Finite Group Databasebeta

← #34198573

Subgroups — #34198573

Order 20224.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#23766668C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761426C212yesmaximal, complemented
5C79×#1742C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098616—21no—
8#19098502C2211yescharacteristic
9#19098499C411yescharacteristic
10#19098496—21no—
11#19098486—21no—
12#19098483C2211yescharacteristic
13#19098475C411yescharacteristic
14C4×Dic79⋊C4—21no—
15C4×C4⋊Dic79C411yescharacteristic
16C4×C22⋊Dic79—21no—
17#19098289C2212yes—
18#19098287—21no—
19C22×M16×C79—21no—
20C2×#117×C79C2211yescharacteristic
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79—21no—
35C4⋊(C79⋊C8)—41no—
36C22⋊(C79⋊C8)—22no—
37C4×C79⋊C8—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C2×M16×C79—24no—
46C2×M16×C79—22no—
47C2×M16×C79—41no—
48C2×M16×C79—42no—
49C22×C8×C79—41no—
50#117×C79—41no—
51#117×C79—41no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×Dic158—41no—
59C2×Dic158—42no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—41no—
66C4⋊Dic79—42no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C4×Dic79—41no—
73C4×Dic79—41no—
74C4×Dic79—42no—
75C24×C79C22⋊C411yescharacteristic
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—22no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79SD1612yes—
83C22×C4×C79Q1612yes—
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87M16×C79—24no—
88M16×C79—44no—
89M16×C79—42no—
90C2×C8×C79—42no—
91C2×C8×C79—44no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79—41no—
94C22⋊C4×C79—41no—
95C22⋊C4×C79—41no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—21no—
99C42×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79#10312yes—
105C23×C79C2×C22⋊C411yescharacteristic
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79#9911yescharacteristic
121C2×C4×C79#9911yescharacteristic
122C2×C4×C79#10312yes—
123C2×C4×C79#10412yes—
124C2×C4×C79C2×SD1611yescharacteristic
125C2×C4×C79C2×Q1611yescharacteristic
126C2×Dic79—22no—
127C2×Dic79—22no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138Dic158—42no—
139Dic158—42no—
140C8×C79—44no—
141C79⋊C8—82no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—22no—
145C22×C79—42no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—81no—
149C22×C79C2×#9911yescharacteristic
150C22×C79C2×#10311yescharacteristic
151C22×C79C2×#10411yescharacteristic
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79#32812yes—
159C4×C79#32812yes—
160Dic79—42no—
161Dic79—42no—
162Dic79—81no—
163Dic79—81no—
164#11721—791nomaximal, Sylow
165C2×C79—21no—
166C2×C79—21no—
167C2×C79—42no—
168C2×C79C2×#32811yescharacteristic
169C2×C79#212011yescharacteristic
170C2×C79#213211yescharacteristic
171#2396—791no—
172#1742D7911yescharacteristic
173#1458—791no—
174#1354—792no—
175#1353—792no—
176C79#1172111yescharacteristic, complemented, Sylow
177C22×M16—21no—
178C2×C4×Q8—791no—
179C2×#117C2×D7911yescharacteristic
180#431—1581no—
181#400—791no—
182#399—791no—
183#398—1581no—
184#396—1581no—
185#395—791no—
186#393—791no—
187C4×C4⋊C4—791no—
188C4×C4⋊C4—1581no—
189C4×C22⋊C4—1581no—
190#343—21no—
191#343C2×D7912yes—
192#324—792no—
193#323—1581no—
194C22×Q8—791no—
195C23×C4C22⋊D7911yescharacteristic
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C22×C8—41no—
201C4×Q8—3161no—
202C4×Q8—3161no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—791no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C22⋊D7911yescharacteristic
211C2×C4⋊C4C22×D7911yescharacteristic
212C2×C22⋊C4—1581no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4C4×D7911yescharacteristic
215C2×C22⋊C4D31611yescharacteristic
216C2×C42—1581no—
217C2×C42—1581no—
218C2×C42C4×D7911yescharacteristic
219C2×C42D31611yescharacteristic
220C4⋊C8—3161no—
221C22⋊C8—1582no—
222C4×C8—3161no—
223#95—791no—
224#95—791no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228#95—1581no—
229C24(C2×D79)⋊C411yescharacteristic
230C2×Q8—3161no—
231C2×Q8—3162no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4#1192961212yes—
243C22×C4#1192961412yes—
244C22×C4C2×C4×D7911yescharacteristic
245C22×C4C2×C22⋊D7911yescharacteristic
246C22×C4C2×D31611yescharacteristic
247M16—24no—
248M16—44no—
249M16—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—3162no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3161no—
256C4⋊C4—3162no—
257C4⋊C4—3162no—
258C4⋊C4—3162no—
259C4⋊C4—3162no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—3162no—
263C22⋊C4—3162no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—3161no—
269C42—3161no—
270C42—3162no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1457908412yes—
276C23C2×(C2×D79)⋊C411yescharacteristic
277Q8—3162no—
278Q8—3162no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—1582no—
304C2×C4—1582no—
305C2×C4#1457907511yescharacteristic
306C2×C4#1457907511yescharacteristic
307C2×C4#1457908412yes—
308C2×C4#1457908912yes—
309C2×C4C2×#1192961211yescharacteristic
310C2×C4C2×#1192961411yescharacteristic
311C8—44no—
312C8—6322no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1457907511yescharacteristic
321C22C2×#1457908411yescharacteristic
322C22C2×#1457908911yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—3162no—
330C4—3162no—
331C4—6321no—
332C4—6321no—
333C4#1909829912yes—
334C4#1909829912yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1909829911yescharacteristic
339C2#2376504211yescharacteristic
340C2#2376508311yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1742C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1742D79
hypocenterC79#11721
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761426C2semidirect
176. C79#11721semidirect

Maximal subgroups

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

subgroupindex
#237666682
#237617142
#237614292
#237614262
C79×#17422
#1172179