Finite Group Databasebeta

← #13257556

Subgroups — #13257556

Order 1792.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#11740795C211yesmaximal, characteristic
2C7×#1542C211yesmaximal, characteristic, complemented
3#11729592C211yesmaximal, characteristic, complemented
4#11728537C211yesmaximal, characteristic
5#11728234C211yesmaximal, characteristic
6#11728175C211yesmaximal, characteristic, complemented
7#11725544C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×D28C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×#6395—21no—
13#89859—21no—
14#89856—21no—
15#89838—22no—
16#89773—21no—
17#89767—21no—
18#89708C2211yescharacteristic
19#89697C2211yescharacteristic
20#89696C2211yescharacteristic
21C7×#333—21no—
22C7×#332—21no—
23#89346—21no—
24#89333—22no—
25#89275—22no—
26#89263—22no—
27#10468—71nomaximal, Sylow
28C22×D28D411yescharacteristic
29C22×C4×D7—21no—
30C22×C7×C8—21no—
31C7×C4⋊Q8D411yescharacteristic
32C7×C4⋊Q8D411yescharacteristic
33C7×#125—21no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C4×C7×Q8—21no—
37C4×C7×Q8—21no—
38C7×#117—21no—
39C2×C42×C7C2311yescharacteristic
40C2×(C2×D7)⋊C4—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7D411yescharacteristic
43C2×#1136—21no—
44C4×D28—21no—
45C4×D28—21no—
46C4×D28—21no—
47C4×D28—21no—
48C7×C4⋊C8—21no—
49C7×C4⋊C8—21no—
50C7×C4⋊C8—21no—
51C7×C22⋊C8—21no—
52C4×C7×C8—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C22⋊(C7⋊C8)D412yes—
56#6395—42no—
57#6383—22no—
58#6383—22no—
59#6382—22no—
60#6381—22no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)D412yes—
64C4×C7⋊C8—21no—
65#3077—71no—
66#1804—71no—
67#1570—71no—
68#1542D711yescharacteristic, complemented
69#1518—71no—
70#1460—71no—
71#1286—71no—
72C23×D7—21no—
73C2×C7×Q8—21no—
74C2×C7×Q8—21no—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7#4111yescharacteristic
78C22×Dic7—21no—
79C22×Dic7—41no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28—21no—
83C2×D28—21no—
84C2×D28—21no—
85C2×D28SD1612yes—
86C2×C4×D7—41no—
87C2×C4×D7—41no—
88C2×C4×D7—82no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C2×C7×C8—41no—
92C2×C7×C8—41no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—21no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C22⋊C4—41no—
100C42×C7—41no—
101C42×C7C2×D411yescharacteristic
102C42×C7C2×D411yescharacteristic
103C42×C7#4111yescharacteristic
104C42×C7#4111yescharacteristic
105(C2×D7)⋊C4—41no—
106(C2×D7)⋊C4—41no—
107C4⋊Dic7—21no—
108C4⋊Dic7—21no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111#1136—41no—
112#1136—41no—
113C2×C7⋊C8—22no—
114C2×C7⋊C8—22no—
115#533C2×D711yescharacteristic
116C2×C4×D4—71no—
117#478—142no—
118#476—141no—
119#474—141no—
120#431—71no—
121#431C2×D711yescharacteristic
122#428—141no—
123#423—71no—
124C2×C4⋊C8C2×D711yescharacteristic
125#418—141no—
126C2×#102—141no—
127#382—71no—
128#340—141no—
129#334—142no—
130#333—21no—
131#332—21no—
132#328—142no—
133#323—142no—
134C23×C7#12111yescharacteristic
135C22×D7—22no—
136C22×D7—41no—
137C22×D7—41no—
138C22×D7—81no—
139C22×D7—81no—
140C7×Q8—81no—
141C7×Q8—81no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7#12011yescharacteristic
147C2×C4×C7#12111yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12311yescharacteristic
150C2×C4×C7#12311yescharacteristic
151C2×C4×C7#12411yescharacteristic
152C2×C4×C7C2×SD1611yescharacteristic
153C2×C4×C7#13511yescharacteristic
154C2×C4×C7#13711yescharacteristic
155C2×C4×C7#13711yescharacteristic
156C2×Dic7—41no—
157C2×Dic7—81no—
158C2×Dic7—81no—
159C2×Dic7—81no—
160C2×Dic7—81no—
161D28—22no—
162D28—22no—
163D28—41no—
164D28—41no—
165C4×D7—82no—
166C4×D7—82no—
167C7×C8—81no—
168C7×C8—81no—
169C7⋊C8—42no—
170C7⋊C8—42no—
171C22×D4—71no—
172C23×C4—141no—
173C2×M16—141no—
174C22×C8—21no—
175C4⋊Q8C22⋊D711yescharacteristic
176C4⋊Q8C22⋊D711yescharacteristic
177#125—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×D4—141no—
183C4×D4—141no—
184C4×D4—141no—
185C4×D4—141no—
186#117—21no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—141no—
189C2×C22⋊C4—141no—
190C2×C42C22×D711yescharacteristic
191#107—142no—
192#106—142no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—72no—
197C4⋊C8—141no—
198C4⋊C8—141no—
199#102—142no—
200#102—142no—
201#102—282no—
202C22⋊C8—21no—
203C22⋊C8—72no—
204C4×C8—21no—
205C4×C8—141no—
206#95—141no—
207#95—141no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7#39411yescharacteristic
211C22×C7#46511yescharacteristic
212C22×C7#47011yescharacteristic
213C2×D7—42no—
214C2×D7—82no—
215C2×D7—81no—
216C2×D7—81no—
217C2×D7—82no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—81no—
223C4×C7—81no—
224C4×C7#44811yescharacteristic
225C4×C7#45211yescharacteristic
226C4×C7#48311yescharacteristic
227C4×C7#48511yescharacteristic
228Dic7—81no—
229Dic7—161no—
230C24—141no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×D4—72no—
234C2×D4—141no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C22×C4—141no—
240C22×C4—281no—
241C22×C4—281no—
242C22×C4—281no—
243C22×C4—562no—
244C22×C4#115711yescharacteristic
245C22×C4D4×D711yescharacteristic
246C22×C4D4×D711yescharacteristic
247M16—281no—
248M16—281no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—142no—
254C2×C8—142no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—281no—
264C4⋊C4—281no—
265C22⋊C4—41no—
266C22⋊C4—281no—
267C22⋊C4—281no—
268C42—41no—
269C42#115711yescharacteristic
270C42D4×D711yescharacteristic
271C42#115911yescharacteristic
272C42C2×C22⋊D711yescharacteristic
273C2×C7—21no—
274C2×C7—21no—
275C2×C7#203511yescharacteristic
276C2×C7#204811yescharacteristic
277C2×C7#205311yescharacteristic
278D7—82no—
279C23—142no—
280C23—281no—
281C23—281no—
282C23—561no—
283C23—561no—
284C23#644711yescharacteristic
285Q8—81no—
286Q8—81no—
287D4—142no—
288D4—142no—
289D4—281no—
290D4—281no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—281no—
296C2×C4—562no—
297C2×C4—562no—
298C2×C4—561no—
299C2×C4—561no—
300C2×C4—561no—
301C2×C4—561no—
302C2×C4#644611yescharacteristic
303C2×C4#644711yescharacteristic
304C2×C4#644811yescharacteristic
305C2×C4D7×SD1611yescharacteristic
306C2×C4#647811yescharacteristic
307C2×C4#648111yescharacteristic
308C2×C4#648211yescharacteristic
309C2×C4#649211yescharacteristic
310C2×C4#650011yescharacteristic
311C2×C4#650211yescharacteristic
312C8—81no—
313C8—81no—
314C8—282no—
315C8—282no—
316C7#1046811yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22—282no—
320C22—562no—
321C22—562no—
322C22—561no—
323C22—561no—
324C22#8957911yescharacteristic
325C22#8958811yescharacteristic
326C22#8973111yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—81no—
333C4—561no—
334C4—1121no—
335C4#8957711yescharacteristic
336C4#8958611yescharacteristic
337C4#8994211yescharacteristic
338C4#8995911yescharacteristic
339C2—21no—
340C2—21no—
341C2—562no—
342C2#1172789411yescharacteristic
343C2#1172802811yescharacteristic
344C2#1172805011yescharacteristic
subgroupG/H
centerC22#89731
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1542C2
socleC22×C7#394
prime radicalC22×D28D4
hypercenter#1542D7
hypocenterC7#10468
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. C7×#1542C2semidirect
3. #11729592C2semidirect
6. #11728175C2semidirect
7. #11725544C2semidirect
68. #1542D7semidirect
316. C7#10468semidirect

Maximal subgroups

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

subgroupindex
#117407952
C7×#15422
#117295922
#117285372
#117282342
#117281752
#117255442
#104687