Finite Group Databasebeta

← #13251463

Subgroups — #13251463

Order 1792.

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

normal subgroups only

This group has 490 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#11739831C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3C7×#1466C211yesmaximal, characteristic
4#11728568C212yesmaximal, complemented
5#11725556C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6433C2211yescharacteristic
8C7×#432C2212yes—
9#89993C411yescharacteristic
10#89993C411yescharacteristic
11#89982—21no—
12#89982—21no—
13#89979—21no—
14C2×C22⋊(C7⋊C8)—21no—
15#89783—24no—
16#89749C411yescharacteristic
17#89748C411yescharacteristic
18#89720—21no—
19#89720—21no—
20C4×C4⋊Dic7C2211yescharacteristic
21#89698C2212yes—
22C7×#323—21no—
23C7×#323—21no—
24#89260—24no—
25#9449—71nomaximal, Sylow
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4D411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—21no—
38C2×C22⋊Dic7—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×C4×Dic7C2×C411yescharacteristic
47C2×C4×Dic7C2×C411yescharacteristic
48C2×#1136—22no—
49C22×C7⋊C8—22no—
50#6433—22no—
51#6433—22no—
52C7×C4⋊C8—22no—
53C7×C22⋊C8—22no—
54C7×#97—22no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C22⋊(C7⋊C8)—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C4⋊(C7⋊C8)—22no—
62#6377D414yes—
63#2973—71no—
64#2566—71no—
65#1534—72no—
66#1466D711yescharacteristic
67#1285—72no—
68C24×C7C22⋊C411yescharacteristic
69C24×C7C22⋊C411yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—41no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—81no—
78C22×C4×C7—81no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×Dic7—21no—
85C22×Dic7—21no—
86C22×Dic7—21no—
87C22×Dic7—21no—
88C2×C7×C8—42no—
89C7×C4⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—22no—
92C42×C7C2×D412yes—
93C42×C7#4112yes—
94C22⋊Dic7—42no—
95C22⋊Dic7—42no—
96C22⋊Dic7—81no—
97C22⋊Dic7—81no—
98C4⋊Dic7—22no—
99C4⋊Dic7—22no—
100C4⋊Dic7—42no—
101Dic7⋊C4—41no—
102Dic7⋊C4—42no—
103Dic7⋊C4—41no—
104Dic7⋊C4—81no—
105Dic7⋊C4—81no—
106C4×Dic7—41no—
107C4×Dic7—41no—
108#1136—82no—
109C2×C7⋊C8—24no—
110C2×C7⋊C8—82no—
111C2×#125—71no—
112C2×C4×D4C2×D711yescharacteristic
113#435—144no—
114#432—72no—
115#432C2×D712yes—
116#407—141no—
117C2×C22⋊C8—141no—
118#396—141no—
119#396—141no—
120#390—71no—
121#390—71no—
122#385—141no—
123#385—141no—
124#384—71no—
125#382—71no—
126C4×C4⋊C4—71no—
127#330—144no—
128#323—21no—
129#323—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7#10112yes—
140C23×C7C2×C22⋊C411yescharacteristic
141C7×D4—44no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—81no—
153C2×C4×C7—81no—
154C2×C4×C7C4×D412yes—
155C2×C4×C7#12011yescharacteristic
156C2×C4×C7#12112yes—
157C2×C4×C7#12311yescharacteristic
158C2×Dic7—22no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C7×C8—82no—
171C7⋊C8—44no—
172C22×D4C4×D711yescharacteristic
173C23×C4D2811yescharacteristic
174C23×C4D2811yescharacteristic
175C2×M16—142no—
176C22×C8—142no—
177#125—142no—
178#125—142no—
179C4×D4—22no—
180C4×D4—22no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—141no—
184C2×C4⋊C4—141no—
185C2×C4⋊C4—141no—
186C2×C4⋊C4—141no—
187C2×C4⋊C4—141no—
188C2×C4⋊C4C4×D711yescharacteristic
189C2×C22⋊C4—141no—
190C2×C22⋊C4—141no—
191C2×C22⋊C4—141no—
192C2×C22⋊C4—141no—
193C2×C22⋊C4C22⋊D711yescharacteristic
194C2×C22⋊C4C22⋊D711yescharacteristic
195C2×C42—71no—
196C2×C42—71no—
197C2×C42C22×D711yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—142no—
200C22⋊C8—22no—
201C22⋊C8—142no—
202C22⋊C8—142no—
203C22⋊C8—142no—
204#97—22no—
205#97—74no—
206#95—141no—
207#95—141no—
208#95—141no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7#38611yescharacteristic
222C22×C7C2×#10111yescharacteristic
223C22×C7#41311yescharacteristic
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230Dic7—42no—
231Dic7—81no—
232Dic7—81no—
233Dic7—81no—
234C24(C2×D7)⋊C411yescharacteristic
235C24(C2×D7)⋊C411yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4(C2×D7)⋊C411yescharacteristic
250C22×C4(C2×D7)⋊C411yescharacteristic
251C22×C4C2×C4×D711yescharacteristic
252C22×C4C2×D2811yescharacteristic
253C22×C4C2×C22⋊D711yescharacteristic
254M16—562no—
255C2×C8—42no—
256C2×C8—144no—
257C2×C8—562no—
258C4⋊C4—22no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—282no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4—282no—
265C4⋊C4—561no—
266C4⋊C4—561no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—282no—
270C22⋊C4—282no—
271C22⋊C4—561no—
272C22⋊C4—561no—
273C42—281no—
274C42—281no—
275C42D4×D712yes—
276C42#115912yes—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—42no—
280C2×C7—81no—
281C2×C7#188411yescharacteristic
282C2×C7#190111yescharacteristic
283C2×C7#191111yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#639812yes—
294C23C2×(C2×D7)⋊C411yescharacteristic
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—142no—
309C2×C4—281no—
310C2×C4—281no—
311C2×C4—281no—
312C2×C4—281no—
313C2×C4—281no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—281no—
317C2×C4—281no—
318C2×C4—281no—
319C2×C4—281no—
320C2×C4(C22⋊D7)⋊C412yes—
321C2×C4#644511yescharacteristic
322C2×C4#644911yescharacteristic
323C2×C4#650212yes—
324C8—82no—
325C8—284no—
326C7#944911yescharacteristic, complemented, Sylow
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#8973011yescharacteristic
340C22#8990411yescharacteristic
341C22C2×#639811yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—282no—
349C4—561no—
350C4—561no—
351C4—561no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#1172774011yescharacteristic
357C2#1172779911yescharacteristic
358C2#1172782511yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1466C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1466D7
hypocenterC7#9449
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
4. #11728568C2semidirect
5. #11725556C2semidirect
326. C7#9449semidirect

Maximal subgroups

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

subgroupindex
#117398312
#117397472
C7×#14662
#117285682
#117255562
#94497