Finite Group Databasebeta

← #21425132

Subgroups — #21425132

Order 7552.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#14273758C211yesmaximal, characteristic
2#16837976C211yesmaximal, characteristic
3C2×#14273655C211yesmaximal, characteristic
4#16837411C211yesmaximal, characteristic, complemented
5C2×#14273646C211yesmaximal, characteristic, complemented
6#16837162C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C59C211yesmaximal, characteristic, complemented
8C2×#11745685C2211yescharacteristic
9#14273758—22no—
10#14273758—22no—
11C2×#11745672—21no—
12C2×#11745671—21no—
13C2×#11745664C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15#14273725—21no—
16#14273716—21no—
17#14273710—22no—
18#14273703—22no—
19#14273686C2211yescharacteristic
20#14273655—22no—
21#14273651—22no—
22#14273646—22no—
23C22×Q8×C59C2211yescharacteristic
24C22×Q8×C59C2211yescharacteristic
25C4⋊Q8×C59—22no—
26C4⋊Q8×C59—22no—
27C2×C4⋊C4×C59—22no—
28C2×C42×C59C2211yescharacteristic
29C2×Q8×C59—21no—
30C2×Q8×C59—21no—
31C2×Q8×C59—21no—
32C2×Q8×C59—21no—
33C2×Q8×C59—21no—
34C2×Q8×C59—21no—
35C2×Q8×C59—21no—
36C2×Q8×C59—21no—
37C2×Q8×C59—21no—
38C2×Q8×C59—21no—
39C2×Q8×C59—21no—
40C2×Q8×C59D412yes—
41C22×C4×C59—21no—
42C22×C4×C59—21no—
43C22×C4×C59D412yes—
44C22×C4×C59C2311yescharacteristic
45C2×C22⋊D59—21no—
46#11745685—21no—
47#11745685—21no—
48#11745685—21no—
49#11745685—21no—
50C2×D236D411yescharacteristic
51C2×C4×D59—21no—
52C2×Dic118D411yescharacteristic
53C4⋊C4×C59—28no—
54C42×C59—21no—
55C42×C59D412yes—
56C22⋊Dic59—21no—
57#11745672—24no—
58#11745671—24no—
59(C2×D59)⋊C4—24no—
60C4⋊Dic59D412yes—
61Dic59⋊C4—24no—
62C4×Dic59—21no—
63#11745664—21no—
64#11745664—21no—
65#11745664—21no—
66#11745664D412yes—
67C2×C59⋊C8—21no—
68C2×C59⋊C8—21no—
69C23×C59C2×D411yescharacteristic
70C22×D59—21no—
71Q8×C59—22no—
72Q8×C59—22no—
73Q8×C59—41no—
74Q8×C59—41no—
75Q8×C59—41no—
76Q8×C59—41no—
77Q8×C59—41no—
78Q8×C59—41no—
79C2×C4×C59—24no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59C2×D411yescharacteristic
88C2×C4×C59C2×D411yescharacteristic
89C2×C4×C59C2×D411yescharacteristic
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2×D411yescharacteristic
92C2×C4×C59#4111yescharacteristic
93C22⋊D59—41no—
94C22⋊D59—41no—
95C2×Dic59—22no—
96C2×Dic59—21no—
97D236—21no—
98D236—21no—
99C4×D59—41no—
100C4×D59—41no—
101Dic118—21no—
102Dic118—21no—
103C59⋊C8—22no—
104C59⋊C8—41no—
105C22×C59—22no—
106C22×C59#12011yescharacteristic
107C22×C59#12011yescharacteristic
108C22×C59#12111yescharacteristic
109C2×D59—41no—
110C2×D59—42no—
111C4×C59—24no—
112C4×C59—42no—
113C4×C59—42no—
114C4×C59#12011yescharacteristic
115C4×C59#12111yescharacteristic
116C4×C59#12111yescharacteristic
117C4×C59#12711yescharacteristic
118Dic59—41no—
119Dic59—42no—
120#2005—591nomaximal, Sylow
121C2×C59—21no—
122C2×C59—21no—
123C2×C59#39211yescharacteristic
124C2×C59#45612yes—
125D59—81no—
126C2×#137—591no—
127#536—591no—
128C2×C4⋊Q8D5911yescharacteristic, complemented
129C2×#104—591no—
130#419—591no—
131C2×#101—591no—
132#339—591no—
133C59#200511yescharacteristic, complemented, Sylow
134C2×#41—591no—
135C22×Q8C2×D5911yescharacteristic
136C22×Q8C2×D5911yescharacteristic
137#137—1182no—
138#137—1182no—
139C2×Q16—1181no—
140C2×SD16—1181no—
141C2×M16—591no—
142C2×M16—591no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—1181no—
146#124—1181no—
147#123—1182no—
148C22⋊Q8—1182no—
149#117—591no—
150C2×C4⋊C4—22no—
151C2×C42C2×D5911yescharacteristic
152#104—1182no—
153#103—1182no—
154#101—1182no—
155#41—1181no—
156#41—1181no—
157#41—1181no—
158#41—1181no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—591no—
171C2×Q8C22⋊D5912yes—
172C2×D4—591no—
173C2×D4—1181no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1181no—
177C22×C4C22⋊D5912yes—
178C22×C4C22×D5911yescharacteristic
179Q16—1184no—
180SD16—1184no—
181M16—592no—
182M16—1181no—
183M16—1181no—
184M16—1181no—
185C2×C8—1181no—
186C2×C8—1181no—
187C4⋊C4—28no—
188C4⋊C4—592no—
189C4⋊C4—1184no—
190C22⋊C4—1184no—
191C22⋊C4—1181no—
192C42—21no—
193C42—1181no—
194C42C22⋊D5912yes—
195C23—1181no—
196C23C2×C22⋊D5911yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—1181no—
206Q8—1181no—
207D4—1181no—
208D4—1181no—
209D4—2361no—
210D4—2361no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—1182no—
220C2×C4—1181no—
221C2×C4—2361no—
222C2×C4—2361no—
223C2×C4D4×D5911yescharacteristic
224C2×C4D4×D5911yescharacteristic
225C2×C4D4×D5911yescharacteristic
226C2×C4#1174568711yescharacteristic
227C2×C4C2×C22⋊D5911yescharacteristic
228C2×C4C2×C22⋊D5911yescharacteristic
229C8—1182no—
230C8—2361no—
231C22—22no—
232C22—2361no—
233C22—2362no—
234C22#1427369511yescharacteristic
235C22#1427369611yescharacteristic
236C22#1427369911yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—2361no—
241C4—2362no—
242C4#1427369511yescharacteristic
243C4#1427369811yescharacteristic
244C4#1427369911yescharacteristic
245C4#1427372311yescharacteristic
246C2—21no—
247C2—21no—
248C2—4721no—
249C2#1683731011yescharacteristic
250C2#1683758412yes—
subgroupG/H
centerC22#14273695
commutatorC22×C4×C59C23
FrattiniC22×C4C22×D59
FittingC2×C4⋊Q8×C59C2
socleC22×C59#120
hypercenterC2×C4⋊Q8D59
hypocenterC59#2005
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #16837411C2semidirect
5. C2×#14273646C2semidirect
6. #16837162C2semidirect
7. C2×C4⋊Q8×C59C2semidirect
128. C2×C4⋊Q8D59semidirect
133. C59#2005semidirect