Finite Group Databasebeta

← #13243127

Subgroups — #13243127

Order 1792.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C7×#2805C211yesmaximal, characteristic
2#11727438C211yesmaximal, characteristic
3#11727366C211yesmaximal, characteristic
4#11727366C211yesmaximal, characteristic
5#11725448C211yesmaximal, characteristic
6#11725444C211yesmaximal, characteristic
7#11725444C211yesmaximal, characteristic
8C7×#400—21no—
9C7×#400—21no—
10C7×#395—21no—
11C7×#391C2211yescharacteristic
12C7×#389—21no—
13C7×#389C2211yescharacteristic
14C7×#389C2211yescharacteristic
15C7×#387—21no—
16C7×#387—21no—
17C2×#6409—21no—
18C2×#6381—21no—
19#89707C2211yescharacteristic
20#89324—21no—
21#89324—21no—
22#89324—21no—
23#89324—21no—
24#89319C2211yescharacteristic
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7835—71nomaximal, Sylow
28C22×C7×Q8D411yescharacteristic
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C42×C7—21no—
39C2×C42×C7C2311yescharacteristic
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—21no—
43C22×C7⋊C8—21no—
44C22×C7⋊C8—21no—
45C22×C7⋊C8—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55#6409—42no—
56#6381—42no—
57#2805D711yescharacteristic
58#2100—71no—
59#2100—71no—
60#2095—71no—
61#1272—71no—
62#1271—71no—
63#1271—71no—
64C2×C7×Q8—21no—
65C2×C7×Q8—21no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—41no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—81no—
89C7×C4⋊C4—81no—
90C7×C4⋊C4Q1612yes—
91C42×C7—41no—
92C42×C7—81no—
93C4⋊Dic7—41no—
94C4⋊Dic7—42no—
95C4⋊Dic7—42no—
96C4⋊Dic7—41no—
97C2×C7⋊C8—22no—
98C2×C7⋊C8—41no—
99C2×C7⋊C8—41no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8—41no—
102C2×C7⋊C8—41no—
103C2×C7⋊C8—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8—41no—
106C2×#107—141no—
107C2×#103—141no—
108#400—21no—
109#400—21no—
110#395—21no—
111#391C2×D711yescharacteristic
112#389—21no—
113#389C2×D711yescharacteristic
114#389C2×D711yescharacteristic
115#387—21no—
116#387—21no—
117#384—71no—
118#340—141no—
119#340—141no—
120#340—141no—
121#340—141no—
122#336—71no—
123#336—71no—
124#336—71no—
125C23×C7#12011yescharacteristic
126C7×Q8—42no—
127C7×Q8—81no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7C2×Q1611yescharacteristic
153C2×C4×C7#13511yescharacteristic
154C2×C4×C7#13511yescharacteristic
155C2×C4×C7#13711yescharacteristic
156C2×C4×C7#13711yescharacteristic
157C2×C4×C7#13711yescharacteristic
158C2×Dic7—81no—
159C2×Dic7—81no—
160C2×Dic7—81no—
161C7⋊C8—42no—
162C7⋊C8—81no—
163C7⋊C8—81no—
164C22×Q8C22⋊D711yescharacteristic
165C22×C8—141no—
166C22×C8—141no—
167C22×C8—141no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4C22⋊D711yescharacteristic
176C2×C4⋊C4C22⋊D711yescharacteristic
177C2×C4⋊C4C22⋊D711yescharacteristic
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C4⋊C4C22⋊D711yescharacteristic
180C2×C42—21no—
181C2×C42C22×D711yescharacteristic
182#107—282no—
183#103—282no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193C22×C7#45111yescharacteristic
194C22×C7#45211yescharacteristic
195C22×C7#45211yescharacteristic
196C22×C7#45311yescharacteristic
197C22×C7#45411yescharacteristic
198C22×C7#45411yescharacteristic
199C22×C7#45811yescharacteristic
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—42no—
207C4×C7—81no—
208C4×C7—81no—
209C4×C7—81no—
210C4×C7—81no—
211C4×C7—81no—
212Dic7—161no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—281no—
224C22×C4C2×C22⋊D711yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C22×C4C2×C22⋊D711yescharacteristic
227C2×C8—142no—
228C2×C8—281no—
229C2×C8—281no—
230C2×C8—281no—
231C2×C8—281no—
232C2×C8—281no—
233C2×C8—281no—
234C2×C8—281no—
235C2×C8—281no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—282no—
251C4⋊C4—282no—
252C4⋊C4#114412yes—
253C42—41no—
254C42—81no—
255C2×C7#161511yescharacteristic
256C2×C7#161711yescharacteristic
257C2×C7#161811yescharacteristic
258C2×C7#163211yescharacteristic
259C2×C7#163611yescharacteristic
260C2×C7#163811yescharacteristic
261C2×C7#165711yescharacteristic
262C23#651611yescharacteristic
263Q8—42no—
264Q8—81no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—561no—
290C2×C4—561no—
291C2×C4—561no—
292C2×C4#650511yescharacteristic
293C2×C4C2×#114411yescharacteristic
294C2×C4#651311yescharacteristic
295C2×C4#651311yescharacteristic
296C2×C4#651411yescharacteristic
297C2×C4#651411yescharacteristic
298C8—282no—
299C8—561no—
300C8—561no—
301C7#783511yescharacteristic, complemented, Sylow
302C22#9000011yescharacteristic
303C22#9001111yescharacteristic
304C22#9001111yescharacteristic
305C22#9001311yescharacteristic
306C22#9001511yescharacteristic
307C22#9001711yescharacteristic
308C22#9001711yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—1121no—
322C2#1172963711yescharacteristic
323C2#1172964111yescharacteristic
324C2#1172965311yescharacteristic
325C2#1172967811yescharacteristic
326C2#1172968611yescharacteristic
327C2#1172969011yescharacteristic
328C2#1172970911yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2805C2
socle, prime radicalC23×C7#120
hypercenter#2805D7
hypocenterC7#7835
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
301. C7#7835semidirect

Maximal subgroups

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

subgroupindex
C7×#28052
#117274382
#117273662
#117254482
#117254442
#78357