Finite Group Databasebeta

← #13255114

Subgroups — #13255114

Order 1792.

484 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2822C211yesmaximal, characteristic
2C2×#89837C211yesmaximal, characteristic
3C2×#89841C211yesmaximal, characteristic
4C2×#89338C211yesmaximal, characteristic
5C2×#89338C211yesmaximal, characteristic
6#11727309C211yesmaximal, characteristic
7#11725436C211yesmaximal, characteristic
8C2×C7×#125C2211yescharacteristic
9C2×C7×#125C2211yescharacteristic
10C2×#6432C2211yescharacteristic
11C7×#400—21no—
12C7×#400—21no—
13C7×#398—22no—
14C7×#398—21no—
15C7×#398—21no—
16C7×#382C2211yescharacteristic
17#89841—22no—
18#89841—22no—
19#89837—22no—
20#89837—22no—
21C2×#6384—21no—
22C2×#6384—21no—
23C2×#6382—21no—
24C2×#6381—21no—
25C2×#6377C2211yescharacteristic
26C2×#6377C2211yescharacteristic
27C2×#6377C2211yescharacteristic
28#89338—22no—
29#89338—22no—
30#89324—22no—
31#10084—71nomaximal, Sylow
32C22×Dic14—21no—
33C7×#125—21no—
34C7×#125—21no—
35C7×#125—21no—
36C7×#125—21no—
37C7×#125—21no—
38C7×#125—21no—
39C7×#125—21no—
40C7×#125—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C4⋊C4D412yes—
51C2×C42×C7C2311yescharacteristic
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C22×C7⋊C8—21no—
55C22×C7⋊C8—21no—
56C22×C7⋊C8—21no—
57#6432—21no—
58#6432—21no—
59#6432—21no—
60#6432D412yes—
61C7×#95—22no—
62C7×#95—22no—
63C7×#95—22no—
64C7×#95—21no—
65C7×#95—21no—
66#6384—24no—
67#6384—24no—
68#6382—24no—
69#6381—24no—
70#6377—21no—
71#6377—21no—
72#6377—21no—
73#6377—21no—
74#6377D412yes—
75#6377D412yes—
76C2×#501—71no—
77C2×#491—71no—
78#2822D711yescharacteristic
79#1993—71no—
80C2×#330—71no—
81C2×#330—71no—
82#1295—71no—
83C22×C4×C7—22no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×Dic7—41no—
92C2×Dic14—22no—
93C2×Dic14—44no—
94C2×Dic14—41no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—82no—
113C42×C7C2×D411yescharacteristic
114C42×C7C2×D411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7#4111yescharacteristic
117C4⋊Dic7—22no—
118C4⋊Dic7—22no—
119C4⋊Dic7—41no—
120C4⋊Dic7—41no—
121C2×C7⋊C8—24no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—22no—
125C2×C7⋊C8—42no—
126C2×C7⋊C8—41no—
127C2×C7⋊C8—41no—
128C2×C4⋊Q8—71no—
129C2×#125C2×D711yescharacteristic
130C2×#125C2×D711yescharacteristic
131#501—142no—
132#501—142no—
133#491—142no—
134#491—142no—
135C2×#107—141no—
136C2×#106—141no—
137C2×#103—141no—
138C2×#103—141no—
139C2×#97—71no—
140C2×#97—71no—
141C2×#97—71no—
142#400—21no—
143#400—21no—
144#398—22no—
145#398—21no—
146#398—21no—
147#382C2×D711yescharacteristic
148#340—142no—
149#330—142no—
150#330—142no—
151C23×C7#12011yescharacteristic
152C2×C4×C7—22no—
153C2×C4×C7—21no—
154C2×C4×C7—22no—
155C2×C4×C7—21no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12711yescharacteristic
175C2×Dic7—42no—
176C2×Dic7—82no—
177Dic14—44no—
178Dic14—82no—
179C7⋊C8—24no—
180C7⋊C8—42no—
181C7⋊C8—42no—
182C22×Q8—141no—
183C22×C8—141no—
184C22×C8—141no—
185C22×C8—141no—
186C4⋊Q8—72no—
187C4⋊Q8—141no—
188C4⋊Q8—141no—
189C4⋊Q8—141no—
190#125—21no—
191#125—21no—
192#125—21no—
193#125—21no—
194#125—21no—
195#125—21no—
196#125—21no—
197#125—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—141no—
204C2×C4⋊C4—141no—
205C2×C4⋊C4C22⋊D711yescharacteristic
206C2×C4⋊C4C22⋊D711yescharacteristic
207C2×C4⋊C4C22⋊D711yescharacteristic
208C2×C4⋊C4C22⋊D711yescharacteristic
209C2×C4⋊C4C22⋊D712yes—
210C2×C42C22×D711yescharacteristic
211#107—144no—
212#106—144no—
213#103—144no—
214#103—144no—
215#97—72no—
216#97—72no—
217#97—141no—
218#97—141no—
219#97—141no—
220#97—141no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226C22×C7#39211yescharacteristic
227C22×C7#45412yes—
228C22×C7#45511yescharacteristic
229C22×C7#45611yescharacteristic
230C22×C7#45612yes—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—82no—
238C4×C7—81no—
239C4×C7—81no—
240C4×C7—81no—
241C4×C7—81no—
242Dic7—82no—
243C2×Q8—142no—
244C2×Q8—284no—
245C2×Q8—281no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—281no—
252C22×C4C2×C22⋊D711yescharacteristic
253C22×C4C2×C22⋊D711yescharacteristic
254C22×C4C2×C22⋊D711yescharacteristic
255C2×C8—144no—
256C2×C8—142no—
257C2×C8—142no—
258C2×C8—142no—
259C2×C8—281no—
260C2×C8—282no—
261C2×C8—281no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C42D4×D711yescharacteristic
285C42D4×D711yescharacteristic
286C42D4×D711yescharacteristic
287C42#115911yescharacteristic
288C2×C7#166312yes—
289C2×C7#166412yes—
290C2×C7#200311yescharacteristic
291C2×C7#200511yescharacteristic
292C2×C7#200711yescharacteristic
293C23#651611yescharacteristic
294Q8—284no—
295Q8—562no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—282no—
314C2×C4—562no—
315C2×C4#650011yescharacteristic
316C2×C4#650011yescharacteristic
317C2×C4#650111yescharacteristic
318C2×C4#650211yescharacteristic
319C2×C4#650211yescharacteristic
320C2×C4#650311yescharacteristic
321C8—144no—
322C8—282no—
323C8—282no—
324C7#1008411yescharacteristic, complemented, Sylow
325C22#8999511yescharacteristic
326C22#9001411yescharacteristic
327C22#9001712yes—
328C22#9001812yes—
329C22#9001811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—82no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—562no—
342C2#1172962111yescharacteristic
343C2#1172962211yescharacteristic
344C2#1172963511yescharacteristic
345C2#1172971112yes—
346C2#1172971512yes—
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2822C2
socle, prime radicalC23×C7#120
hypercenter#2822D7
hypocenterC7#10084
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
324. C7#10084semidirect

Maximal subgroups

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

subgroupindex
C7×#28222
C2×#898372
C2×#898412
C2×#893382
#117273092
#117254362
#100847