Finite Group Databasebeta

← #13257578

Subgroups — #13257578

Order 1792.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#11740812C211yesmaximal, characteristic
2C7×#1518C211yesmaximal, characteristic
3#11729582C211yesmaximal, characteristic
4#11728540C211yesmaximal, characteristic
5#11728504C211yesmaximal, characteristic
6#11727313C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C4×Dic14C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#382C2211yescharacteristic
12#89866—21no—
13#89863—21no—
14#89835—22no—
15#89776—21no—
16#89767—21no—
17C2×#6384—21no—
18#89697C2211yescharacteristic
19#89696C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21C7×#328—22no—
22#89342—21no—
23#89341—21no—
24#89324—21no—
25#89264—22no—
26#89253—22no—
27#10471—71nomaximal, Sylow
28C22×Dic14D411yescharacteristic
29C7×C4⋊Q8D411yescharacteristic
30C7×C4⋊Q8D411yescharacteristic
31C7×#125—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C7×#117—21no—
37C2×C7×C4⋊C4—21no—
38C2×C42×C7C2311yescharacteristic
39C2×C4⋊Dic7D411yescharacteristic
40C2×Dic7⋊C4—21no—
41C2×C4×Dic7—21no—
42C2×#1136—21no—
43C22×C7⋊C8—21no—
44C4×Dic14—21no—
45C4×Dic14—21no—
46C4×Dic14—21no—
47C4×Dic14—21no—
48C7×C4⋊C8—21no—
49C7×C22⋊C8D412yes—
50C4×C7×C8—21no—
51C7×#95—21no—
52C7×#95—21no—
53C22⋊(C7⋊C8)—21no—
54#6384—22no—
55#6384—22no—
56#6384—42no—
57#6382—22no—
58#6381—22no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)D412yes—
64C4×C7⋊C8—21no—
65#3078—71no—
66#1805—71no—
67#1571—71no—
68#1542—71no—
69#1518D711yescharacteristic
70#1461—71no—
71#1286—71no—
72C2×C7×Q8—21no—
73C2×C7×Q8—21no—
74C22×C4×C7—41no—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7#4111yescharacteristic
78C22×Dic7—21no—
79C22×Dic7—21no—
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14—21no—
83C2×Dic14—21no—
84C2×Dic14—21no—
85C2×Dic14Q1612yes—
86C2×C7×C8—22no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—21no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C22⋊C4—41no—
96C42×C7—41no—
97C42×C7C2×D411yescharacteristic
98C42×C7C2×D411yescharacteristic
99C42×C7#4111yescharacteristic
100C42×C7#4111yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105C4×Dic7—41no—
106C4×Dic7—41no—
107#1136—41no—
108#1136—41no—
109C2×C7⋊C8—22no—
110C2×C7⋊C8—41no—
111C2×C7⋊C8—41no—
112C2×C7⋊C8—41no—
113C2×C7⋊C8—41no—
114#533C2×D711yescharacteristic
115C2×C4×Q8—71no—
116#479—142no—
117#477—141no—
118#475—141no—
119#431—71no—
120#431C2×D711yescharacteristic
121#428—141no—
122#423—71no—
123C2×C4⋊C8—71no—
124#419—141no—
125C2×#103—141no—
126#382C2×D711yescharacteristic
127#340—141no—
128#335—142no—
129#333—141no—
130#332—141no—
131#328—22no—
132#324—142no—
133C23×C7#12111yescharacteristic
134C7×Q8—81no—
135C7×Q8—81no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—81no—
141C2×C4×C7—81no—
142C2×C4×C7—81no—
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7#12111yescharacteristic
145C2×C4×C7#12111yescharacteristic
146C2×C4×C7#12311yescharacteristic
147C2×C4×C7#12311yescharacteristic
148C2×C4×C7#12411yescharacteristic
149C2×C4×C7C2×Q1611yescharacteristic
150C2×C4×C7#13511yescharacteristic
151C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×Dic7—22no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—81no—
158Dic14—22no—
159Dic14—22no—
160Dic14—41no—
161Dic14—41no—
162C7×C8—42no—
163C7⋊C8—42no—
164C7⋊C8—81no—
165C7⋊C8—81no—
166C22×Q8—71no—
167C2×M16—141no—
168C22×C8—141no—
169C4⋊Q8D2811yescharacteristic
170C4⋊Q8D2811yescharacteristic
171#125—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—141no—
177C4×Q8—141no—
178C4×Q8—141no—
179C4×Q8—141no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—141no—
184C2×C42—141no—
185C2×C42C22×D711yescharacteristic
186#107—142no—
187#106—142no—
188C4⋊C8—21no—
189C4⋊C8—72no—
190C4⋊C8—141no—
191C4⋊C8—141no—
192C4⋊C8—141no—
193C4⋊C8—141no—
194#103—142no—
195#103—142no—
196#103—282no—
197C22⋊C8—141no—
198C22⋊C8C22⋊D712yes—
199C4×C8—21no—
200C4×C8—141no—
201#95—21no—
202#95—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7#39411yescharacteristic
206C22×C7#46711yescharacteristic
207C22×C7#46811yescharacteristic
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—81no—
213C4×C7—81no—
214C4×C7—161no—
215C4×C7#44911yescharacteristic
216C4×C7#45111yescharacteristic
217C4×C7#48411yescharacteristic
218C4×C7#48511yescharacteristic
219Dic7—42no—
220Dic7—81no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—72no—
224C2×Q8—141no—
225C2×Q8—141no—
226C2×Q8—141no—
227C2×Q8—141no—
228C2×Q8—141no—
229C22×C4—41no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4D4×D711yescharacteristic
233C22×C4#115911yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235M16—281no—
236M16—281no—
237C2×C8—22no—
238C2×C8—142no—
239C2×C8—281no—
240C2×C8—281no—
241C2×C8—281no—
242C2×C8—281no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C22⋊C4—41no—
256C42—41no—
257C42—281no—
258C42—281no—
259C42C2×D2811yescharacteristic
260C42#115711yescharacteristic
261C42D4×D711yescharacteristic
262C42#115911yescharacteristic
263C2×C7—21no—
264C2×C7—21no—
265C2×C7#203611yescharacteristic
266C2×C7#204911yescharacteristic
267C2×C7#205311yescharacteristic
268C23#650111yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—142no—
272Q8—142no—
273Q8—281no—
274Q8—281no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—142no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—561no—
287C2×C4#644511yescharacteristic
288C2×C4#644611yescharacteristic
289C2×C4#644811yescharacteristic
290C2×C4#644911yescharacteristic
291C2×C4#647711yescharacteristic
292C2×C4#647811yescharacteristic
293C2×C4D7×Q1611yescharacteristic
294C2×C4#648211yescharacteristic
295C2×C4#649311yescharacteristic
296C2×C4#650211yescharacteristic
297C8—42no—
298C8—282no—
299C8—561no—
300C8—561no—
301C7#1047111yescharacteristic, complemented, Sylow
302C22—21no—
303C22—21no—
304C22#8973311yescharacteristic
305C22#8994611yescharacteristic
306C22#8996011yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—81no—
313C4—161no—
314C4—282no—
315C4—561no—
316C4#8958011yescharacteristic
317C4#8958111yescharacteristic
318C4#8958311yescharacteristic
319C4#8958611yescharacteristic
320C2—21no—
321C2—21no—
322C2#1172779711yescharacteristic
323C2#1172799811yescharacteristic
324C2#1172815711yescharacteristic
subgroupG/H
centerC22#89733
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1518C2
socleC22×C7#394
prime radicalC23×C7#121
hypercenter#1518D7
hypocenterC7#10471
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
301. C7#10471semidirect

Maximal subgroups

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

subgroupindex
#117408122
C7×#15182
#117295822
#117285402
#117285042
#117273132
#117254372
#104717