Finite Group Databasebeta

← #13328702

Subgroups — #13328702

Order 1792.

870 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 207 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×#3502C211yesmaximal, characteristic
2#11740769C212yesmaximal
3#11739924C211yesmaximal, characteristic
4#11729653C218yesmaximal
5#11729589C212yesmaximal
6#11729378C211yesmaximal, characteristic
7C7×Q82—22no—
8C7×#557—22no—
9C7×#557—24no—
10C7×#544—21no—
11C7×#544C2214yes—
12C7×#533—22no—
13C7×#533C2212yes—
14C7×#519—21no—
15C2×C4×C7×Q8C2211yescharacteristic
16C2×#6432C2211yescharacteristic
17#90002—21no—
18#89853—22no—
19#89850—22no—
20#89816C2218yes—
21#89793—24no—
22#89758C2214yes—
23#89749—21no—
24C4×C4⋊Dic7C2211yescharacteristic
25#89701—22no—
26#89697C2212yes—
27#89333C2218yes—
28#89323C2214yes—
29#18411—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×Dic14—21no—
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—22no—
34C7×C4⋊Q8—24no—
35C7×C4⋊Q8—28no—
36C7×C4⋊Q8D412yes—
37C7×C4⋊Q8D412yes—
38C7×#125—24no—
39C7×#125—24no—
40C7×#125—28no—
41C7×#125—22no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8C2314yes—
45C4×C7×Q8—24no—
46C4×C7×Q8—24no—
47C4×C7×Q8—24no—
48C4×C7×Q8—24no—
49C4×C7×Q8—22no—
50C4×C7×Q8—22no—
51C4×C7×Q8—22no—
52C4×C7×Q8D414yes—
53C7×#117—24no—
54C7×#117—22no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7D412yes—
58C2×C42×C7C2311yescharacteristic
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2312yes—
62C2×C4×Dic7—21no—
63#6432—22no—
64#6432—22no—
65#6409—28no—
66C7⋊#95—24no—
67C7⋊#95—24no—
68C7⋊#95C2312yes—
69C22⋊(C7⋊C8)C2314yes—
70#6381—28no—
71C4⋊(C7⋊C8)—22no—
72C4×C7⋊C8—22no—
73#3502D711yescharacteristic
74#3230—72no—
75#2568—71no—
76#1617—78no—
77#1503—72no—
78#1461—71no—
79C2×C7×Q8—24no—
80C2×C7×Q8—22no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8Q1614yes—
84C2×C7×Q8C2×D414yes—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D412yes—
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2411yescharacteristic
90C22×Dic7—22no—
91C22×Dic7—22no—
92C2×Dic14—44no—
93C2×Dic14—41no—
94C2×Dic14—41no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—28no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—24no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4C2×D414yes—
107C7×C22⋊C4—24no—
108C42×C7—28no—
109C42×C7—22no—
110C42×C7—22no—
111C42×C7—24no—
112C42×C7C2×D412yes—
113C42×C7C2×D412yes—
114C4⋊Dic7—24no—
115C4⋊Dic7—42no—
116C4⋊Dic7—42no—
117C4×Dic7—42no—
118C2×C7⋊C8—24no—
119Q82—22no—
120#557—22no—
121#557—24no—
122#544—21no—
123#544C2×D714yes—
124#533—22no—
125#533C2×D712yes—
126C2×C4⋊Q8—71no—
127#519—21no—
128C2×C4×Q8C2×D711yescharacteristic
129#500—142no—
130#487—142no—
131#484—78no—
132#477—144no—
133#431—72no—
134#400—74no—
135#391—141no—
136#391—142no—
137#382—141no—
138C4×C4⋊C4—71no—
139#328—78no—
140#324—74no—
141C23×C7C22×D411yescharacteristic
142C7×Q8—28no—
143C7×Q8—44no—
144C7×Q8—44no—
145C7×Q8—84no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—24no—
150C2×C4×C7—24no—
151C2×C4×C7—22no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7#12011yescharacteristic
155C2×C4×C7#12012yes—
156C2×C4×C7#12011yescharacteristic
157C2×C4×C7C2×Q1614yes—
158C2×C4×C7C2×Q1612yes—
159C2×C4×C7#13712yes—
160C2×C4×C7C22×D412yes—
161C2×Dic7—42no—
162C2×Dic7—44no—
163C2×Dic7—42no—
164C2×Dic7—42no—
165C2×Dic7—42no—
166Dic14—82no—
167Dic14—82no—
168C7⋊C8—44no—
169C22×Q8—141no—
170C22×Q8C22⋊D711yescharacteristic
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—24no—
174C4⋊Q8—28no—
175C4⋊Q8—142no—
176C4⋊Q8—142no—
177C4⋊Q8C22⋊D712yes—
178C4⋊Q8C22⋊D712yes—
179#125—24no—
180#125—24no—
181#125—28no—
182#125—22no—
183C22⋊Q8—22no—
184C22⋊Q8—24no—
185C22⋊Q8C22×D714yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8C22⋊D714yes—
194#117—24no—
195#117—22no—
196C2×C4⋊C4—72no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—141no—
199C2×C4⋊C4C22⋊D711yescharacteristic
200C2×C4⋊C4C22×D712yes—
201C2×C42—141no—
202C2×C42C22⋊D712yes—
203C2×C42C22×D711yescharacteristic
204#107—148no—
205C4⋊C8—142no—
206#103—148no—
207C22⋊C8—74no—
208C4×C8—142no—
209#95—72no—
210#95—144no—
211#95—144no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7C2×#12011yescharacteristic
215C22×C7C22×Q1611yescharacteristic
216C22×C7C2×#13711yescharacteristic
217C4×C7—24no—
218C4×C7—21no—
219C4×C7—22no—
220C4×C7—44no—
221C4×C7—42no—
222C4×C7—44no—
223C4×C7#45114yes—
224C4×C7#45412yes—
225Dic7—82no—
226Dic7—82no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—284no—
232C2×Q8—281no—
233C2×Q8—281no—
234C2×Q8#114414yes—
235C2×Q8C2×C22⋊D714yes—
236C22×C4—142no—
237C22×C4—142no—
238C22×C4C2×C22⋊D712yes—
239C22×C4C2×C22⋊D711yescharacteristic
240C22×C4C2×C22⋊D712yes—
241C22×C4C2×C22⋊D711yescharacteristic
242C22×C4C23×D711yescharacteristic
243C2×C8—144no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—28no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—144no—
256C4⋊C4—282no—
257C4⋊C4—282no—
258C4⋊C4C2×C22⋊D714yes—
259C22⋊C4—24no—
260C42—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42—282no—
265C42C2×C22⋊D712yes—
266C42C2×C22⋊D712yes—
267C2×C7—21no—
268C2×C7—21no—
269C2×C7C2×#45111yescharacteristic
270C2×C7C2×#45411yescharacteristic
271C2×C7#302811yescharacteristic
272C23C22×C22⋊D711yescharacteristic
273Q8—28no—
274Q8—44no—
275Q8—44no—
276Q8—84no—
277Q8—562no—
278Q8—562no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—282no—
288C2×C4—282no—
289C2×C4—284no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4#650512yes—
293C2×C4C2×#114414yes—
294C2×C4C2×#114412yes—
295C2×C4#651611yescharacteristic
296C2×C4#651612yes—
297C2×C4#651611yescharacteristic
298C2×C4C22×C22⋊D712yes—
299C8—284no—
300C7#1841111yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22C2×#650511yescharacteristic
304C22C22×#114411yescharacteristic
305C22C2×#651611yescharacteristic
306C4—24no—
307C4—21no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—562no—
313C4—562no—
314C4#9000014yes—
315C4#9001712yes—
316C2—21no—
317C2—21no—
318C2C2×#9000011yescharacteristic
319C2C2×#9001711yescharacteristic
320C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C2×#120
prime radicalC23×C7C22×D4
hypercenter#3502D7
hypocenterC7#18411
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
300. C7#18411semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117407692
#117399242
#117296532
#117295892
#117293782
#184117