Finite Group Databasebeta

← #13274071

Subgroups — #13274071

Order 1792.

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

normal subgroups only

This group has 486 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#11729564C211yesmaximal, 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#89696—21no—
16#89343C2212yescomplemented
17#89341C2212yescomplemented
18#89339—21no—
19#89337—21no—
20#13574—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—
45C2×#1136—41no—
46C7×#104—22no—
47C7×#104—22no—
48C7×#101C2×C412yes—
49C7×#100—21no—
50#6410—22no—
51#6407—41no—
52C4⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—41no—
54#3018D711yescharacteristic
55#2132—71no—
56#1560—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—82no—
109#1136—81no—
110C2×C7⋊C8—42no—
111C2×C7⋊C8—81no—
112#585Dic711yescharacteristic
113#575—21no—
114#532C2×D711yescharacteristic
115#456C2×D712yes—
116#455—21no—
117#454Dic712yes—
118#453—21no—
119#423—141no—
120C2×#104—21no—
121#413—141no—
122#413Dic711yescharacteristic
123#356—72no—
124#355—141no—
125#333—72no—
126#331—141no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—42no—
131C23×C7#9911yescharacteristic
132C7×Q8—22no—
133C7×Q8—41no—
134C7×Q8—41no—
135C7×Q8—42no—
136C7×Q8—41no—
137C7×D4—22no—
138C7×D4—41no—
139C7×D4—41no—
140C7×D4—42no—
141C7×D4—42no—
142C7×D4—42no—
143C7×D4—44no—
144C7×D4—42no—
145C7×D4—41no—
146C7×D4—81no—
147C7×D4—81no—
148C7×D4—81no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—44no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7#9911yescharacteristic
167C2×C4×C7C2×C22⋊C411yescharacteristic
168C7×C8—42no—
169C7⋊C8—82no—
170C7⋊C8—82no—
1712−1+4—41no—
1722−1+4C2×Dic712yes—
173H(5,𝔽2)—21no—
174H(5,𝔽2)—41no—
175H(5,𝔽2)—41no—
176C2×#41—21no—
177C2×#41—41no—
178C2×#41—42no—
179C2×#41C2×Dic711yescharacteristic
180C2×#41C22⋊D711yescharacteristic
181#137—22no—
182#137—41no—
183AGL(1,ℤ/8)—22no—
184AGL(1,ℤ/8)—41no—
185#135—42no—
186C2×M16—21no—
187C2×M16—281no—
188C2×M16—281no—
189C4⋊Q8C22×D711yescharacteristic
190#127C22⋊D711yescharacteristic
191#124C2×Dic711yescharacteristic
192#124C22⋊D711yescharacteristic
193#121—41no—
194C4×D4—41no—
195C2×C42C22⋊D711yescharacteristic
196C4⋊C8—142no—
197C4⋊C8—281no—
198#104—22no—
199#104—22no—
200#101—142no—
201#101C2×Dic712yes—
202#100—21no—
203#100—281no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—42no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7—161no—
212C22×C7—162no—
213C22×C7C2×#9911yescharacteristic
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—42no—
221C4×C7AΣL(1,𝔽16)11yescharacteristic
222C4×C7AΣL(1,𝔽16)11yescharacteristic
223#41—22no—
224#41—22no—
225#41—41no—
226#41—42no—
227#41—41no—
228#41—41no—
229#41—41no—
230#41—41no—
231#41—42no—
232#41—42no—
233#41—44no—
234#41—42no—
235#41—41no—
236#41—42no—
237C2×Q8—21no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8C22×Dic711yescharacteristic
241C2×Q8C2×C22⋊D711yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4C22⋊Dic711yescharacteristic
252C2×D4C22⋊Dic711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4C22⋊Dic711yescharacteristic
259Q16—42no—
260SD16—42no—
261SD16—42no—
262D8—42no—
263M16—22no—
264M16—41no—
265M16—282no—
266M16—562no—
267M16—561no—
268C2×C8—41no—
269C2×C8—282no—
270C2×C8—561no—
271C4⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—21no—
275C42C22⋊Dic711yescharacteristic
276C42C2×C22⋊D711yescharacteristic
277C2×C7—21no—
278C2×C7—41no—
279C2×C7—81no—
280C2×C7—81no—
281C2×C7—82no—
282C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
283C23—21no—
284C23—21no—
285C23—41no—
286C23—42no—
287C23#640811yescharacteristic
288Q8—22no—
289Q8—41no—
290Q8—41no—
291Q8—42no—
292Q8—41no—
293D4—22no—
294D4—41no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—42no—
299D4—44no—
300D4—42no—
301D4—41no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4#640811yescharacteristic
323C2×C4C2×C22⋊Dic711yescharacteristic
324C8—42no—
325C8—562no—
326C8—562no—
327C7#1357411yescharacteristic, complemented, Sylow
328C22—21no—
329C22—41no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—161no—
336C22—162no—
337C22C2×#640811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4#8933111yescharacteristic
346C4#8933111yescharacteristic
347C2—21no—
348C2—41no—
349C2—81no—
350C2—81no—
351C2—82no—
352C2C2×#8933111yescharacteristic
subgroupG/H
centerC2C2×#89331
commutatorC2×C7×Q8C22×C4
FrattiniC4⋊Q8C22×D7
Fitting, prime radicalC7×#3018C2
socleC2×C7C2×AΣL(1,𝔽16)
cosocleC7×C4⋊Q8C23
hypercenter#3018D7
hypocenterC7#13574
LCSG ▷ C2×C7×Q8 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3018

Complements

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

NH ≅ G/Ntype
2. #11729574C2semidirect
3. #11729564C2semidirect
4. #11725579C2semidirect
16. #89343C22semidirect
17. #89341C22semidirect
327. C7#13574semidirect

Maximal subgroups

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

subgroupindex
C7×#30182
#117295742
#117295642
#117255792
#135747