Finite Group Databasebeta

← #34014352

Subgroups — #34014352

Order 18688.

398 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 60 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#22626224C212yesmaximal
2#22625068C214yesmaximal, complemented
3#22620388C211yesmaximal, characteristic
4#17989573—22no—
5#17989498C2214yes—
6#17989496—22no—
7C2×#14556622—22no—
8#17989137C2211yescharacteristic
9#17988979—22no—
10#17988966C2212yes—
11#17988961C412yes—
12C4×#11924012C411yescharacteristic
13C2×C4⋊Q8×C73C411yescharacteristic
14D73⋊M16—42no—
15#14556652—24no—
16#14556651—42no—
17C2×#11924028D412yes—
18#14556622—44no—
19#14556622—44no—
20C2×#11924018—24no—
21C2×#11924015—24no—
22C2×#11924012—21no—
23#14556591C2311yescharacteristic
24#14556588C2×C411yescharacteristic
25#14556579D412yes—
26#14556565—41no—
27C4×D292—41no—
28#14556502—21no—
29C4×#10596831C2×C412yes—
30C22×Q8×C73—21no—
31C4⋊Q8×C73—21no—
32C4⋊Q8×C73—41no—
33C4⋊Q8×C73C2×C412yes—
34C2×C4⋊C4×C73—41no—
35C2×C42×C73C2×C411yescharacteristic
36#11924038—82no—
37#11924037—44no—
38#11924036—82no—
39#11924028—42no—
40#11924028—42no—
41C2×Dic146C2×D412yes—
42C2×D292—41no—
43C2×D292C22⋊C412yes—
44C2×C22⋊D73—22no—
45C2×C4×D73—22no—
46#11924018—48no—
47#11924015—48no—
48#11924012—24no—
49C2×#10596831—22no—
50C4⋊Dic73—41no—
51#11924005—42no—
52C2×Q8×C73—22no—
53C2×Q8×C73—42no—
54C2×Q8×C73—42no—
55C2×Q8×C73—41no—
56C22×C4×C73—21no—
57C22×C4×C73—41no—
58C22×C4×C73C22⋊C411yescharacteristic
59C4⋊C4×C73—42no—
60C4⋊C4×C73—42no—
61C42×C73—21no—
62C42×C73C22⋊C411yescharacteristic
63C42×C73C22×C411yescharacteristic
64C23×C73#9911yescharacteristic
65C22×D73—22no—
66Q8×C73—44no—
67Q8×C73—82no—
68C2×C4×C73—21no—
69C2×C4×C73—22no—
70C2×C4×C73—21no—
71C2×C4×C73—42no—
72C2×C4×C73—42no—
73C2×C4×C73—81no—
74C2×C4×C73—81no—
75C2×C4×C73#9911yescharacteristic
76C2×C4×C73C2×C22⋊C411yescharacteristic
77C22⋊D73—82no—
78C2×Dic73—22no—
79D292—42no—
80D292—82no—
81C4×D73—42no—
82C4×D73—42no—
83Dic146—42no—
84#10596833—84no—
85#10596831—24no—
86C22×C73—21no—
87C22×C73—41no—
88C22×C73C2×#9911yescharacteristic
89C2×D73—42no—
90C2×D73—82no—
91C4×C73—22no—
92C4×C73—21no—
93C4×C73—41no—
94C4×C73—82no—
95C4×C73#35312yes—
96Dic73—42no—
97#13592—731nomaximal, Sylow
98C2×C73—21no—
99C2×C73—41no—
100C2×C73C2×#35311yescharacteristic
101D73—82no—
102#3152—731no—
103#2132—732no—
104#1411—734no—
105C73#1359211yescharacteristic, complemented, Sylow
106C2×#137—1462no—
107#532—731no—
108C2×C4⋊Q8#6440811yescharacteristic
109#497—1462no—
110#494—732no—
111#492—732no—
112#413—1462no—
113C4×M16—731no—
114#356—734no—
115#355—1462no—
116C2×#41—732no—
117C22×Q8—21no—
118#137—2924no—
119#137—2924no—
120C2×Q16—1464no—
121C2×SD16—1464no—
122C2×M16—2922no—
123C2×M16—1461no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8—731no—
127C4⋊Q8C2×#6440812yes—
128#127—731no—
129#124—732no—
130#121—2921no—
131C4×D4—2921no—
132C2×C4⋊C4—41no—
133C2×C42C2×#6440811yescharacteristic
134#101—1464no—
135#100—2922no—
136#97—1461no—
137C4×C8—732no—
138#41—2922no—
139#41—2922no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8—732no—
145C2×D4—2921no—
146C2×D4—732no—
147C2×D4—1462no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—1462no—
151C22×C4C22⋊#6440811yescharacteristic
152Q16—2928no—
153SD16—2928no—
154M16—2924no—
155M16—5842no—
156M16—1464no—
157C2×C8—5842no—
158C2×C8—1462no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—2921no—
162C22⋊C4—2922no—
163C42—21no—
164C42C22⋊#6440811yescharacteristic
165C42C22×#6440811yescharacteristic
166C23—1462no—
167C23#1455664911yescharacteristic
168Q8—44no—
169Q8—82no—
170Q8—2922no—
171D4—2922no—
172D4—5842no—
173D4—5842no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—2922no—
182C2×C4—2922no—
183C2×C4—1462no—
184C2×C4#1455664911yescharacteristic
185C2×C4C2×C22⋊#6440811yescharacteristic
186C8—5844no—
187C8—1464no—
188C22—21no—
189C22—41no—
190C22—2922no—
191C22—5842no—
192C22C2×#1455664911yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—82no—
197C4—2922no—
198C4#1798949112yes—
199C2—21no—
200C2—41no—
201C2—5842no—
202C2C2×#1798949111yescharacteristic
subgroupG/H
centerC2C2×#17989491
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC2×C73C2×#353
prime radical#17989137C22
cosocle#14556591C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#13592
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22625068C2semidirect
105. C73#13592semidirect

Maximal subgroups

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

subgroupindex
#226262242
#226250682
#226203882
#1359273