Finite Group Databasebeta

← #13250700

Subgroups — #13250700

Order 1792.

536 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 115 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×#3113C211yesmaximal, characteristic
2#11739895C211yesmaximal, characteristic
3#11729595C211yesmaximal, characteristic
4#11729590C211yesmaximal, characteristic
5#11729377C211yesmaximal, characteristic
6#11725556C211yesmaximal, characteristic
7#11725554C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×Q8×Dic7—21no—
11C7×#483—21no—
12C7×#475C412yescomplemented
13C7×#462C412yescomplemented
14C7×#451—21no—
15C7×#441—22no—
16C7×#432C2211yescharacteristic
17C4⋊C4×Dic7—21no—
18C4⋊C4×Dic7—22no—
19C4×C4⋊Dic7—21no—
20C4×C4⋊Dic7C2211yescharacteristic
21#89704—21no—
22#89704C2211yescharacteristic
23#89698C2211yescharacteristic
24C4×#1136C2211yescharacteristic
25#89333—21no—
26#89333—21no—
27Q8⋊(C7⋊C8)—22no—
28#89327—22no—
29#9324—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C2×C7×Q16—22no—
32C7×C4⋊Q8C2×C411yescharacteristic
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×#125—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C4×C7×Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8C2×C412yes—
42C4×C7×Q8D412yes—
43C7×#117—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C42×C7—21no—
47C2×C42×C7C2311yescharacteristic
48Q8×Dic7—28no—
49C2×C4⋊Dic7—22no—
50C2×C4⋊Dic7—21no—
51C2×C4×Dic7—22no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—22no—
54C2×C4×Dic7D412yes—
55C2×#1136—21no—
56C7×#106—22no—
57C7×C4⋊C8C2×C412yes—
58C7×#103—22no—
59C7×#103—22no—
60C7×C22⋊C8—21no—
61C7×#97—21no—
62C7⋊#95—22no—
63C7⋊#95—21no—
64C7⋊#95—21no—
65C7⋊#95D411yescharacteristic
66C7⋊#95D411yescharacteristic
67C22⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69#6377—21no—
70#6377—21no—
71C4×C7⋊C8—21no—
72#3113D711yescharacteristic
73#2276—71no—
74#1986—71no—
75#1506—71no—
76#1491—71no—
77#1300—71no—
78#1285—71no—
79C2×C7×Q8—24no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8C22⋊C412yes—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×Dic7—21no—
90C22×Dic7—22no—
91C22×Dic7—21no—
92C7×Q16—44no—
93C2×C7×C8—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4C22⋊C412yes—
104C7×C22⋊C4—41no—
105C42×C7—22no—
106C42×C7—22no—
107C42×C7—41no—
108C42×C7C22×C411yescharacteristic
109C42×C7C2×D411yescharacteristic
110C42×C7#4111yescharacteristic
111C42×C7#4111yescharacteristic
112C4⋊Dic7—28no—
113C4⋊Dic7—24no—
114C4×Dic7—28no—
115C4×Dic7—24no—
116C4×Dic7—42no—
117C4×Dic7—82no—
118#1136—22no—
119#1136—41no—
120C2×C7⋊C8—41no—
121C2×C7⋊C8—41no—
122#533C2×D711yescharacteristic
123C2×C4×Q8—141no—
124C2×C4×Q8C2×D711yescharacteristic
125#483—21no—
126#475Dic712yescomplemented
127#462Dic712yescomplemented
128#451—21no—
129#441—22no—
130#432—71no—
131#432C2×D711yescharacteristic
132C4×M16—71no—
133C4×C4⋊C4—71no—
134C4×C4⋊C4—141no—
135C4×C4⋊C4—142no—
136C4×C4⋊C4—141no—
137#376—71no—
138#376—141no—
139#339—142no—
140#328—141no—
141#328—141no—
142Q8⋊C8—142no—
143C23×C7#12011yescharacteristic
144C7×Q8—24no—
145C7×Q8—22no—
146C7×Q8—42no—
147C7×Q8—81no—
148C7×Q8#10414yes—
149C2×C4×C7—24no—
150C2×C4×C7—22no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7C2×C22⋊C411yescharacteristic
161C2×C4×C7C4×D411yescharacteristic
162C2×C4×C7C4×D411yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12311yescharacteristic
166C2×Dic7—22no—
167C2×Dic7—24no—
168C2×Dic7—22no—
169C2×Dic7—41no—
170C2×Dic7—42no—
171C2×Dic7—41no—
172C2×Dic7—42no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C7×C8—42no—
176C7⋊C8—42no—
177C7⋊C8—81no—
178C22×Q8C22⋊D711yescharacteristic
179C2×Q16—22no—
180C2×M16—141no—
181C4⋊Q8C2×Dic711yescharacteristic
182C4⋊Q8C2×Dic711yescharacteristic
183#125—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—148no—
191C4×Q8C2×Dic712yes—
192C4×Q8C22⋊D712yes—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—142no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4C22⋊D711yescharacteristic
198C2×C42—21no—
199C2×C42—72no—
200C2×C42—142no—
201C2×C42—141no—
202C2×C42—142no—
203C2×C42C22×D711yescharacteristic
204#106—22no—
205C4⋊C8—141no—
206C4⋊C8C2×Dic712yes—
207#103—22no—
208#103—22no—
209C22⋊C8—21no—
210C22⋊C8—141no—
211#97—21no—
212#97—141no—
213#97—141no—
214C4×C8—141no—
215#95—71no—
216#95—71no—
217#95—142no—
218#95—141no—
219#95—141no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7#38611yescharacteristic
223C22×C7#45511yescharacteristic
224C22×C7#45611yescharacteristic
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—24no—
228C4×C7—21no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—42no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7C2×#10412yes—
235Dic7—44no—
236Dic7—42no—
237Dic7—42no—
238C2×Q8—24no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8C22⋊Dic712yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—142no—
248C22×C4D4×D711yescharacteristic
249C22×C4D4×D711yescharacteristic
250C22×C4C2×C22⋊D711yescharacteristic
251Q16—44no—
252M16—142no—
253M16—281no—
254C2×C8—22no—
255C2×C8—281no—
256C2×C8—281no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—148no—
267C4⋊C4—144no—
268C4⋊C4C22⋊Dic712yes—
269C22⋊C4—41no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—148no—
274C42—144no—
275C42—282no—
276C42—562no—
277C42#115911yescharacteristic
278C42#115911yescharacteristic
279C42C22×Dic711yescharacteristic
280C42C2×C22⋊D711yescharacteristic
281C2×C7—21no—
282C2×C7—21no—
283C2×C7#188211yescharacteristic
284C2×C7#189611yescharacteristic
285C2×C7#190511yescharacteristic
286C23#650011yescharacteristic
287Q8—24no—
288Q8—22no—
289Q8—42no—
290Q8—81no—
291Q8#641114yes—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—142no—
304C2×C4—142no—
305C2×C4—144no—
306C2×C4—281no—
307C2×C4—281no—
308C2×C4—282no—
309C2×C4—282no—
310C2×C4—281no—
311C2×C4—281no—
312C2×C4D4×Dic711yescharacteristic
313C2×C4D4×Dic711yescharacteristic
314C2×C4#649811yescharacteristic
315C2×C4#650211yescharacteristic
316C2×C4#650211yescharacteristic
317C2×C4C2×C22⋊Dic711yescharacteristic
318C8—42no—
319C8—282no—
320C8—561no—
321C7#932411yescharacteristic, complemented, Sylow
322C22—21no—
323C22—21no—
324C22#8997411yescharacteristic
325C22#8997711yescharacteristic
326C22#8999211yescharacteristic
327C4—24no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—81no—
335C4—81no—
336C4—284no—
337C4—282no—
338C4—282no—
339C4C2×#641112yes—
340C2—21no—
341C2—21no—
342C2#1172946511yescharacteristic
343C2#1172985411yescharacteristic
344C2#1172985511yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3113C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#3113D7
hypocenterC7#9324
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
12. C7×#475C4semidirect
13. C7×#462C4semidirect
126. #475Dic7semidirect
127. #462Dic7semidirect
321. C7#9324semidirect

Maximal subgroups

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

subgroupindex
C7×#31132
#117398952
#117295952
#117295902
#117293772
#117255562
#117255542
#93247