Finite Group Databasebeta

← #32997179

Subgroups — #32997179

Order 18688.

274 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 61 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#22620310C211yesmaximal, characteristic
2#22611741C212yesmaximal
3#22607966C211yesmaximal, characteristic, complemented
4#22607965C212yesmaximal, complemented
5C73×#1356C211yesmaximal, characteristic, complemented
6#17989143C411yescharacteristic
7C4⋊Q8×D73C411yescharacteristic
8#17988957C411yescharacteristic
9#17988956C2211yescharacteristic
10#17988955C411yescharacteristic
11#17988936—22no—
12#17988594—22no—
13#17988539C2211yescharacteristic
14#17988407—22no—
15#17988368—22no—
16#17988365C2212yes—
17#491×C73C2211yescharacteristic
18#347×C73C2212yes—
19C2×Q8×D73—21no—
20C2×#11924011D412yes—
21#14556591C2×C411yescharacteristic
22#14556590—21no—
23#14556590—41no—
24#14556589—21no—
25#14556583C2×C411yescharacteristic
26#14556569—41no—
27#14556556—41no—
28#14556549—21no—
29C4⋊C4×D73—41no—
30C42×D73C2×C411yescharacteristic
31#14556531—22no—
32#14556516—22no—
33#14556515—22no—
34#14556514—22no—
35#14556502C2×C411yescharacteristic
36C8×Dic73D412yes—
37C4⋊Q8×C73C2×C411yescharacteristic
38#125×C73C2×C411yescharacteristic
39M32×C73—22no—
40#103×C73—22no—
41#97×C73C2311yescharacteristic
42C2×Dic146—21no—
43C2×Dic146—41no—
44C2×Dic146—41no—
45Q8×D73—44no—
46C2×C4×D73—21no—
47C2×C4×D73—41no—
48C2×C4×D73C22⋊C411yescharacteristic
49#11924011—42no—
50C2×#10596831C22⋊C412yes—
51C4⋊Dic73—21no—
52C4⋊Dic73—41no—
53#11924007—42no—
54#11924007—41no—
55#11924007—41no—
56#11924005—41no—
57C4×Dic73—21no—
58C4×Dic73—41no—
59C4×Dic73C22⋊C411yescharacteristic
60#11924003—42no—
61C2×Q8×C73—21no—
62C2×C8×C73C2×D412yes—
63C4⋊C4×C73—21no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—41no—
66C42×C73C22×C411yescharacteristic
67C16×C73—42no—
68C22×D73#10011yescharacteristic
69Q8×C73—42no—
70C2×C4×C73—21no—
71C2×C4×C73—41no—
72C2×C4×C73—41no—
73C2×C4×C73C2×C22⋊C411yescharacteristic
74C2×Dic73—21no—
75C2×Dic73—41no—
76C2×Dic73—41no—
77C2×Dic73#10011yescharacteristic
78C4×D73—22no—
79C4×D73—42no—
80C4×D73—82no—
81Dic146—42no—
82Dic146—82no—
83Dic146—82no—
84C8×C73—22no—
85#10596831—22no—
86C22×C73C2×#10011yescharacteristic
87C2×D73—21no—
88C2×D73—41no—
89C4×C73—41no—
90C4×C73—81no—
91C4×C73—81no—
92C4×C73#42112yes—
93Dic73—22no—
94Dic73—21no—
95Dic73—81no—
96Dic73—81no—
97#11916—731nomaximal, Sylow
98C2×C73—21no—
99C2×C73#153311yescharacteristic
100D73—41no—
101#3139—731no—
102#2165—732no—
103#1356—731no—
104#1356—732no—
105#1356D7311yescharacteristic, complemented
106C73#1191611yescharacteristic, complemented, Sylow
107#533—731no—
108C2×C4⋊Q8—731no—
109#491—731no—
110#491—731no—
111#491—731no—
112#491C2×D7311yescharacteristic
113#487—1462no—
114#419—1462no—
115C4×M16—731no—
116#364—1462no—
117#349—1462no—
118#347—732no—
119#347C2×D7312yes—
120C22×Q8—1461no—
121C2×M16—732no—
122C4⋊Q8—2921no—
123C4⋊Q8—731no—
124C4⋊Q8—1461no—
125C4⋊Q8—1461no—
126C4⋊Q8C4×D7311yescharacteristic
127#125—731no—
128#125C4×D7311yescharacteristic
129C22⋊Q8—2921no—
130C4×Q8—2921no—
131#117—1461no—
132C2×C4⋊C4—2921no—
133C2×C42—731no—
134M32—22no—
135M32—1462no—
136#103—22no—
137#103—1462no—
138#103—1462no—
139#103—1462no—
140#97—731no—
141#97C22×D7311yescharacteristic
142C4×C8—732no—
143C2×Q8—21no—
144C2×Q8—2921no—
145C2×Q8—2921no—
146C2×Q8—2924no—
147C2×Q8—1461no—
148C22×C4—2921no—
149C22×C4—731no—
150C22×C4—1461no—
151M16—2922no—
152C2×C8—732no—
153C2×C8D4×D7312yes—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—2921no—
158C4⋊C4—2921no—
159C4⋊C4—2922no—
160C4⋊C4—2921no—
161C4⋊C4—1461no—
162C22⋊C4—2921no—
163C42—2921no—
164C42—731no—
165C42—1461no—
166C42C2×C4×D7311yescharacteristic
167C16—42no—
168C16—2922no—
169C23—731no—
170Q8—42no—
171Q8—2922no—
172Q8—5842no—
173Q8—5842no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2922no—
178C2×C4—2921no—
179C2×C4—2921no—
180C2×C4—5842no—
181C2×C4—731no—
182C2×C4—1461no—
183C2×C4—1462no—
184C2×C4C22⋊C4×D7311yescharacteristic
185C8—22no—
186C8—1462no—
187C22—2921no—
188C22—1461no—
189C22#100×D7311yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—5841no—
194C4—5841no—
195C4—1462no—
196C4—1461no—
197C4#1798860212yes—
198C2—21no—
199C2—2921no—
200C2#2260855311yescharacteristic
subgroupG/H
centerC2#22608553
commutatorC42×C73C22×C4
Frattini#97C22×D73
FittingC73×#1356C2
socleC2×C73#1533
prime radicalC22×D73#100
cosocle#97×C73C23
hypercenter#1356D73
hypocenterC73#11916
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
3. #22607966C2semidirect
4. #22607965C2semidirect
5. C73×#1356C2semidirect
105. #1356D73semidirect
106. C73#11916semidirect

Maximal subgroups

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

subgroupindex
#226203102
#226117412
#226079662
#226079652
C73×#13562
#1191673