Finite Group Databasebeta

← #13243131

Subgroups — #13243131

Order 1792.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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#11734043C211yesmaximal, characteristic
2C7×#2095C211yesmaximal, characteristic
3#11727344C211yesmaximal, characteristic
4#11727326C211yesmaximal, characteristic
5#11725448C211yesmaximal, characteristic
6#11725286C211yesmaximal, characteristic
7#11725284C211yesmaximal, characteristic
8C2×C7×#107—21no—
9C2×C7×#103—21no—
10C7×#391C2211yescharacteristic
11C7×#384C2211yescharacteristic
12#90001—21no—
13#89744—21no—
14C2×#6384—21no—
15C2×#6381—21no—
16#89701C2211yescharacteristic
17#89700C2211yescharacteristic
18#89429—21no—
19#89429—21no—
20#89425—21no—
21#89418—21no—
22C7×#336C2211yescharacteristic
23#89324—21no—
24#89324—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7836—71nomaximal, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×Dic14D411yescharacteristic
30C22×C7×C8—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36C2×C4⋊Dic7D411yescharacteristic
37C2×C4⋊Dic7D411yescharacteristic
38C2×C4⋊Dic7D411yescharacteristic
39C2×Dic7⋊C4—21no—
40C2×Dic7⋊C4—21no—
41C2×Dic7⋊C4—21no—
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7—21no—
45C22×C7⋊C8—21no—
46C22×C7⋊C8—21no—
47C7×#107—42no—
48C7×#103—42no—
49C7×#95—21no—
50C7×#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57#6384—42no—
58#6381—42no—
59#2806—71no—
60#2100—71no—
61#2095—71no—
62#2095D711yescharacteristic
63#1272—71no—
64#1272—71no—
65#1271—71no—
66C2×C7×Q8—21no—
67C2×C7×Q8—21no—
68C2×C7×Q8—41no—
69C2×C7×Q8—41no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7—21no—
80C2×Dic14—21no—
81C2×Dic14—41no—
82C2×Dic14—41no—
83C2×Dic14Q1612yes—
84C2×C7×C8—22no—
85C2×C7×C8—41no—
86C2×C7×C8—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C42×C7—41no—
94C4⋊Dic7—21no—
95C4⋊Dic7—41no—
96C4⋊Dic7—41no—
97C4⋊Dic7Q1612yes—
98Dic7⋊C4—41no—
99Dic7⋊C4—41no—
100Dic7⋊C4—81no—
101Dic7⋊C4—81no—
102Dic7⋊C4—81no—
103Dic7⋊C4—81no—
104C4×Dic7—81no—
105C2×C7⋊C8—22no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8—41no—
109C2×C7⋊C8—41no—
110C2×C7⋊C8—41no—
111C2×#107—21no—
112C2×#107—141no—
113C2×#103—21no—
114C2×#103—141no—
115#398—141no—
116#398—141no—
117#395—141no—
118#391—71no—
119#391—141no—
120#391C2×D711yescharacteristic
121#389—71no—
122#389—141no—
123#387—141no—
124#384C2×D711yescharacteristic
125#340—141no—
126#340—141no—
127#336—71no—
128#336—71no—
129#336C2×D711yescharacteristic
130C23×C7#12011yescharacteristic
131C7×Q8—42no—
132C7×Q8—81no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7C2×Q1611yescharacteristic
149C2×C4×C7C2×Q1611yescharacteristic
150C2×C4×C7#13511yescharacteristic
151C2×C4×C7#13711yescharacteristic
152C2×C4×C7#13711yescharacteristic
153C2×C4×C7#13711yescharacteristic
154C2×Dic7—22no—
155C2×Dic7—22no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166Dic14—42no—
167Dic14—81no—
168C7×C8—42no—
169C7⋊C8—42no—
170C7⋊C8—81no—
171C22×Q8—71no—
172C22×Q8C22⋊D711yescharacteristic
173C22×C8—21no—
174C22×C8—141no—
175C22×C8—141no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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⋊C4C22⋊D711yescharacteristic
188C2×C42—141no—
189C2×C42C22×D711yescharacteristic
190#107—42no—
191#107—282no—
192#103—42no—
193#103—282no—
194#95—21no—
195#95—21no—
196#95—141no—
197#95—141no—
198#95—141no—
199#95—141no—
200#95—141no—
201#95—141no—
202C22×C7#45111yescharacteristic
203C22×C7#45111yescharacteristic
204C22×C7#45211yescharacteristic
205C22×C7#45311yescharacteristic
206C22×C7#45311yescharacteristic
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—81no—
216C4×C7—81no—
217C4×C7—161no—
218Dic7—42no—
219Dic7—42no—
220Dic7—81no—
221Dic7—81no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—72no—
227C2×Q8—141no—
228C2×Q8—281no—
229C2×Q8—281no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4D4×D711yescharacteristic
238C22×C4D4×D711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—142no—
244C2×C8—281no—
245C2×C8—281no—
246C2×C8—281no—
247C2×C8—281no—
248C2×C8—281no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—72no—
256C4⋊C4—141no—
257C4⋊C4—281no—
258C4⋊C4—281no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C4⋊C4—561no—
262C4⋊C4—561no—
263C4⋊C4—561no—
264C4⋊C4—561no—
265C42—41no—
266C42—561no—
267C2×C7#161411yescharacteristic
268C2×C7#161711yescharacteristic
269C2×C7#161811yescharacteristic
270C2×C7#163211yescharacteristic
271C2×C7#163311yescharacteristic
272C2×C7#163611yescharacteristic
273C2×C7#165611yescharacteristic
274C23#650011yescharacteristic
275Q8—42no—
276Q8—81no—
277Q8—282no—
278Q8—561no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4D7×Q1611yescharacteristic
307C2×C4D7×Q1611yescharacteristic
308C2×C4#648111yescharacteristic
309C2×C4#648111yescharacteristic
310C2×C4#650511yescharacteristic
311C2×C4#651311yescharacteristic
312C8—42no—
313C8—282no—
314C8—561no—
315C7#783611yescharacteristic, complemented, Sylow
316C22#8981911yescharacteristic
317C22#8982511yescharacteristic
318C22#8984911yescharacteristic
319C22#8995811yescharacteristic
320C22#8995811yescharacteristic
321C22#8997011yescharacteristic
322C22#8997311yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—282no—
333C4—282no—
334C4—561no—
335C4—561no—
336C2#1172746911yescharacteristic
337C2#1172891011yescharacteristic
338C2#1172891611yescharacteristic
339C2#1172893011yescharacteristic
340C2#1172893311yescharacteristic
341C2#1172906111yescharacteristic
342C2#1172908111yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2095C2
socle, prime radicalC23×C7#120
hypercenter#2095D7
hypocenterC7#7836
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
315. C7#7836semidirect

Maximal subgroups

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

subgroupindex
#117340432
C7×#20952
#117273442
#117273262
#117254482
#117252862
#117252842
#78367