Finite Group Databasebeta

← #13242304

Subgroups — #13242304

Order 1792.

390 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 95 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#11733637C211yesmaximal, characteristic
2#11726351C211yesmaximal, characteristic
3#11726350C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic
5#11725444C211yesmaximal, characteristic
6#11725286C211yesmaximal, characteristic
7#11725286C211yesmaximal, characteristic
8C7×#389C2211yescharacteristic
9C2×#6389—21no—
10C2×C8×Dic7—21no—
11#89757C411yescharacteristic
12#89753—21no—
13#89750—21no—
14#89746C411yescharacteristic
15#89746C411yescharacteristic
16#89745C411yescharacteristic
17C42×Dic7C2211yescharacteristic
18#89701C2211yescharacteristic
19#89426—21no—
20#89423—21no—
21C7×#336C2211yescharacteristic
22C7×#336C2211yescharacteristic
23#89345—21no—
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7693—71nomaximal, Sylow
28C22×Dic14C2×C411yescharacteristic
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—21no—
38C2×C4×Dic7C2×C411yescharacteristic
39C2×C4×Dic7C2×C411yescharacteristic
40C2×C4×Dic7D411yescharacteristic
41C2×C4×Dic7D411yescharacteristic
42C2×C4×Dic7D411yescharacteristic
43C2×C4×Dic7D411yescharacteristic
44C22×C7⋊C8—21no—
45C22×C7⋊C8—21no—
46C7×#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55#6389—24no—
56C8×Dic7—24no—
57#2863—71no—
58#1745—71no—
59#1744—71no—
60#1272—71no—
61#1272—71no—
62#1271—71no—
63#1271D711yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7C22×C411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×Dic7—21no—
70C22×Dic7—21no—
71C22×Dic7C22⋊C411yescharacteristic
72C22×Dic7C22⋊C411yescharacteristic
73C22×Dic7C22⋊C411yescharacteristic
74C22×Dic7C22⋊C411yescharacteristic
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×Dic14—41no—
78C2×Dic14—41no—
79C2×C7×C8—22no—
80C2×C7×C8—22no—
81C2×C7×C8—41no—
82C2×C7×C8—41no—
83C2×C7×C8—41no—
84C2×C7×C8—41no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C42×C7—41no—
91C4⋊Dic7—42no—
92C4⋊Dic7—41no—
93Dic7⋊C4—42no—
94C4×Dic7—21no—
95C4×Dic7—21no—
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—22no—
101C4×Dic7—22no—
102C4×Dic7—41no—
103C2×C7⋊C8—22no—
104C2×C7⋊C8—22no—
105C2×C7⋊C8—41no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—41no—
109C2×#97—141no—
110C2×C4×C8—141no—
111#391—71no—
112#391—71no—
113#391—71no—
114#389—71no—
115#389—71no—
116#389C2×D711yescharacteristic
117#387—141no—
118#387—141no—
119#387—141no—
120#383—141no—
121C43—71no—
122#336—71no—
123#336—71no—
124#336—141no—
125#336—141no—
126#336C2×D711yescharacteristic
127#336C2×D711yescharacteristic
128C23×C7C2×C22⋊C411yescharacteristic
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7#13111yescharacteristic
142C2×C4×C7#13111yescharacteristic
143C2×C4×C7#13511yescharacteristic
144C2×C4×C7#13511yescharacteristic
145C2×C4×C7#13711yescharacteristic
146C2×C4×C7#13711yescharacteristic
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7#10412yes—
164C2×Dic7#10412yes—
165Dic14—81no—
166Dic14—81no—
167C7×C8—42no—
168C7×C8—42no—
169C7⋊C8—42no—
170C7⋊C8—42no—
171C22×Q8—71no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—141no—
175C22×C8—141no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4C4×D711yescharacteristic
181C2×C4⋊C4C4×D711yescharacteristic
182C2×C42—71no—
183C2×C42—71no—
184C2×C42—71no—
185C2×C42—71no—
186C2×C42—71no—
187C2×C42—71no—
188C2×C42C22×D711yescharacteristic
189#97—144no—
190C4×C8—144no—
191#95—21no—
192#95—141no—
193#95—141no—
194#95—141no—
195#95—141no—
196#95—141no—
197#95—141no—
198#95—141no—
199#95—141no—
200C22×C7#40811yescharacteristic
201C22×C7#41011yescharacteristic
202C22×C7#41311yescharacteristic
203C22×C7#41611yescharacteristic
204C22×C7#41911yescharacteristic
205C22×C7C2×#10411yescharacteristic
206C22×C7C2×#10411yescharacteristic
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—81no—
214C4×C7—81no—
215Dic7—22no—
216Dic7—22no—
217Dic7—41no—
218Dic7—41no—
219Dic7—81no—
220Dic7—81no—
221C2×Q8—141no—
222C2×Q8—141no—
223C2×Q8—281no—
224C2×Q8—281no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4C2×C4×D711yescharacteristic
234C22×C4D4×D711yescharacteristic
235C22×C4D4×D711yescharacteristic
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—142no—
243C2×C8—142no—
244C2×C8—281no—
245C2×C8—281no—
246C2×C8—281no—
247C2×C8—281no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—281no—
254C4⋊C4—282no—
255C4⋊C4—282no—
256C42—41no—
257C42—142no—
258C42—142no—
259C42—141no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42—281no—
266C2×C7#147311yescharacteristic
267C2×C7#148511yescharacteristic
268C2×C7#149411yescharacteristic
269C2×C7#150011yescharacteristic
270C2×C7#150611yescharacteristic
271C2×C7#152411yescharacteristic
272C2×C7#155411yescharacteristic
273C23D7×C22⋊C411yescharacteristic
274Q8—561no—
275Q8—561no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—72no—
289C2×C4—72no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4#646411yescharacteristic
307C2×C4#647011yescharacteristic
308C2×C4#647511yescharacteristic
309C2×C4#647811yescharacteristic
310C2×C4#647911yescharacteristic
311C2×C4#648111yescharacteristic
312C8—42no—
313C8—42no—
314C8—282no—
315C8—282no—
316C7#769311yescharacteristic, complemented, Sylow
317C22#8949911yescharacteristic
318C22#8951211yescharacteristic
319C22#8952311yescharacteristic
320C22#8954411yescharacteristic
321C22#8957411yescharacteristic
322C22D7×#10411yescharacteristic
323C22D7×#10411yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—142no—
333C4—142no—
334C4—281no—
335C4—281no—
336C4—561no—
337C4—561no—
338C2#1172588311yescharacteristic
339C2#1172593111yescharacteristic
340C2#1172598911yescharacteristic
341C2#1172603011yescharacteristic
342C2#1172606411yescharacteristic
343C2#1172611511yescharacteristic
344C2#1172618111yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socle, prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1271D7
hypocenterC7#7693
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
316. C7#7693semidirect

Maximal subgroups

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

subgroupindex
#117336372
#117263512
#117263502
C7×#12712
#117254442
#117252862
#76937