteristic
144C2×C4×C7#13711yescharacteristic
145C2×C4×C7#13711yescharacteristic
146C2×Dic7—22no—
147C2×Dic7—22no—
148C2×Dic7—22no—
149C2×Dic7—22no—
150C2×Dic7—42no—
151C2×Dic7—44no—
152C2×Dic7—42no—
153Dic14—41no—
154Dic14—41no—
155Dic14—41no—
156Dic14—41no—
157Dic14—82no—
158Dic14—82no—
159C7×C8—42no—
160C7×C8—81no—
161C7⋊C8—42no—
162C7⋊C8—81no—
163C22×Q8—71no—
164C22×Q8—72no—
165C2×M16—21no—
166C2×M16—141no—
167C4⋊Q8—141no—
168C4⋊Q8—141no—
169C4⋊Q8—141no—
170C4⋊Q8—141no—
171C4×Q8—141no—
172C4×Q8—141no—
173C4×Q8—141no—
174C4×Q8—141no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—72no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4—144no—
181C2×C42—21no—
182C2×C42—72no—
183C2×C42C22×D711yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—72no—
187C4⋊C8—144no—
188C4⋊C8—141no—
189C4⋊C8—141no—
190C4⋊C8D2812yes—
191#101—282no—
192C22⋊C8—21no—
193C22⋊C8—141no—
194#97—144no—
195C4×C8—21no—
196C4×C8—141no—
197#95—144no—
198#95C4×D712yes—
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—44no—
208C4×C7#45612yes—
209C4×C7#46811yescharacteristic
210C4×C7#46811yescharacteristic
211C4×C7#47011yescharacteristic
212C4×C7#47011yescharacteristic
213Dic7—42no—
214Dic7—42no—
215Dic7—82no—
216C2×Q8—141no—
217C2×Q8—141no—
218C2×Q8—141no—
219C2×Q8—141no—
220C2×Q8—141no—
221C2×Q8—141no—
222C2×Q8—142no—
223C2×Q8—142no—
224C2×Q8—562no—
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—24no—
241C4⋊C4—142no—
242C4⋊C4—141no—
243C4⋊C4—142no—
244C4⋊C4—141no—
245C4⋊C4—281no—
246C4⋊C4—281no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C4⋊C4—564no—
250C42—41no—
251C42—41no—
252C42—142no—
253C42—142no—
254C42C2×D2811yescharacteristic
255C42#115711yescharacteristic
256C42D4×D711yescharacteristic
257C42#116111yescharacteristic
258C2×C7—21no—
259C2×C7—21no—
260C2×C7#188611yescharacteristic
261C2×C7#189411yescharacteristic
262C2×C7#191411yescharacteristic
263C23D7×C22⋊C411yescharacteristic
264Q8—281no—
265Q8—281no—
266Q8—281no—
267Q8—281no—
268Q8—562no—
269Q8—562no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—142no—
276C2×C4—142no—
277C2×C4—142no—
278C2×C4—142no—
279C2×C4—282no—
280C2×C4—284no—
281C2×C4—282no—
282C2×C4C4×D2811yescharacteristic
283C2×C4#644511yescharacteristic
284C2×C4#644711yescharacteristic
285C2×C4#645611yescharacteristic
286C2×C4#645711yescharacteristic
287C2×C4#645811yescharacteristic
288C2×C4#647411yescharacteristic
289C2×C4#647811yescharacteristic
290C2×C4#647811yescharacteristic
291C2×C4#648111yescharacteristic
292C8—42no—
293C8—81no—
294C8—282no—
295C8—561no—
296C7#940011yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22#8943711yescharacteristic
300C22#8954311yescharacteristic
301C22#8957411yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—44no—
306C4—282no—
307C4—282no—
308C4—562no—
309C4#8955211yescharacteristic
310C4#8958811yescharacteristic
311C4#8959812yes—
312C4#8964011yescharacteristic
313C4#8965811yescharacteristic
314C2—21no—
315C2—21no—
316C2#1172651911yescharacteristic
317C2#1172663811yescharacteristic
318C2#1172679411yescharacteristic
subgroupG/H
centerC22#89437
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1303C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1303D7
hypocenterC7#9400
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

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
#117367092
#117311662
#117285332
#117282822
#117282612
C7×#13032
#117254262
#94007