Finite Group Databasebeta

← #13243098

Subgroups — #13243098

Order 1792.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#11733580C211yesmaximal, characteristic
2#11727366C211yesmaximal, characteristic
3#11726445C211yesmaximal, characteristic
4#11726397C211yesmaximal, characteristic
5C7×#1271C211yesmaximal, characteristic
6#11725286C211yesmaximal, characteristic
7#11725284C211yesmaximal, characteristic
8C2×#6451—21no—
9C7×#389C2211yescharacteristic
10C2×#6392—21no—
11C2×#6390—21no—
12#89756—21no—
13#89753—21no—
14#89751—21no—
15#89707C2211yescharacteristic
16#89701C2211yescharacteristic
17#89700C2211yescharacteristic
18#89428—21no—
19#89426—21no—
20C7×#336C2211yescharacteristic
21C7×#336C2211yescharacteristic
22#89346—21no—
23#89346—21no—
24#89324—21no—
25#89324—21no—
26#89319C2211yescharacteristic
27#7829—71nomaximal, Sylow
28C22×Dic14—21no—
29C22×Dic14D411yescharacteristic
30C22×C7×C8—21no—
31C22×C7×C8—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7D411yescharacteristic
41C2×C4⋊Dic7D411yescharacteristic
42C2×C4⋊Dic7D411yescharacteristic
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×C4×Dic7—21no—
47C22×C7⋊C8—21no—
48#6451—42no—
49#6451—42no—
50C7×#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57#6392—42no—
58#6390—42no—
59#2734—71no—
60#2100—71no—
61#2100—71no—
62#2095—71no—
63#1272—71no—
64#1271—71no—
65#1271D711yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×Dic7—21no—
72C22×Dic7—21no—
73C22×Dic7—21no—
74C22×Dic7—21no—
75C22×Dic7—41no—
76C2×Dic14—21no—
77C2×Dic14—41no—
78C2×Dic14—41no—
79C2×Dic14—42no—
80C2×Dic14—81no—
81C2×Dic14—81no—
82C2×Dic14SD1612yes—
83C2×C7×C8—22no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C2×C7×C8—41no—
89C7×C4⋊C4—21no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4Q1612yes—
94C42×C7—41no—
95C4⋊Dic7—21no—
96C4⋊Dic7—42no—
97C4⋊Dic7—41no—
98C4⋊Dic7—41no—
99C4⋊Dic7—42no—
100C4⋊Dic7—42no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103C4⋊Dic7Q1612yes—
104Dic7⋊C4—42no—
105Dic7⋊C4—41no—
106Dic7⋊C4—41no—
107Dic7⋊C4—81no—
108C4×Dic7—42no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111C2×C7⋊C8—41no—
112C2×C4⋊Q8—141no—
113C2×#107—141no—
114C2×#103—141no—
115#398—141no—
116#395—141no—
117#395—141no—
118#391—71no—
119#389—71no—
120#389C2×D711yescharacteristic
121#387—141no—
122#387—141no—
123#384—71no—
124#340—141no—
125#340—141no—
126#340—141no—
127#340—141no—
128#336—71no—
129#336C2×D711yescharacteristic
130#336C2×D711yescharacteristic
131C23×C7#12011yescharacteristic
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—22no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7C2×SD1611yescharacteristic
145C2×C4×C7C2×Q1611yescharacteristic
146C2×C4×C7C2×Q1611yescharacteristic
147C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C7#13711yescharacteristic
149C2×C4×C7#13711yescharacteristic
150C2×Dic7—22no—
151C2×Dic7—22no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—81no—
163C2×Dic7—81no—
164C2×Dic7—81no—
165Dic14—42no—
166Dic14—81no—
167Dic14—84no—
168C7×C8—42no—
169C7×C8—81no—
170C7⋊C8—81no—
171C22×Q8—71no—
172C22×Q8—141no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—141no—
176C4⋊Q8—282no—
177C4⋊Q8—282no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—141no—
183C2×C4⋊C4—141no—
184C2×C4⋊C4—141no—
185C2×C4⋊C4—141no—
186C2×C4⋊C4—141no—
187C2×C4⋊C4—141no—
188C2×C4⋊C4—141no—
189C2×C4⋊C4D2811yescharacteristic
190C2×C4⋊C4D2811yescharacteristic
191C2×C42—141no—
192C2×C42C22×D711yescharacteristic
193#107—282no—
194#103—282no—
195#95—21no—
196#95—141no—
197#95—141no—
198#95—141no—
199#95—141no—
200#95—141no—
201#95—141no—
202C22×C7#45011yescharacteristic
203C22×C7#45111yescharacteristic
204C22×C7#45111yescharacteristic
205C22×C7#45311yescharacteristic
206C22×C7#45411yescharacteristic
207C22×C7#45411yescharacteristic
208C22×C7#45811yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—42no—
216C4×C7—81no—
217Dic7—42no—
218Dic7—42no—
219Dic7—81no—
220Dic7—81no—
221Dic7—161no—
222C2×Q8—72no—
223C2×Q8—141no—
224C2×Q8—281no—
225C2×Q8—281no—
226C2×Q8—282no—
227C2×Q8—561no—
228C2×Q8—561no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—281no—
236C22×C4C2×D2811yescharacteristic
237C22×C4D4×D711yescharacteristic
238C22×C4D4×D711yescharacteristic
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—281no—
246C2×C8—281no—
247C2×C8—281no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—72no—
253C4⋊C4—141no—
254C4⋊C4—281no—
255C4⋊C4—281no—
256C4⋊C4—281no—
257C4⋊C4—281no—
258C4⋊C4—282no—
259C4⋊C4—282no—
260C4⋊C4—282no—
261C4⋊C4—282no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4—561no—
265C4⋊C4Dic2812yes—
266C42—41no—
267C42—282no—
268C2×C7#161311yescharacteristic
269C2×C7#161711yescharacteristic
270C2×C7#161711yescharacteristic
271C2×C7#163311yescharacteristic
272C2×C7#163711yescharacteristic
273C2×C7#163711yescharacteristic
274C2×C7#165711yescharacteristic
275C23#644511yescharacteristic
276Q8—282no—
277Q8—561no—
278Q8—564no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—281no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—561no—
304C2×C4—561no—
305C2×C4—561no—
306C2×C4C2×Dic2811yescharacteristic
307C2×C4#647211yescharacteristic
308C2×C4#647411yescharacteristic
309C2×C4D7×SD1611yescharacteristic
310C2×C4#647811yescharacteristic
311C2×C4D7×Q1611yescharacteristic
312C8—42no—
313C8—81no—
314C8—561no—
315C7#782911yescharacteristic, complemented, Sylow
316C22#8951111yescharacteristic
317C22#8951411yescharacteristic
318C22#8952511yescharacteristic
319C22#8954811yescharacteristic
320C22#8958011yescharacteristic
321C22#8959411yescharacteristic
322C22#8959911yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—81no—
331C4—282no—
332C4—282no—
333C4—561no—
334C4—561no—
335C4—1121no—
336C2#1172589511yescharacteristic
337C2#1172594311yescharacteristic
338C2#1172602111yescharacteristic
339C2#1172604411yescharacteristic
340C2#1172608511yescharacteristic
341C2#1172612511yescharacteristic
342C2#1172621311yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socle, prime radicalC23×C7#120
hypercenter#1271D7
hypocenterC7#7829
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
315. C7#7829semidirect

Maximal subgroups

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

subgroupindex
#117335802
#117273662
#117264452
#117263972
C7×#12712
#117252862
#117252842
#78297