Finite Group Databasebeta

← #32995196

Subgroups — #32995196

Order 18688.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#22618571C211yesmaximal, characteristic
2#22611731C212yesmaximal, complemented
3#22607872C212yesmaximal, complemented
4#22607857C211yesmaximal, characteristic
5C73×#1315C211yesmaximal, characteristic
6#41×Dic73C411yescharacteristic
7#17988645—22no—
8#17988637—22no—
9#17988615C411yescharacteristic
10#17988613C2211yescharacteristic
11#17988554C411yescharacteristic
12M16×Dic73C411yescharacteristic
13C2×C8×Dic73C2211yescharacteristic
14#17988405—22no—
15C22⋊#11924003C2212yes—
16#17988362—22no—
17#408×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C2×#11924012C2×C411yescharacteristic
20C22×#10596831C2×C411yescharacteristic
21Q8×Dic73C812yes—
22D4×Dic73—22no—
23D4×Dic73C812yes—
24#14556550—21no—
25#14556550C2×C411yescharacteristic
26C2×C4×Dic73—21no—
27C2×C4×Dic73C2×C411yescharacteristic
28C2×#11924003—22no—
29#14556520—22no—
30#14556520—22no—
31C22⋊#10596831—22no—
32#14556501—21no—
33C8×Dic73—21no—
34C8×Dic73D412yes—
35C8×Dic73D412yes—
36C2×#41×C73C2×C411yescharacteristic
37C2×M16×C73C2×C411yescharacteristic
38C22×C8×C73C2311yescharacteristic
39C2×C16×C73—22no—
40C22⋊C8×C73—22no—
41C22×Dic73—22no—
42C22×Dic73M1612yes—
43#11924012—41no—
44C2×#10596831—21no—
45C2×#10596831—41no—
46C2×#10596831C22⋊C412yes—
47C4⋊Dic73—22no—
48C4⋊Dic73C2×C812yes—
49C22⋊Dic73—21no—
50C22⋊Dic73—22no—
51C4×Dic73—21no—
52C4×Dic73—22no—
53C4×Dic73C22⋊C411yescharacteristic
54C4×Dic73C22⋊C411yescharacteristic
55C4×Dic73C2×C812yes—
56#11924003—42no—
57#41×C73—41no—
58#41×C73—41no—
59C2×Q8×C73C2×C811yescharacteristic
60C2×D4×C73—21no—
61C2×D4×C73C2×C811yescharacteristic
62C22×C4×C73—21no—
63C22×C4×C73C22×C411yescharacteristic
64M16×C73—41no—
65C2×C8×C73—21no—
66C2×C8×C73—41no—
67C2×C8×C73C2×D412yes—
68C16×C73—42no—
69C23×C73—21no—
70C23×C73C2×M1611yescharacteristic
71Q8×C73—41no—
72D4×C73—41no—
73D4×C73—42no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—41no—
78C2×C4×C73—41no—
79C2×C4×C73C2×C22⋊C411yescharacteristic
80C2×C4×C73C22×C811yescharacteristic
81C2×Dic73—22no—
82C2×Dic73—42no—
83C2×Dic73—44no—
84C2×Dic73C22⋊C812yes—
85C2×Dic73C22⋊C812yes—
86C8×C73—22no—
87C8×C73—41no—
88#10596831—22no—
89#10596831—41no—
90C22×C73—22no—
91C22×C73—21no—
92C22×C73—42no—
93C22×C73—41no—
94C22×C73C2×C22⋊C811yescharacteristic
95C4×C73—21no—
96C4×C73—41no—
97C4×C73#41011yescharacteristic
98C4×C73#41311yescharacteristic
99Dic73—22no—
100Dic73—22no—
101Dic73—42no—
102#11463—731nomaximal, Sylow
103C2×C73—41no—
104C2×C73—81no—
105C2×C73#145811yescharacteristic
106C2×C73#211712yes—
107#2922—731no—
108#2163—732no—
109#1323—732no—
110#1315—731no—
111#1315D7311yescharacteristic
112C73#1146311yescharacteristic, complemented, Sylow
113C4×#41—731no—
114#433—1462no—
115#431—1462no—
116#424—731no—
117#424—731no—
118#408—731no—
119#408C2×D7311yescharacteristic
120C4×M16—731no—
121C2×C4×C8—731no—
122C4⋊C16—1462no—
123C22⋊C16—732no—
124C22⋊C16C2×D7312yes—
125#346—1462no—
126C2×#41C4×D7311yescharacteristic
127C2×M16—731no—
128C2×M16C4×D7311yescharacteristic
129C22×C8—731no—
130C22×C8C22×D7311yescharacteristic
131C4×Q8—732no—
132C4×D4—732no—
133C4×D4—1462no—
134#117—731no—
135#117—1461no—
136C2×C42—731no—
137C2×C42—1461no—
138C2×C16—22no—
139C2×C16—1462no—
140C4⋊C8—1462no—
141C4⋊C8—1462no—
142C22⋊C8—22no—
143C22⋊C8—1462no—
144#97—1461no—
145C4×C8—732no—
146C4×C8—732no—
147C4×C8—1461no—
148#41—41no—
149#41—41no—
150C2×Q8C8×D7311yescharacteristic
151C2×D4—21no—
152C2×D4C8×D7311yescharacteristic
153C22×C4—21no—
154C22×C4—732no—
155C22×C4—1462no—
156C22×C4C2×C4×D7311yescharacteristic
157M16—41no—
158M16—2921no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—2921no—
162C2×C8—732no—
163C2×C8—1461no—
164C2×C8D4×D7312yes—
165C4⋊C4—732no—
166C4⋊C4—1462no—
167C22⋊C4—1462no—
168C22⋊C4—1461no—
169C42—731no—
170C42—731no—
171C42—732no—
172C42—1462no—
173C42—1461no—
174C16—42no—
175C16—2922no—
176C23—21no—
177C23M16×D7311yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2924no—
187C2×C4—2922no—
188C2×C4—732no—
189C2×C4—732no—
190C2×C4—1462no—
191C2×C4C22⋊C4×D7311yescharacteristic
192C2×C4C2×C8×D7311yescharacteristic
193C8—22no—
194C8—41no—
195C8—2921no—
196C8—1462no—
197C22—22no—
198C22—21no—
199C22—42no—
200C22—41no—
201C22C22⋊C8×D7311yescharacteristic
202C4—21no—
203C4—41no—
204C4—2922no—
205C4—1462no—
206C4—1462no—
207C4#1798855911yescharacteristic
208C4#1798856811yescharacteristic
209C2—41no—
210C2—81no—
211C2#2260824211yescharacteristic
212C2#2261155712yes—
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1315C2
socleC22×C73C2×C22⋊C8
prime radicalC2×#41×C73C2×C4
cosocleC22×C8×C73C23
hypercenter#1315D73
hypocenterC73#11463
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #22611731C2semidirect
3. #22607872C2semidirect
112. C73#11463semidirect

Maximal subgroups

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

subgroupindex
#226185712
#226117312
#226078722
#226078572
C73×#13152
#1146373