Finite Group Databasebeta

← #34176990

Subgroups — #34176990

Order 20224.

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

normal subgroups only

This group has 604 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#23764634C212yesmaximal, complemented
2#23764470C211yesmaximal, characteristic
3#23761223C212yesmaximal, complemented
4#23761221C211yesmaximal, characteristic, complemented
5C79×#2683C211yesmaximal, characteristic, complemented
6C22×#11929609—21no—
7C2×#14579158C2211yescharacteristic
8C2×#14579081—22no—
9C2×#14579081—22no—
10C2×#14579081—21no—
11#19098318C2212yes—
12#19098318C2211yescharacteristic
13C2×C22⋊Q8×C79—22no—
14#396×C79—22no—
15#395×C79—21no—
16#390×C79C2211yescharacteristic
17#389×C79C2212yes—
18#386×C79—21no—
19C22×D316—22no—
20C22×D316—21no—
21C2×#11929609—42no—
22C2×#11929609—42no—
23C2×#11929609—42no—
24C22×C79⋊C8—21no—
25C22×C79⋊C8—22no—
26#14579158—42no—
27#14579081—42no—
28#14579081—44no—
29#14579081—44no—
30C22×Q8×C79—21no—
31C22×D4×C79D411yescharacteristic
32C23×C4×C79—21no—
33C22⋊Q8×C79—44no—
34C22⋊Q8×C79—44no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79D412yes—
38C2×C4⋊C4×C79D412yes—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—21no—
43C2×C42×C79C2311yescharacteristic
44#95×C79—21no—
45#95×C79—22no—
46C23×D79—41no—
47C2×D316—24no—
48C2×D316—22no—
49C2×D316—42no—
50C2×D316—41no—
51C2×D316—41no—
52C2×D316—41no—
53C2×D316—42no—
54C2×D316—42no—
55C2×D316—42no—
56C2×D316—41no—
57C2×D316—42no—
58C2×D316—41no—
59#11929609—48no—
60C2×C79⋊C8—22no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—42no—
64C2×C79⋊C8—42no—
65C2×C79⋊C8—41no—
66C24×C79—21no—
67C2×Q8×C79—44no—
68C2×Q8×C79—81no—
69C2×D4×C79—21no—
70C2×D4×C79—41no—
71C2×D4×C79—41no—
72C2×D4×C79SD1612yes—
73C22×C4×C79—22no—
74C22×C4×C79—22no—
75C22×C4×C79—21no—
76C22×C4×C79—44no—
77C22×C4×C79—81no—
78C22×C4×C79C2×D412yes—
79C22×C4×C79C2×D411yescharacteristic
80C4⋊C4×C79—22no—
81C4⋊C4×C79—44no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—42no—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79D814yescomplemented
87C22⋊C4×C79—44no—
88C22⋊C4×C79—42no—
89C22⋊C4×C79—42no—
90C22⋊C4×C79—81no—
91C42×C79—41no—
92C23×C79—22no—
93C23×C79—42no—
94C23×C79—41no—
95C23×C79—42no—
96C23×C79—41no—
97C23×C79#12011yescharacteristic
98C22×D79—42no—
99C22×D79—82no—
100C22×D79—81no—
101C22×D79—81no—
102C22×D79—82no—
103Q8×C79—84no—
104D4×C79—42no—
105D4×C79—81no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—21no—
110C2×C4×C79—24no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—84no—
119C2×C4×C79C2×D812yes—
120C2×C4×C79C2×SD1611yescharacteristic
121C2×C4×C79AGL(1,ℤ/8)12yes—
122C2×C4×C79#13711yescharacteristic
123D316—42no—
124D316—42no—
125D316—44no—
126D316—44no—
127D316—81no—
128D316—81no—
129D316—82no—
130D316—82no—
131C79⋊C8—42no—
132C79⋊C8—82no—
133C22×C79—42no—
134C22×C79—44no—
135C22×C79—82no—
136C22×C79—81no—
137C22×C79—81no—
138C22×C79#44712yes—
139C22×C79#44811yescharacteristic
140C22×C79#45311yescharacteristic
141C22×C79#45412yes—
142C22×C79B(4,𝔽2)11yescharacteristic
143C2×D79—82no—
144C2×D79—84no—
145C2×D79—161no—
146C2×D79—162no—
147C2×D79—161no—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—22no—
151C4×C79—22no—
152C4×C79—44no—
153C4×C79—82no—
154C4×C79—81no—
155#7738—791nomaximal, Sylow
156C2×C79—82no—
157C2×C79#159611yescharacteristic
158C2×C79#159812yes—
159C2×C79#161911yescharacteristic
160C2×C79#162712yes—
161C2×C79#165711yescharacteristic
162D79—162no—
163#2683D7911yescharacteristic, complemented
164#2041—792no—
165#2011—791no—
166#1271—792no—
167#1270—791no—
168C79#773811yescharacteristic, complemented, Sylow
169C22×D8—1581no—
170C2×#127—791no—
171C2×C22⋊Q8—22no—
172C2×#102—1581no—
173C2×#102—1582no—
174C2×#102—1582no—
175#396—22no—
176#395—21no—
177#390C2×D7911yescharacteristic
178#389C2×D7912yes—
179#386—21no—
180#336—791no—
181#336—792no—
182C22×Q8—21no—
183C22×D4—1581no—
184C22×D4—1582no—
185C22×D4C22⋊D7911yescharacteristic
186C23×C4—21no—
187C2×D8—3162no—
188C2×D8—3162no—
189C2×D8—3162no—
190C22×C8—1582no—
191C22×C8—1581no—
192#127—3162no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D7912yes—
198C2×C4⋊C4C22⋊D7912yes—
199C2×C4⋊C4C22⋊D7911yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C42C22×D7911yescharacteristic
204#102—3164no—
205#102—3162no—
206#102—3164no—
207#95—21no—
208#95—22no—
209C24—21no—
210C24—3161no—
211C2×Q8—44no—
212C2×Q8—81no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—3161no—
217C2×D4—3161no—
218C2×D4—3162no—
219C2×D4—3162no—
220C2×D4—3162no—
221C2×D4—3161no—
222C2×D4—3162no—
223C2×D4—3161no—
224C2×D4—3162no—
225C2×D4—3161no—
226C2×D4—1584no—
227C2×D4—1582no—
228C2×D4#1192961112yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—44no—
233C22×C4—81no—
234C22×C4C2×C22⋊D7912yes—
235C22×C4C2×C22⋊D7911yescharacteristic
236D8—3168no—
237C2×C8—3161no—
238C2×C8—3161no—
239C2×C8—3162no—
240C2×C8—3162no—
241C2×C8—3162no—
242C2×C8—1582no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4#1192960914yescomplemented
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C42—41no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—3162no—
261C23—6322no—
262C23—6321no—
263C23—6321no—
264C23—6322no—
265C23#1457913011yescharacteristic
266Q8—84no—
267D4—42no—
268D4—81no—
269D4—3162no—
270D4—3162no—
271D4—3164no—
272D4—3164no—
273D4—6321no—
274D4—6321no—
275D4—6322no—
276D4—6322no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4C2×#1192960912yes—
291C2×C4C2×#1192961111yescharacteristic
292C2×C4#1457918712yes—
293C2×C4#1457919211yescharacteristic
294C8—3162no—
295C8—6322no—
296C22—44no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—6322no—
302C22—6324no—
303C22—12641no—
304C22—12641no—
305C22—12642no—
306C22#1909869712yes—
307C22#1909870211yescharacteristic
308C22#1909873111yescharacteristic
309C22#1909873612yes—
310C22#1909875011yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—81no—
317C4—82no—
318C2—82no—
319C2—12642no—
320C2#2376232311yescharacteristic
321C2#2376233512yes—
322C2#2376245411yescharacteristic
323C2#2376250312yes—
324C2#2376269811yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2683C2
socleC23×C79#120
prime radical#23764470C2
hypercenter#2683D79
hypocenterC79#7738
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #23764634C2semidirect
3. #23761223C2semidirect
4. #23761221C2semidirect
5. C79×#2683C2semidirect
86. C4⋊C4×C79D8semidirect
163. #2683D79semidirect
168. C79#7738semidirect
249. C4⋊C4#11929609semidirect

Maximal subgroups

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

subgroupindex
#237646342
#237644702
#237612232
#237612212
C79×#26832
#773879