Finite Group Databasebeta

← #13242232

Subgroups — #13242232

Order 1792.

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

normal subgroups only

This group has 396 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
1#11733638C211yesmaximal, characteristic
2#11726356C211yesmaximal, characteristic
3#11726351C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic
5#11725444C211yesmaximal, characteristic
6#11725284C211yesmaximal, characteristic
7#11725283C211yesmaximal, characteristic
8C7×#389C2211yescharacteristic
9C2×#6389—21no—
10C2×Dic7⋊C8—21no—
11C2×C8×Dic7—21no—
12#89757C411yescharacteristic
13#89753—21no—
14#89750—21no—
15#89749C411yescharacteristic
16#89749C411yescharacteristic
17#89745C411yescharacteristic
18C42×Dic7C2211yescharacteristic
19#89700C2211yescharacteristic
20#89425—21no—
21#89424—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7677—71nomaximal, Sylow
28C22×C7×C8—21no—
29C22×C7×C8—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7C2×C411yescharacteristic
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—21no—
38C2×Dic7⋊C4—21no—
39C2×C4×Dic7C2×C411yescharacteristic
40C2×C4×Dic7C2×C411yescharacteristic
41C2×C4×Dic7D411yescharacteristic
42C2×C4×Dic7D411yescharacteristic
43C2×C4×Dic7D411yescharacteristic
44C2×C4×Dic7D411yescharacteristic
45C22×C7⋊C8—21no—
46C22×C7⋊C8—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—
56Dic7⋊C8—24no—
57C8×Dic7—42no—
58#2864—71no—
59C4×C4⋊C8—71no—
60#1745—71no—
61#1271—71no—
62#1271—71no—
63#1271—71no—
64#1271D711yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—21no—
71C22×Dic7—21no—
72C22×Dic7C22⋊C411yescharacteristic
73C22×Dic7C22⋊C411yescharacteristic
74C22×Dic7C22⋊C411yescharacteristic
75C22×Dic7C22⋊C411yescharacteristic
76C2×C7×C8—22no—
77C2×C7×C8—22no—
78C2×C7×C8—41no—
79C2×C7×C8—41no—
80C2×C7×C8—41no—
81C2×C7×C8—41no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C42×C7—41no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—42no—
91C4⋊Dic7—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—42no—
95C4×Dic7—21no—
96C4×Dic7—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7—22no—
104C2×C7⋊C8—22no—
105C2×C7⋊C8—22no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—41no—
109C2×C7⋊C8—41no—
110C2×C4⋊C8—141no—
111C2×#97—141no—
112C2×C4×C8—141no—
113#389—71no—
114#389—71no—
115#389—71no—
116#389C2×D711yescharacteristic
117#387—141no—
118#387—141no—
119#387—141no—
120#387—141no—
121#382—71no—
122#382—71no—
123C43—71no—
124#336—71no—
125#336—71no—
126#336—141no—
127#336C2×D711yescharacteristic
128#336C2×D711yescharacteristic
129C23×C7C2×C22⋊C411yescharacteristic
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7#13111yescharacteristic
143C2×C4×C7#13111yescharacteristic
144C2×C4×C7#13511yescharacteristic
145C2×C4×C7#13511yescharacteristic
146C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C7#13711yescharacteristic
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#10112yes—
164C2×Dic7#10412yes—
165C2×Dic7#10412yes—
166C7×C8—42no—
167C7×C8—42no—
168C7⋊C8—42no—
169C7⋊C8—42no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—141no—
173C22×C8—141no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—141no—
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
189C4⋊C8—144no—
190#97—144no—
191C4×C8—282no—
192#95—21no—
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×C7C2×#10111yescharacteristic
203C22×C7#41611yescharacteristic
204C22×C7#41711yescharacteristic
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—22no—
218Dic7—41no—
219Dic7—81no—
220Dic7—81no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4C2×C4×D711yescharacteristic
230C22×C4D4×D711yescharacteristic
231C22×C4D4×D711yescharacteristic
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—142no—
239C2×C8—142no—
240C2×C8—281no—
241C2×C8—281no—
242C2×C8—281no—
243C2×C8—281no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—281no—
252C4⋊C4—282no—
253C4⋊C4—282no—
254C4⋊C4—281no—
255C4⋊C4—281no—
256C42—41no—
257C42—142no—
258C42—142no—
259C42—142no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42—141no—
266C2×C7#147011yescharacteristic
267C2×C7#148311yescharacteristic
268C2×C7#149411yescharacteristic
269C2×C7#150011yescharacteristic
270C2×C7#150911yescharacteristic
271C2×C7#152611yescharacteristic
272C2×C7#155211yescharacteristic
273C23D7×C22⋊C411yescharacteristic
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—72no—
287C2×C4—72no—
288C2×C4—72no—
289C2×C4—141no—
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—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4#646411yescharacteristic
305C2×C4#647011yescharacteristic
306C2×C4#647511yescharacteristic
307C2×C4#647711yescharacteristic
308C2×C4#647911yescharacteristic
309C2×C4#648111yescharacteristic
310C8—42no—
311C8—42no—
312C8—282no—
313C8—282no—
314C7#767711yescharacteristic, complemented, Sylow
315C22#8949911yescharacteristic
316C22#8951211yescharacteristic
317C22D7×#10111yescharacteristic
318C22#8954411yescharacteristic
319C22#8957511yescharacteristic
320C22D7×#10411yescharacteristic
321C22D7×#10411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—81no—
329C4—81no—
330C4—142no—
331C4—142no—
332C4—142no—
333C4—281no—
334C4—561no—
335C4—561no—
336C2#1172588211yescharacteristic
337C2#1172593111yescharacteristic
338C2#1172598811yescharacteristic
339C2#1172603011yescharacteristic
340C2#1172606511yescharacteristic
341C2#1172611611yescharacteristic
342C2#1172617811yescharacteristic
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#7677
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
314. C7#7677semidirect

Maximal subgroups

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

subgroupindex
#117336382
#117263562
#117263512
C7×#12712
#117254442
#117252842
#117252832
#76777