Finite Group Databasebeta

← #13274063

Subgroups — #13274063

Order 1792.

482 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 79 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×#3018C211yesmaximal, characteristic
2#11729574C211yesmaximal, characteristic, complemented
3#11729559C211yesmaximal, characteristic, complemented
4#11725579C214yesmaximal, complemented
5C7×#585C411yescharacteristic
6C7×#575—21no—
7C7×#532C2211yescharacteristic
8C7×#456C2212yes—
9C7×#455—21no—
10C7×#454C412yes—
11C7×#453—21no—
12C2×C7×#104—21no—
13C7×#413C411yescharacteristic
14#90009—21no—
15#89697—21no—
16#89343C2212yescomplemented
17#89341C2212yescomplemented
18#89339—21no—
19#89337—21no—
20#13572—71nomaximal, Sylow
21C7×2−1+4—41no—
22C7×2−1+4C2×C412yes—
23C7×H(5,𝔽2)—21no—
24C7×H(5,𝔽2)—41no—
25C7×H(5,𝔽2)—41no—
26C2×C7×#41—21no—
27C2×C7×#41—41no—
28C2×C7×#41—42no—
29C2×C7×#41C2×C411yescharacteristic
30C2×C7×#41D411yescharacteristic
31C7×#137—22no—
32C7×#137—41no—
33C7×AGL(1,ℤ/8)—22no—
34C7×AGL(1,ℤ/8)—41no—
35C7×#135—42no—
36C2×C7×M16—21no—
37C7×C4⋊Q8C2311yescharacteristic
38C7×#127D411yescharacteristic
39C7×#124C2×C411yescharacteristic
40C7×#124D411yescharacteristic
41C7×#121—41no—
42C4×C7×D4—41no—
43C2×C42×C7D411yescharacteristic
44C2×#1136—41no—
45C7×#104—22no—
46C7×#104—22no—
47C7×#101C2×C412yes—
48C7×#100—21no—
49#6410—22no—
50#6407—41no—
51C22⋊(C7⋊C8)—41no—
52C4⋊(C7⋊C8)—22no—
53C4×C7⋊C8—41no—
54#3018D711yescharacteristic
55#2132—71no—
56#1543—71no—
57#1405—74no—
58C7×#41—22no—
59C7×#41—22no—
60C7×#41—41no—
61C7×#41—42no—
62C7×#41—41no—
63C7×#41—41no—
64C7×#41—41no—
65C7×#41—41no—
66C7×#41—42no—
67C7×#41—42no—
68C7×#41—44no—
69C7×#41—42no—
70C7×#41—41no—
71C7×#41—42no—
72C2×C7×Q8—21no—
73C2×C7×Q8—42no—
74C2×C7×Q8—41no—
75C2×C7×Q8C22×C411yescharacteristic
76C2×C7×Q8C2×D411yescharacteristic
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—21no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—44no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4C22⋊C411yescharacteristic
87C2×C7×D4C22⋊C411yescharacteristic
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—41no—
92C22×C4×C7—42no—
93C22×C4×C7C22⋊C411yescharacteristic
94C7×Q16—42no—
95C7×SD16—42no—
96C7×SD16—42no—
97C7×D8—42no—
98C7×M16—22no—
99C7×M16—41no—
100C2×C7×C8—41no—
101C7×C4⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C42×C7—21no—
105C42×C7C22⋊C411yescharacteristic
106C42×C7C2×D411yescharacteristic
107#1136—42no—
108#1136—81no—
109C2×C7⋊C8—42no—
110C2×C7⋊C8—81no—
111#585Dic711yescharacteristic
112#575—21no—
113#532C2×D711yescharacteristic
114#456C2×D712yes—
115#455—21no—
116#454Dic712yes—
117#453—21no—
118#431—141no—
119C2×#104—21no—
120#413—141no—
121#413Dic711yescharacteristic
122#356—72no—
123#355—141no—
124#333—72no—
125#331—141no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—42no—
130C23×C7#9911yescharacteristic
131C7×Q8—22no—
132C7×Q8—41no—
133C7×Q8—41no—
134C7×Q8—42no—
135C7×Q8—41no—
136C7×D4—22no—
137C7×D4—41no—
138C7×D4—41no—
139C7×D4—42no—
140C7×D4—42no—
141C7×D4—42no—
142C7×D4—44no—
143C7×D4—42no—
144C7×D4—41no—
145C7×D4—82no—
146C7×D4—81no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—44no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7#9911yescharacteristic
164C2×C4×C7C2×C22⋊C411yescharacteristic
165C7×C8—42no—
166C7⋊C8—82no—
167C7⋊C8—82no—
1682−1+4—41no—
1692−1+4C2×Dic712yes—
170H(5,𝔽2)—21no—
171H(5,𝔽2)—41no—
172H(5,𝔽2)—41no—
173C2×#41—21no—
174C2×#41—41no—
175C2×#41—42no—
176C2×#41C2×Dic711yescharacteristic
177C2×#41C22⋊D711yescharacteristic
178#137—22no—
179#137—41no—
180AGL(1,ℤ/8)—22no—
181AGL(1,ℤ/8)—41no—
182#135—42no—
183C2×M16—21no—
184C2×M16—281no—
185C4⋊Q8C22×D711yescharacteristic
186#127C22⋊D711yescharacteristic
187#124C2×Dic711yescharacteristic
188#124C22⋊D711yescharacteristic
189#121—41no—
190C4×D4—41no—
191C2×C42C22⋊D711yescharacteristic
192C4⋊C8—142no—
193#104—22no—
194#104—22no—
195#101—142no—
196#101C2×Dic712yes—
197#100—21no—
198#100—281no—
199C22⋊C8—281no—
200C4×C8—281no—
201C22×C7—21no—
202C22×C7—41no—
203C22×C7—42no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—81no—
207C22×C7—81no—
208C22×C7—161no—
209C22×C7—162no—
210C22×C7C2×#9911yescharacteristic
211C4×C7—22no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—42no—
217C4×C7AΣL(1,𝔽16)11yescharacteristic
218C4×C7AΣL(1,𝔽16)11yescharacteristic
219#41—22no—
220#41—22no—
221#41—41no—
222#41—42no—
223#41—41no—
224#41—41no—
225#41—41no—
226#41—41no—
227#41—42no—
228#41—42no—
229#41—44no—
230#41—42no—
231#41—41no—
232#41—42no—
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8—41no—
236C2×Q8C22×Dic711yescharacteristic
237C2×Q8C2×C22⋊D711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4C22⋊Dic711yescharacteristic
248C2×D4C22⋊Dic711yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4C22⋊Dic711yescharacteristic
255Q16—42no—
256SD16—42no—
257SD16—42no—
258D8—42no—
259M16—22no—
260M16—41no—
261M16—282no—
262M16—561no—
263C2×C8—41no—
264C2×C8—282no—
265C2×C8—561no—
266C4⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—21no—
270C42C22⋊Dic711yescharacteristic
271C42C2×C22⋊D711yescharacteristic
272C2×C7—21no—
273C2×C7—41no—
274C2×C7—81no—
275C2×C7—81no—
276C2×C7—82no—
277C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
278C23—21no—
279C23—21no—
280C23—41no—
281C23—42no—
282C23#640811yescharacteristic
283Q8—22no—
284Q8—41no—
285Q8—41no—
286Q8—42no—
287Q8—41no—
288D4—22no—
289D4—41no—
290D4—41no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—44no—
295D4—42no—
296D4—41no—
297D4—82no—
298D4—81no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4#640811yescharacteristic
316C2×C4C2×C22⋊Dic711yescharacteristic
317C8—42no—
318C8—562no—
319C8—562no—
320C7#1357211yescharacteristic, complemented, Sylow
321C22—21no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22—161no—
329C22—162no—
330C22C2×#640811yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—42no—
337C4#8933111yescharacteristic
338C4#8933111yescharacteristic
339C2—21no—
340C2—41no—
341C2—81no—
342C2—81no—
343C2—82no—
344C2C2×#8933111yescharacteristic
subgroupG/H
centerC4#89331
commutatorC2×C7×Q8C22×C4
FrattiniC4⋊Q8C22×D7
Fitting, prime radicalC7×#3018C2
socleC2×C7C2×AΣL(1,𝔽16)
cosocleC7×C4⋊Q8C23
hypercenter#3018D7
hypocenterC7#13572
LCSG ▷ C2×C7×Q8 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3018

Complements

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

NH ≅ G/Ntype
2. #11729574C2semidirect
3. #11729559C2semidirect
4. #11725579C2semidirect
16. #89343C22semidirect
17. #89341C22semidirect
320. C7#13572semidirect

Maximal subgroups

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

subgroupindex
C7×#30182
#117295742
#117295592
#117255792
#135727