Finite Group Databasebeta

← #13258191

Subgroups — #13258191

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
1C7×#3189C211yesmaximal, characteristic
2#11729455C212yesmaximal
3#11729450C212yesmaximal
4#11727704C211yesmaximal, characteristic
5#11725482C211yesmaximal, characteristic
6C2×C7×C22⋊Q8C2212yes—
7C23×C7⋊C8C2211yescharacteristic
8C7×#484—22no—
9C7×#467—21no—
10C7×#467—21no—
11C7×#451—22no—
12C2×C7×C22⋊C8C2211yescharacteristic
13C2×#6409—22no—
14#89984C2212yes—
15C4×C22⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C8—21no—
17#89345—21no—
18#89324—22no—
19#89321—22no—
20#89321—22no—
21#10566—71nomaximal, Sylow
22C22×C7×Q8D412yes—
23C23×C4×C7C2311yescharacteristic
24C2×C7×Q16—21no—
25C2×C7×Q16—21no—
26C22×C7×C8D411yescharacteristic
27C22×C7×C8D411yescharacteristic
28C7×C22⋊Q8—22no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4D412yes—
34C2×C7×C22⋊C4—22no—
35C2×C22⋊Dic7—22no—
36C2×C22⋊Dic7D411yescharacteristic
37C2×C22⋊Dic7D411yescharacteristic
38C2×C4⋊Dic7—22no—
39C2×C4×Dic7—21no—
40C22×C7⋊C8—21no—
41C22×C7⋊C8—22no—
42C22×C7⋊C8—42no—
43C7×#107—21no—
44C7×#107—21no—
45C7×#103—22no—
46C7×#103—22no—
47C7×C22⋊C8—22no—
48#6409—44no—
49C7⋊#95—21no—
50C7⋊#95—22no—
51Dic7⋊C8—42no—
52#3189D711yescharacteristic
53#1996—72no—
54#1892—72no—
55#1850—71no—
56#1289—71no—
57C24×C7C2×D411yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—42no—
61C2×C7×Q8—42no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—41no—
65C22×C4×C7—41no—
66C22×C4×C7C2×D412yes—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7#4111yescharacteristic
69C22×C4×C7#4112yes—
70C22×Dic7—21no—
71C22×Dic7—21no—
72C22×Dic7—42no—
73C7×Q16—81no—
74C7×Q16—81no—
75C2×C7×C8—21no—
76C2×C7×C8—21no—
77C2×C7×C8—41no—
78C2×C7×C8—41no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—42no—
85C22⋊Dic7—21no—
86C22⋊Dic7—22no—
87C22⋊Dic7—42no—
88C22⋊Dic7—42no—
89C22⋊Dic7Q1612yescomplemented
90C4⋊Dic7—42no—
91C4⋊Dic7—42no—
92C4×Dic7—41no—
93C4×Dic7—41no—
94C2×C7⋊C8—24no—
95C2×C7⋊C8—22no—
96C2×C7⋊C8—41no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8—41no—
99C23×C8—71no—
100C2×C22⋊Q8C2×D712yes—
101#484—22no—
102#467—21no—
103#467—21no—
104#451—22no—
105C2×C4⋊C8—141no—
106C2×#103—142no—
107C2×C22⋊C8C2×D711yescharacteristic
108#380—72no—
109C4×C22⋊C4—71no—
110#342—142no—
111#342—142no—
112#340—142no—
113#336—141no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7#12111yescharacteristic
117C23×C7#12112yes—
118C23×C7C2×Q1611yescharacteristic
119C23×C7#13511yescharacteristic
120C23×C7#13712yes—
121C7×Q8—44no—
122C7×Q8—42no—
123C7×Q8—42no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—82no—
133C2×C4×C7—82no—
134C2×C4×C7#12011yescharacteristic
135C2×C4×C7#12312yes—
136C2×C4×C7#12411yescharacteristic
137C2×Dic7—22no—
138C2×Dic7—41no—
139C2×Dic7—41no—
140C2×Dic7—41no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143C2×Dic7—82no—
144C2×Dic7—82no—
145C2×Dic7—82no—
146C7×C8—41no—
147C7×C8—41no—
148C7⋊C8—42no—
149C22×Q8C22⋊D712yes—
150C23×C4C22×D711yescharacteristic
151C2×Q16—21no—
152C2×Q16—21no—
153C22×C8—141no—
154C22×C8—142no—
155C22×C8—282no—
156C22×C8C22⋊D711yescharacteristic
157C22×C8C22⋊D711yescharacteristic
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—142no—
164C2×C4⋊C4C22⋊D712yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—71no—
167C2×C22⋊C4—71no—
168C2×C22⋊C4—142no—
169C2×C42—141no—
170#107—21no—
171#107—21no—
172C4⋊C8—282no—
173#103—22no—
174#103—22no—
175#103—284no—
176C22⋊C8—22no—
177#95—141no—
178#95—142no—
179C22×C7—21no—
180C22×C7—22no—
181C22×C7—21no—
182C22×C7—22no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—42no—
186C22×C7#39411yescharacteristic
187C22×C7#46212yes—
188C22×C7#46412yes—
189C22×C7#46711yescharacteristic
190C22×C7#47011yescharacteristic
191C4×C7—22no—
192C4×C7—21no—
193C4×C7—82no—
194C4×C7—82no—
195C4×C7B(4,𝔽2)11yescharacteristic
196C4×C7#45811yescharacteristic
197Dic7—42no—
198Dic7—81no—
199Dic7—162no—
200C24D4×D711yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—141no—
210C22×C4—141no—
211C22×C4—282no—
212C22×C4#115911yescharacteristic
213C22×C4#115912yes—
214C22×C4C2×C22⋊D712yes—
215C22×C4C2×C22⋊D711yescharacteristic
216Q16—81no—
217Q16—81no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—144no—
223C2×C8—142no—
224C2×C8—281no—
225C2×C8—282no—
226C2×C8—281no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—282no—
232C4⋊C4—282no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—72no—
236C22⋊C4—141no—
237C22⋊C4—142no—
238C22⋊C4—282no—
239C22⋊C4—282no—
240C42—281no—
241C42—281no—
242C2×C7—21no—
243C2×C7—21no—
244C2×C7—22no—
245C2×C7—22no—
246C2×C7#204411yescharacteristic
247C2×C7#205311yescharacteristic
248C2×C7#205611yescharacteristic
249C23—41no—
250C23—41no—
251C23D7×Q1611yescharacteristic
252C23#648112yes—
253C23#648211yescharacteristic
254C23#650111yescharacteristic
255C23#650212yes—
256Q8—44no—
257Q8—42no—
258Q8—42no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—142no—
270C2×C4—281no—
271C2×C4—281no—
272C2×C4—281no—
273C2×C4—281no—
274C2×C4—281no—
275C2×C4—562no—
276C2×C4—562no—
277C2×C4—562no—
278C2×C4#649812yes—
279C2×C4#649911yescharacteristic
280C2×C4#651611yescharacteristic
281C8—41no—
282C8—41no—
283C8—282no—
284C7#1056611yescharacteristic, complemented, Sylow
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22#8995412yes—
293C22#8995712yes—
294C22#8996011yescharacteristic
295C22#8996111yescharacteristic
296C22#8999411yescharacteristic
297C4—22no—
298C4—21no—
299C4—82no—
300C4—82no—
301C4—282no—
302C4—561no—
303C4—1122no—
304C4#9001511yescharacteristic
305C4#9001611yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#1172960411yescharacteristic
311C2#1172982211yescharacteristic
312C2#1172986911yescharacteristic
subgroupG/H
centerC22#89994
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#3189C2
socleC22×C7#394
prime radicalC24×C7C2×D4
hypercenter#3189D7
hypocenterC7#10566
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
89. C22⋊Dic7Q16semidirect
284. C7#10566semidirect

Maximal subgroups

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

subgroupindex
C7×#31892
#117294552
#117294502
#117277042
#117254822
#105667