Finite Group Databasebeta

← #27157821

Subgroups — #27157821

Order 12416.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261746C211yesmaximal, characteristic
2#20261728C211yesmaximal, characteristic
3C2×#15690873C211yesmaximal, characteristic
4#20261701C211yesmaximal, characteristic
5#20261614C211yesmaximal, characteristic
6#20261610C211yesmaximal, characteristic
7#101×D97C211yesmaximal, characteristic
8D97⋊M16C2211yescharacteristic
9C2×#13095519C2211yescharacteristic
10#15690889C2211yescharacteristic
11C2×C4⋊#87604C2211yescharacteristic
12#15690879—21no—
13#15690878—21no—
14#15690873—21no—
15#15690873—21no—
16#15690873—21no—
17#15690873—21no—
18#15690864—21no—
19C8×#87604—21no—
20C2×Q8×D97C2211yescharacteristic
21M16×D97C2211yescharacteristic
22M16×D97C2211yescharacteristic
23#15690729C411yescharacteristic, complemented
24#15690728C411yescharacteristic, complemented
25#15690728C411yescharacteristic, complemented
26#101×C97C411yescharacteristic, complemented
27C22×#87604—21no—
28#13095521—21no—
29#13095520—21no—
30#13095519—21no—
31#13095519—21no—
32#13095519D411yescharacteristic
33#13095519D411yescharacteristic
34C2×#11715897—21no—
35C4⋊#87604—21no—
36C4⋊#87604—21no—
37C4⋊#87604D411yescharacteristic
38C4⋊#87604D411yescharacteristic
39C22⋊#87604—21no—
40C4×#87604—21no—
41C2×Dic194—21no—
42C2×Dic194C2×C411yescharacteristic
43Q8×D97—21no—
44Q8×D97—21no—
45Q8×D97—21no—
46Q8×D97—21no—
47C2×C4×D97—21no—
48C2×C4×D97C2311yescharacteristic
49#13095495C2×C411yescharacteristic
50#13095495C2×C411yescharacteristic
51#13095494—21no—
52#13095494—21no—
53C8×D97—21no—
54C8×D97—21no—
55C2×Q8×C97C2×C411yescharacteristic
56M16×C97C2×C411yescharacteristic
57M16×C97C2×C411yescharacteristic
58C22×D97#4111yescharacteristic
59C2×#87604—21no—
60C2×#87604—21no—
61C2×#87604—41no—
62C2×#87604—81no—
63Q8×C97—21no—
64Q8×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—21no—
68C2×Dic97#4111yescharacteristic
69C4×D97—41no—
70C4×D97—41no—
71C4×D97C2×D411yescharacteristic
72C4×D97C2×D411yescharacteristic
73C4×D97#4111yescharacteristic
74C4×D97#4111yescharacteristic
75Dic194—21no—
76Dic194—21no—
77Dic194—41no—
78Dic194—41no—
79#11715897—21no—
80#11715897—21no—
81#11715897—41no—
82C8×C97—21no—
83C8×C97—21no—
84#11715895—21no—
85#11715895—21no—
86C22×C97#11711yescharacteristic
87C2×D97—21no—
88C2×D97—21no—
89C2×D97#12111yescharacteristic
90C2×D97#12611yescharacteristic
91#87604—41no—
92#87604—41no—
93#87604—81no—
94C4×C97—41no—
95C4×C97C4×D411yescharacteristic
96C4×C97C4×D411yescharacteristic
97Dic97—41no—
98Dic97#12311yescharacteristic
99Dic97#12411yescharacteristic
100C2×C97—21no—
101C2×C97#38811yescharacteristic
102D97—21no—
103D97#47111yescharacteristic
104D97#47311yescharacteristic
105#1930—971nomaximal, Sylow
106C97#193011yescharacteristic, complemented, Sylow
107#428—971no—
108#419—971no—
109C2×#103—971no—
110C2×#101—971no—
111#405—971no—
112#343—971no—
113#341—971no—
114C22×Q8—971no—
115C2×M16—971no—
116C2×M16—971no—
117C2×M16—971no—
118C22×C8—971no—
119#117—971no—
120C2×C4⋊C4—971no—
121#107—1941no—
122#106—1941no—
123#103—1941no—
124#103—1941no—
125#103—1941no—
126#103—1941no—
127#101—971no—
128#101—971no—
129#101—971no—
130#101#8760411yescharacteristic, complemented
131#97—1941no—
132C4×C8—1941no—
133C2×Q8—971no—
134C2×Q8—1941no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×Q8—1941no—
138C2×Q8—1941no—
139C2×Q8C2×#8760411yescharacteristic
140C22×C4—971no—
141C22×C4—1941no—
142C22×C4—1941no—
143M16—971no—
144M16—971no—
145M16—1941no—
146M16—1941no—
147M16—1941no—
148M16—1941no—
149M16C2×#8760411yescharacteristic
150M16C2×#8760411yescharacteristic
151C2×C8—971no—
152C2×C8—971no—
153C2×C8—1941no—
154C2×C8—1941no—
155C2×C8—1941no—
156C2×C8—1941no—
157C2×C8—1941no—
158C4⋊C4—971no—
159C4⋊C4—971no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C22⋊C4—1941no—
163C42—1941no—
164C23—971no—
165Q8—21no—
166Q8—21no—
167Q8—3881no—
168Q8—3881no—
169Q8—1941no—
170Q8—1941no—
171C2×C4—21no—
172C2×C4—7761no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—971no—
177C2×C4—971no—
178C2×C4—3881no—
179C2×C4—3881no—
180C2×C4—3881no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4—1941no—
184C2×C4C22×#8760411yescharacteristic
185C8—21no—
186C8—21no—
187C8—3881no—
188C8—1941no—
189C8—1941no—
190C8—1941no—
191C8—1941no—
192C22—971no—
193C22—971no—
194C22—1941no—
195C22—1941no—
196C22#1569088911yescharacteristic
197C4—41no—
198C4—7761no—
199C4—971no—
200C4—971no—
201C4—3881no—
202C4—3881no—
203C4—3881no—
204C4D4×#8760411yescharacteristic
205C4D4×#8760411yescharacteristic
206C2—21no—
207C2—971no—
208C2—971no—
209C2—1941no—
210C2#2026168511yescharacteristic
subgroupG/H
centerC2#20261685
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#101×C97C4
socleC2×C97#388
prime radicalC22×D97#41
cosocleC2×C4×D97C23
hypercenter#101#87604
hypocenterC97#1930
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
23. #15690729C4semidirect
24. #15690728C4semidirect
25. #15690728C4semidirect
26. #101×C97C4semidirect
106. C97#1930semidirect
130. #101#87604semidirect

Maximal subgroups

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

subgroupindex
#202617462
#202617282
C2×#156908732
#202617012
#202616142
#202616102
#101×D972
#193097