Finite Group Databasebeta

← #34014121

Subgroups — #34014121

Order 18688.

316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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#22626272C211yesmaximal, characteristic
2#22626154C211yesmaximal, characteristic
3#22624981C212yesmaximal
4#22624976C212yesmaximal
5#22618864C211yesmaximal, characteristic
6C2×#14556665C2212yes—
7C2×#14556665C2212yes—
8#17989509—22no—
9#17989480—22no—
10#17989140C411yescharacteristic
11#17988716—22no—
12#17988711—22no—
13#17988613C411yescharacteristic
14#17988610C411yescharacteristic
15#17988545C2211yescharacteristic
16#17988543C2211yescharacteristic
17C2×C8×Dic73C2211yescharacteristic
18#424×C73C411yescharacteristic
19#14556665—22no—
20#14556665—22no—
21#14556665—22no—
22#14556665Q814yescomplemented
23C2×#11924031—22no—
24C2×#11924031—22no—
25C2×#11924012C2×C411yescharacteristic
26C22×#10596831C2311yescharacteristic
27#14556590—21no—
28#14556582—21no—
29#14556568—22no—
30C4×Dic146—22no—
31#14556550C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33#14556520—22no—
34#14556520—22no—
35#14556502—21no—
36#14556501D412yes—
37C4×#10596831—21no—
38C8×Dic73—21no—
39C8×Dic73—21no—
40C8×Dic73D412yes—
41C2×M16×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43#117×C73C2×C411yescharacteristic
44C4⋊C8×C73—22no—
45#11924031—24no—
46#11924031—42no—
47C2×Dic146—22no—
48C22×Dic73—21no—
49#11924012—22no—
50C2×#10596831—21no—
51C2×#10596831C2×D411yescharacteristic
52C2×#10596831C2×D411yescharacteristic
53C2×#10596831C2×Q812yes—
54C2×#10596831#4112yes—
55C4⋊Dic73—22no—
56C4⋊Dic73—21no—
57#11924007—22no—
58#11924007—22no—
59C22⋊Dic73—21no—
60C4×Dic73—21no—
61C4×Dic73C22⋊C412yes—
62C4×Dic73M1611yescharacteristic
63C4×Dic73M1611yescharacteristic
64C22×C4×C73C22×C411yescharacteristic
65M16×C73—22no—
66C2×C8×C73—21no—
67C2×C8×C73—22no—
68C2×C8×C73C22⋊C412yes—
69C4⋊C4×C73—21no—
70C22⋊C4×C73M1612yes—
71C42×C73—21no—
72C23×C73C2×M1611yescharacteristic
73C2×C4×C73—22no—
74C2×C4×C73C2×C22⋊C411yescharacteristic
75C2×C4×C73C4×Q812yes—
76C2×C4×C73C2×M1611yescharacteristic
77C2×C4×C73#13112yes—
78C2×Dic73—21no—
79C2×Dic73—21no—
80C2×Dic73—22no—
81C2×Dic73—42no—
82Dic146—44no—
83C8×C73—22no—
84C8×C73—41no—
85#10596831—41no—
86#10596831C22⋊Q812yes—
87#10596831#12411yescharacteristic
88#10596831C4⋊Q811yescharacteristic
89C22×C73—41no—
90C22×C73#40711yescharacteristic
91C22×C73#44612yes—
92C4×C73—42no—
93C4×C73#39111yescharacteristic
94C4×C73#40811yescharacteristic
95C4×C73#44612yes—
96Dic73—22no—
97Dic73—42no—
98Dic73—41no—
99#13270—731nomaximal, Sylow
100C2×C73—21no—
101C2×C73—22no—
102C2×C73#198511yescharacteristic
103#2960—731no—
104#2153—731no—
105#2110—731no—
106#1378—732no—
107#1376—732no—
108C73#1327011yescharacteristic, complemented, Sylow
109#533—731no—
110C2×M32—732no—
111C2×M32—732no—
112#446—1462no—
113C8×Q8—1462no—
114#424—731no—
115#424#6440811yescharacteristic
116#423—731no—
117#405—731no—
118#405—731no—
119C2×C4×C8—731no—
120#364—1462no—
121#347—1462no—
122C2×M16—731no—
123C2×M16C2×#6440811yescharacteristic
124C22×C8—731no—
125C22×C8C2×#6440811yescharacteristic
126C4⋊Q8—1461no—
127#125—1461no—
128C22⋊Q8—1462no—
129C4×Q8—1462no—
130#117—731no—
131#117C2×#6440811yescharacteristic
132C2×C42—731no—
133M32—734no—
134M32—1462no—
135M32—1462no—
136M32—1462no—
137C2×C16—1462no—
138C2×C16—1462no—
139C4⋊C8—22no—
140C4⋊C8—1462no—
141C4⋊C8—1462no—
142#97—732no—
143#97—1461no—
144C4×C8—732no—
145C4×C8—1461no—
146C4×C8—1461no—
147C4×C8—1461no—
148C2×Q8—1462no—
149C22×C4—1461no—
150C22×C4C22×#6440811yescharacteristic
151M16—22no—
152M16—1462no—
153C2×C8—22no—
154C2×C8—21no—
155C2×C8—732no—
156C2×C8—732no—
157C2×C8—731no—
158C2×C8—731no—
159C2×C8—1461no—
160C2×C8C22⋊#6440812yes—
161C4⋊C4—21no—
162C4⋊C4—1462no—
163C4⋊C4—1462no—
164C4⋊C4—1462no—
165C4⋊C4—1461no—
166C22⋊C4—1461no—
167C22⋊C4#1192403812yes—
168C42—21no—
169C42—731no—
170C42—731no—
171C42—732no—
172C42—1461no—
173C16—2922no—
174C16—1464no—
175C23C2×#1192403811yescharacteristic
176Q8—2924no—
177C2×C4—22no—
178C2×C4—2922no—
179C2×C4—1461no—
180C2×C4—1462no—
181C2×C4—1461no—
182C2×C4C2×C22⋊#6440811yescharacteristic
183C2×C4Q8×#6440812yes—
184C2×C4D73⋊M1611yescharacteristic
185C2×C4#1455667812yes—
186C8—22no—
187C8—41no—
188C8—2921no—
189C8—731no—
190C8—732no—
191C8—731no—
192C22—41no—
193C22#1798955811yescharacteristic
194C22#1798963712yes—
195C4—42no—
196C4—2921no—
197C4—2922no—
198C4—1462no—
199C4(C2×#64408)⋊Q811yescharacteristic
200C4#1798956011yescharacteristic
201C4#1798963912yes—
202C2—21no—
203C2—22no—
204C2#2262610411yescharacteristic
subgroupG/H
centerC4(C2×#64408)⋊Q8
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#424×C73C4
socleC2×C73#1985
prime radicalC23×C73C2×M16
cosocleC22×#10596831C23
hypercenter#424#64408
hypocenterC73#13270
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
22. #14556665Q8semidirect
108. C73#13270semidirect

Maximal subgroups

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

subgroupindex
#226262722
#226261542
#226249812
#226249762
#226188642
#1327073