Finite Group Databasebeta

← #28374943

Subgroups — #28374943

Order 13952.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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#20295681C211yesmaximal, characteristic
2#20295378C211yesmaximal, characteristic, complemented
3#20295367C212yesmaximal, complemented
4#20295303C211yesmaximal, characteristic, complemented
5#20295265C212yesmaximal, complemented
6#20295236C211yesmaximal, characteristic, complemented
7#20295186C212yesmaximal, complemented
8C4×#13182016C211yesmaximal, characteristic, complemented
9D4×#11724784C211yesmaximal, characteristic, complemented
10#20295020C212yesmaximal, complemented
11#556×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2212yes—
13C2×#13182019C2212yes—
14C2×#13182016C2211yescharacteristic
15C22×#11724784C2212yes—
16#16812699C2211yescharacteristic
17#16812681—22no—
18#16812678C2212yes—
19#16812670—21no—
20C4×D436C2211yescharacteristic
21C4⋊C4×D109—21no—
22#16812632C2212yes—
23#16812630C2211yescharacteristic
24C4⋊#11724784C2211yescharacteristic
25#16812623C2211yescharacteristic
26#16812622C2212yes—
27#16812622C2212yes—
28#16812622C2212yes—
29#16812619C2212yes—
30#16812619C2211yescharacteristic
31C22⋊#11724784C2212yes—
32C4×#11724784C2211yescharacteristic
33C4⋊Q8×C109C2211yescharacteristic
34#126×C109—22no—
35#124×C109—21no—
36C22⋊Q8×C109C2212yes—
37C4×Q8×C109—21no—
38C4×Q8×C109C2211yescharacteristic
39C4×D4×C109C2211yescharacteristic
40#117×C109C2212yes—
41#13182029—24no—
42#13182029—42no—
43C2×Dic218C2312yes—
44C2×D436C2311yescharacteristic
45C2×C22⋊D109—22no—
46C2×C4×D109—21no—
47C2×C4×D109—22no—
48#13182019—42no—
49#13182016—22no—
50C2×#11724784—42no—
51C2×#11724784C2311yescharacteristic
52C2×#11724784C2311yescharacteristic
53C2×#11724784C2312yes—
54C4⋊Dic109—21no—
55C4⋊Dic109—21no—
56C4⋊Dic109C2311yescharacteristic
57#13182008—22no—
58C22⋊Dic109—22no—
59#13182006—21no—
60C2×Q8×C109—21no—
61C2×Q8×C109C2311yescharacteristic
62C2×D4×C109D411yescharacteristic
63C22×C4×C109C2312yes—
64C4⋊C4×C109—22no—
65C4⋊C4×C109—21no—
66C4⋊C4×C109—21no—
67C4⋊C4×C109D411yescharacteristic
68C4⋊C4×C109C2311yescharacteristic
69C4⋊C4×C109C2312yes—
70C22⋊C4×C109—22no—
71C22⋊C4×C109—22no—
72C22⋊C4×C109D412yes—
73C42×C109—22no—
74C42×C109—21no—
75C42×C109C2311yescharacteristic
76C23×C109C2×D412yes—
77C22×D109—21no—
78Q8×C109—21no—
79Q8×C109—21no—
80Q8×C109—42no—
81D4×C109—22no—
82C2×C4×C109—22no—
83C2×C4×C109—22no—
84C2×C4×C109—21no—
85C2×C4×C109—21no—
86C2×C4×C109—22no—
87C2×C4×C109C2×D412yes—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2×D411yescharacteristic
90C2×C4×C109C2411yescharacteristic
91C22⋊D109—44no—
92C2×Dic109—21no—
93C2×Dic109—22no—
94D436—21no—
95D436#4112yes—
96C4×D109—44no—
97C4×D109—42no—
98Dic218—22no—
99Dic218—22no—
100#11724784—21no—
101#11724784—21no—
102#11724784—22no—
103C22×C109—22no—
104C22×C109—22no—
105C22×C109—22no—
106C22×C109C22×D411yescharacteristic
107C2×D109—22no—
108C2×D109—41no—
109C2×D109—41no—
110C4×C109—22no—
111C4×C109—21no—
112C4×C109—41no—
113C4×C109—42no—
114C4×C109—41no—
115C4×C109#13512yes—
116C4×C109C2×#4111yescharacteristic
117C4×C109H(5,𝔽2)11yescharacteristic
118Dic109—42no—
119Dic109—41no—
120C2×C109—42no—
121C2×C109#54611yescharacteristic
122C2×C109C2×#13511yescharacteristic
123C2×C109#57811yescharacteristic
124D109—42no—
125#3302—1091nomaximal, Sylow
126C109#330211yescharacteristic, complemented, Sylow
127#556D10911yescharacteristic, complemented
128#547—1091no—
129#487—1091no—
130#485—1092no—
131#474—1091no—
132#467—1092no—
133#462—1091no—
134#452—1092no—
135C4×SD16—1091no—
136C8×D4—1091no—
137#416—1092no—
138C2×#41—1092no—
139C2×Q16—1092no—
140C2×SD16—1091no—
141C22×C8—1092no—
142C4⋊Q8—1091no—
143C4⋊Q8C2×D10911yescharacteristic
144#126—22no—
145#124—21no—
146#123—2182no—
147C22⋊Q8—1092no—
148C22⋊Q8C2×D10912yes—
149#121—2181no—
150C4×Q8—21no—
151C4×Q8C2×D10911yescharacteristic
152C4×D4—1091no—
153C4×D4C2×D10911yescharacteristic
154#117C2×D10912yes—
155C2×C4⋊C4—2181no—
156#107—1092no—
157#106—1091no—
158C4⋊C8—1091no—
159#103—1091no—
160#103—1092no—
161#103—1092no—
162#103—1092no—
163#102—1091no—
164#102—1092no—
165C22⋊C8—1092no—
166C4×C8—1091no—
167#41—4362no—
168#41—2184no—
169C2×Q8—21no—
170C2×Q8—1092no—
171C2×Q8C22×D10911yescharacteristic
172C2×D4—1091no—
173C2×D4—2182no—
174C2×D4C22⋊D10911yescharacteristic
175C22×C4—2181no—
176C22×C4—2182no—
177C22×C4C22×D10912yes—
178Q16—4362no—
179SD16—2182no—
180C2×C8—1091no—
181C2×C8—1091no—
182C2×C8—1092no—
183C2×C8—4362no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—1091no—
188C4⋊C4—2182no—
189C4⋊C4—2181no—
190C4⋊C4—2181no—
191C4⋊C4C22⋊D10911yescharacteristic
192C4⋊C4C22×D10911yescharacteristic
193C4⋊C4C22×D10912yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—2182no—
197C22⋊C4—2181no—
198C22⋊C4C22⋊D10912yes—
199C42—22no—
200C42—21no—
201C42C22×D10911yescharacteristic
202C23—2181no—
203C23C2×C22⋊D10912yes—
204Q8—21no—
205Q8—21no—
206Q8—42no—
207Q8—2182no—
208Q8—2182no—
209D4—22no—
210D4—1092no—
211D4—4364no—
212D4—2181no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—4364no—
219C2×C4—4362no—
220C2×C4—2181no—
221C2×C4—2182no—
222C2×C4C2×C22⋊D10912yes—
223C2×C4C2×C22⋊D10911yescharacteristic
224C2×C4C2×C22⋊D10911yescharacteristic
225C2×C4C23×D10911yescharacteristic
226C8—2181no—
227C8—2181no—
228C8—2182no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—4361no—
233C22—4361no—
234C22—2182no—
235C22C22×C22⋊D10911yescharacteristic
236C4—21no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—4361no—
242C4—4362no—
243C4#1681271912yes—
244C4#41×D10911yescharacteristic
245C4#1681274511yescharacteristic
246C2—42no—
247C2—4362no—
248C2#2029566411yescharacteristic
249C2C2×#1681271911yescharacteristic
250C2#2029584111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#556×C109C2
socleC22×C109C22×D4
prime radical#20295681C2
hypercenter#556D109
hypocenterC109#3302
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #20295378C2semidirect
3. #20295367C2semidirect
4. #20295303C2semidirect
5. #20295265C2semidirect
6. #20295236C2semidirect
7. #20295186C2semidirect
8. C4×#13182016C2semidirect
9. D4×#11724784C2semidirect
10. #20295020C2semidirect
11. #556×C109C2semidirect
126. C109#3302semidirect
127. #556D109semidirect

Maximal subgroups

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

subgroupindex
#202956812
#202953782
#202953672
#202953032
#202952652
#202952362
#202951862
C4×#131820162
D4×#117247842
#202950202
#556×C1092
#3302109