Finite Group Databasebeta

← #13251165

Subgroups — #13251165

Order 1792.

506 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 117 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#11736708C211yesmaximal, characteristic
2#11731156C211yesmaximal, characteristic
3#11728532C211yesmaximal, characteristic
4#11728282C211yesmaximal, characteristic
5#11728260C211yesmaximal, characteristic
6C7×#1303C211yesmaximal, characteristic
7#11725426C211yesmaximal, characteristic
8C2×#6432—21no—
9C2×#6432C2211yescharacteristic
10C2×C4×Dic14C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13C4×C7×C4⋊C4C2211yescharacteristic
14C2×#6398—21no—
15#89757—22no—
16#89701—21no—
17#89697C2211yescharacteristic
18#89696C2211yescharacteristic
19#89602—22no—
20#89490—22no—
21C8×Dic14—22no—
22#89433C414yes—
23#89349—21no—
24#89281—22no—
25#89277—22no—
26#9394—71nomaximal, Sylow
27C22×Dic14—21no—
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—24no—
35C2×C4⋊Dic7C2×C412yes—
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—
44#6432—42no—
45#6432—42no—
46C4×Dic14—21no—
47C4×Dic14—21no—
48C4×Dic14—21no—
49C4×Dic14—21no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8D412yes—
53C7×C22⋊C8—21no—
54C4×C7×C8—21no—
55C7×#95C2×C412yes—
56C7⋊#95—24no—
57C22⋊(C7⋊C8)—21no—
58#6398—42no—
59Dic7⋊C8—24no—
60C8×Dic7—24no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)D412yes—
64C4×C7⋊C8—21no—
65#2982—71no—
66#2546—71no—
67#1801—71no—
68#1546—71no—
69#1546—71no—
70#1303—71no—
71#1303D711yescharacteristic
72C22×C4×C7—22no—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—22no—
77C22×Dic7C22⋊C412yes—
78C22×Dic7C22⋊C412yes—
79C2×Dic14—21no—
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14—21no—
83C2×Dic14—41no—
84C2×Dic14—41no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C2×Dic14—41no—
88C2×Dic14—41no—
89C2×Dic14D812yes—
90C2×Dic14SD1612yes—
91C7×M16—41no—
92C7×M16—41no—
93C2×C7×C8—22no—
94C2×C7×C8—41no—
95C7×C4⋊C4—24no—
96C42×C7—41no—
97C42×C7—41no—
98C42×C7C2×D411yescharacteristic
99C42×C7C2×D411yescharacteristic
100C42×C7#4111yescharacteristic
101C42×C7#4111yescharacteristic
102C4⋊Dic7—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7—22no—
105C4⋊Dic7—22no—
106C4⋊Dic7—41no—
107C4⋊Dic7—41no—
108C4⋊Dic7—48no—
109Dic7⋊C4—22no—
110Dic7⋊C4—22no—
111C4×Dic7—22no—
112C4×Dic7—22no—
113#1136—41no—
114#1136—41no—
115C2×C7⋊C8—22no—
116C2×C7⋊C8—41no—
117C2×C4⋊Q8—71no—
118C2×C4⋊Q8—141no—
119C2×C4×Q8—71no—
120C8×Q8—142no—
121C8×Q8—142no—
122#431—71no—
123#431—142no—
124#431C2×D711yescharacteristic
125#423—71no—
126#423C2×D711yescharacteristic
127C2×#101—141no—
128#398—74no—
129#391—141no—
130#389—142no—
131C4×C4⋊C4C2×D711yescharacteristic
132#343—141no—
133#332—142no—
134#332—142no—
135C23×C7C2×C22⋊C411yescharacteristic
136C2×C4×C7—22no—
137C2×C4×C7—24no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7C4×D411yescharacteristic
142C2×C4×C7C4×D411yescharacteristic
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7#12111yescharacteristic
145C2×C4×C7#12111yescharacteristic
146C2×C4×C7#12311yescharacteristic
147C2×C4×C7C2×D811yescharacteristic
148C2×C4×C7C2×SD1611yescharacteristic
149C2×C4×C7#13511yescharacteristic
150C2×C4×C7#13511yescharacteristic
151C2×Dic7—22no—
152C2×Dic7—22no—
153C2×Dic7—22no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7#10214yes—
159Dic14—22no—
160Dic14—22no—
161Dic14—22no—
162Dic14—22no—
163Dic14—81no—
164Dic14—81no—
165Dic14—81no—
166Dic14—81no—
167C7×C8—42no—
168C7×C8—81no—
169C7⋊C8—42no—
170C7⋊C8—81no—
171C22×Q8—71no—
172C22×Q8—141no—
173C2×M16—21no—
174C2×M16—141no—
175C4⋊Q8—141no—
176C4⋊Q8—141no—
177C4⋊Q8—141no—
178C4⋊Q8—141no—
179C4⋊Q8—282no—
180C4⋊Q8—282no—
181C4×Q8—141no—
182C4×Q8—141no—
183C4×Q8—141no—
184C4×Q8—141no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—72no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—72no—
189C2×C4⋊C4—144no—
190C2×C4⋊C4—141no—
191C2×C42—21no—
192C2×C42—72no—
193C2×C42C22×D711yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—72no—
197C4⋊C8—144no—
198C4⋊C8—141no—
199C4⋊C8—141no—
200C4⋊C8D2812yes—
201#101—282no—
202C22⋊C8—21no—
203C22⋊C8—141no—
204C4×C8—21no—
205C4×C8—144no—
206C4×C8—141no—
207#95—144no—
208#95C4×D712yes—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7#38611yescharacteristic
212C22×C7C2×#10211yescharacteristic
213C22×C7#41611yescharacteristic
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—44no—
218C4×C7#45612yes—
219C4×C7#46511yescharacteristic
220C4×C7#46511yescharacteristic
221C4×C7#46611yescharacteristic
222C4×C7#46611yescharacteristic
223Dic7—24no—
224Dic7—42no—
225Dic7—82no—
226C2×Q8—72no—
227C2×Q8—72no—
228C2×Q8—141no—
229C2×Q8—141no—
230C2×Q8—141no—
231C2×Q8—141no—
232C2×Q8—281no—
233C2×Q8—281no—
234C2×Q8—281no—
235C2×Q8—281no—
236C2×Q8—281no—
237C2×Q8—281no—
238C22×C4—22no—
239C22×C4—72no—
240C22×C4—72no—
241C22×C4—142no—
242C22×C4C2×C4×D711yescharacteristic
243C22×C4D4×D711yescharacteristic
244C22×C4D4×D711yescharacteristic
245M16—41no—
246M16—41no—
247M16—281no—
248M16—281no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—142no—
252C2×C8—281no—
253C4⋊C4—24no—
254C4⋊C4—142no—
255C4⋊C4—141no—
256C4⋊C4—142no—
257C4⋊C4—141no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—288no—
261C4⋊C4—281no—
262C4⋊C4—281no—
263C42—41no—
264C42—41no—
265C42—142no—
266C42—142no—
267C42C2×D2811yescharacteristic
268C42#115711yescharacteristic
269C42D4×D711yescharacteristic
270C42#116111yescharacteristic
271C2×C7—21no—
272C2×C7—21no—
273C2×C7#188611yescharacteristic
274C2×C7#189311yescharacteristic
275C2×C7#191511yescharacteristic
276C23D7×C22⋊C411yescharacteristic
277Q8—142no—
278Q8—142no—
279Q8—142no—
280Q8—142no—
281Q8—561no—
282Q8—561no—
283Q8—561no—
284Q8—561no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—74no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—142no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4C4×D2811yescharacteristic
299C2×C4#644511yescharacteristic
300C2×C4#644711yescharacteristic
301C2×C4#645611yescharacteristic
302C2×C4#645711yescharacteristic
303C2×C4#645811yescharacteristic
304C2×C4D7×D811yescharacteristic
305C2×C4D7×SD1611yescharacteristic
306C2×C4#647911yescharacteristic
307C2×C4#648211yescharacteristic
308C8—42no—
309C8—81no—
310C8—282no—
311C8—561no—
312C7#939411yescharacteristic, complemented, Sylow
313C22—21no—
314C22—21no—
315C22#8943711yescharacteristic
316C22D7×#10211yescharacteristic
317C22#8957611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—144no—
323C4—282no—
324C4—562no—
325C4#8954711yescharacteristic
326C4#8957911yescharacteristic
327C4#8959812yes—
328C4#8963611yescharacteristic
329C4#8965511yescharacteristic
330C2—21no—
331C2—21no—
332C2#1172651911yescharacteristic
333C2#1172663711yescharacteristic
334C2#1172679511yescharacteristic
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#9394
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
312. C7#9394semidirect

Maximal subgroups

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

subgroupindex
#117367082
#117311562
#117285322
#117282822
#117282602
C7×#13032
#117254262
#93947