Finite Group Databasebeta

← #13256979

Subgroups — #13256979

Order 1792.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#11731188C211yesmaximal, characteristic
2C7×#1986C211yesmaximal, characteristic
3#11729591C211yesmaximal, characteristic
4#11729590C211yesmaximal, characteristic
5#11725472C211yesmaximal, characteristic
6#11725471C211yesmaximal, characteristic
7#11725427C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C4×C7×M16C2211yescharacteristic
10C4×C7×C4⋊C4C2211yescharacteristic
11#89758—21no—
12#89751—21no—
13#89709C2211yescharacteristic
14#89705—21no—
15#89704C2211yescharacteristic
16#89701—21no—
17#89698C2211yescharacteristic
18#89698C2211yescharacteristic
19#89426—21no—
20#89418—21no—
21C7×#339—21no—
22C7×#339—21no—
23#89333—21no—
24#89333—21no—
25#89333—21no—
26#89333—21no—
27#10379—71nomaximal, Sylow
28C22×Dic14—21no—
29C2×C7×M16—21no—
30C7×C4⋊Q8Q811yescharacteristic, complemented
31C7×C4⋊Q8Q811yescharacteristic, complemented
32C7×#125—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C4×C7×Q8—21no—
36C4×C7×Q8—21no—
37C7×#117—21no—
38C2×C7×C4⋊C4—21no—
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C42×C7C2311yescharacteristic
42C2×C4⋊Dic7—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×C4×Dic7—21no—
46C7×#97—21no—
47C4×C7×C8—21no—
48C7×#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95D411yescharacteristic
57C7⋊#95D411yescharacteristic
58C7⋊#95D411yescharacteristic
59C7⋊#95D411yescharacteristic
60C22⋊(C7⋊C8)—21no—
61C22⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64#6377—21no—
65#6377—21no—
66#2605—71no—
67#1986D711yescharacteristic
68#1515—71no—
69#1506—71no—
70#1300—71no—
71#1288—71no—
72#1285—71no—
73C2×C7×Q8—21no—
74C2×C7×Q8—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C22×Dic7—21no—
84C2×Dic14—42no—
85C2×Dic14—42no—
86C2×Dic14—81no—
87C7×M16—22no—
88C7×M16—41no—
89C2×C7×C8—41no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C22⋊C4—41no—
98C42×C7—41no—
99C42×C7—41no—
100C42×C7—41no—
101C42×C7C2×Q811yescharacteristic
102C42×C7#4111yescharacteristic
103C42×C7#4111yescharacteristic
104C42×C7#4111yescharacteristic
105C4⋊Dic7—42no—
106Dic7⋊C4—42no—
107Dic7⋊C4—81no—
108C4×Dic7—42no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111#533C2×D711yescharacteristic
112#432—71no—
113#432—71no—
114C4×M16C2×D711yescharacteristic
115#400—141no—
116#395—141no—
117#391—141no—
118#391—141no—
119#387—141no—
120#387—141no—
121#383—71no—
122C4×C4⋊C4C2×D711yescharacteristic
123#376—71no—
124#339—21no—
125#339—21no—
126#328—141no—
127#328—141no—
128#328—141no—
129#328—141no—
130C23×C7#12011yescharacteristic
131C7×Q8—81no—
132C7×Q8—81no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
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×C7C22⋊Q811yescharacteristic
146C2×C4×C7C22⋊Q811yescharacteristic
147C2×C4×C7C22⋊Q811yescharacteristic
148C2×C4×C7#12411yescharacteristic
149C2×C4×C7#12411yescharacteristic
150C2×C4×C7#12411yescharacteristic
151C2×Dic7—22no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163Dic14—84no—
164C7×C8—42no—
165C7⋊C8—81no—
166C7⋊C8—81no—
167C22×Q8—141no—
168C2×M16—21no—
169C4⋊Q8Dic1411yescharacteristic, complemented
170C4⋊Q8Dic1411yescharacteristic, complemented
171#125—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176#117—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4—141no—
181C2×C42—141no—
182C2×C42C22⋊D711yescharacteristic
183C2×C42C22⋊D711yescharacteristic
184C2×C42C22×D711yescharacteristic
185C4⋊C8—141no—
186C4⋊C8—141no—
187C22⋊C8—141no—
188C22⋊C8—141no—
189#97—21no—
190#97—141no—
191#97—141no—
192C4×C8—21no—
193#95—21no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—141no—
199#95—141no—
200#95—141no—
201#95—141no—
202#95—141no—
203#95—141no—
204#95—141no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7#39311yescharacteristic
208C22×C7#45511yescharacteristic
209C22×C7#45611yescharacteristic
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—42no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—81no—
220Dic7—42no—
221Dic7—81no—
222Dic7—81no—
223Dic7—81no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—282no—
227C2×Q8—282no—
228C2×Q8—561no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4D4×D711yescharacteristic
236C22×C4D4×D711yescharacteristic
237C22×C4C2×C22⋊D711yescharacteristic
238M16—22no—
239M16—41no—
240C2×C8—41no—
241C2×C8—281no—
242C2×C8—281no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—282no—
251C4⋊C4—282no—
252C4⋊C4—561no—
253C22⋊C4—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—282no—
258C42C2×Dic1411yescharacteristic
259C42#115711yescharacteristic
260C42#115911yescharacteristic
261C42#115911yescharacteristic
262C2×C7—21no—
263C2×C7—21no—
264C2×C7#202811yescharacteristic
265C2×C7#203111yescharacteristic
266C2×C7#203111yescharacteristic
267C23#650011yescharacteristic
268Q8—81no—
269Q8—81no—
270Q8—564no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
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—142no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4#644111yescharacteristic
296C2×C4#644111yescharacteristic
297C2×C4#644811yescharacteristic
298C2×C4#644811yescharacteristic
299C2×C4#648711yescharacteristic
300C2×C4#649911yescharacteristic
301C8—42no—
302C8—561no—
303C8—561no—
304C7#1037911yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22#8971911yescharacteristic
308C22#8983311yescharacteristic
309C22#8986711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—282no—
321C4—561no—
322C4—561no—
323C4—561no—
324C2—21no—
325C2—21no—
326C2#1172742711yescharacteristic
327C2#1172812211yescharacteristic
328C2#1172814811yescharacteristic
subgroupG/H
centerC22#89719
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1986C2
socleC22×C7#393
prime radicalC23×C7#120
hypercenter#1986D7
hypocenterC7#10379
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
30. C7×C4⋊Q8Q8semidirect
31. C7×C4⋊Q8Q8semidirect
169. C4⋊Q8Dic14semidirect
170. C4⋊Q8Dic14semidirect
304. C7#10379semidirect

Maximal subgroups

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

subgroupindex
#117311882
C7×#19862
#117295912
#117295902
#117254722
#117254712
#117254272
#103797