Finite Group Databasebeta

← #34010989

Subgroups — #34010989

Order 18688.

316 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 62 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#22626306C211yesmaximal, characteristic, complemented
2#22625652C211yesmaximal, characteristic, complemented
3#22624905C212yesmaximal, complemented
4#22624901C212yesmaximal, complemented
5#22620578C211yesmaximal, characteristic
6#17989550—21no—
7#17989509—21no—
8#17989484C2212yes—
9#17989479—21no—
10#17989456C2211yescharacteristic
11#17989455C2211yescharacteristic
12#17989454—21no—
13#17989191—21no—
14#17988989C411yescharacteristic, complemented
15#17988951—21no—
16#17988837C2212yes—
17#17988820—21no—
18#17988699C412yescomplemented
19D8×Dic73—21no—
20#17988541C2211yescharacteristic
21#493×C73C411yescharacteristic, complemented
22D73⋊M16—21no—
23#14556670—21no—
24#14556665—22no—
25#14556645—22no—
26C8×#64408—22no—
27C2×#11924011C2311yescharacteristic
28C2×C8×D73D411yescharacteristic
29#14556583—21no—
30#14556572—42no—
31#14556563—42no—
32#14556562—22no—
33D4×Dic73—22no—
34#14556548D411yescharacteristic
35#14556525—21no—
36#14556523C2×C412yes—
37#14556512—22no—
38#14556512—22no—
39#14556502D411yescharacteristic
40#14556501C2×C411yescharacteristic
41C8×Dic73D411yescharacteristic
42C2×D8×C73—21no—
43C2×D8×C73C2×C412yes—
44#127×C73—21no—
45C4×C8×C73C2×C411yescharacteristic
46#11924038—41no—
47#11924037—41no—
48#11924036—22no—
49C4⋊#64408—41no—
50C22⋊#64408—41no—
51C4×#64408—22no—
52#11924031—42no—
53C2×C22⋊D73—42no—
54C22×Dic73—42no—
55C2×C4×D73C2×D411yescharacteristic
56#11924011—41no—
57C2×#10596831C22⋊C411yescharacteristic
58C2×#10596831C2×D411yescharacteristic
59C8×D73—21no—
60C8×D73—21no—
61C4⋊Dic73—22no—
62#11924007—21no—
63#11924007—42no—
64C22⋊Dic73—42no—
65C22⋊Dic73—42no—
66#11924005—21no—
67C4×Dic73C22⋊C411yescharacteristic
68C2×D4×C73—22no—
69C2×D4×C73—42no—
70D8×C73—42no—
71D8×C73—41no—
72D8×C73—41no—
73C2×C8×C73C22⋊C411yescharacteristic
74C2×C8×C73C22×C411yescharacteristic
75C42×C73C22⋊C411yescharacteristic
76C23×C73—42no—
77C22×D73AGL(1,ℤ/8)11yescharacteristic
78C2×#64408—42no—
79D4×C73—42no—
80D4×C73—42no—
81D4×C73—162no—
82C2×C4×C73—21no—
83C2×C4×C73C2×C22⋊C411yescharacteristic
84C22⋊D73—162no—
85C2×Dic73—21no—
86C2×Dic73—42no—
87C2×Dic73—162no—
88C2×Dic73AGL(1,ℤ/8)11yescharacteristic
89C4×D73—21no—
90C4×D73—21no—
91#10596833—42no—
92C8×C73—21no—
93C8×C73#10411yescharacteristic
94C8×C73#10411yescharacteristic
95#10596831—21no—
96#10596831—21no—
97C22×C73—82no—
98C22×C73—82no—
99C22×C73—82no—
100C22×C73#41811yescharacteristic
101C2×D73—21no—
102C2×D73—41no—
103#64408—82no—
104C4×C73—41no—
105C4×C73#41011yescharacteristic
106C4×C73C2×#10411yescharacteristic
107Dic73—21no—
108Dic73—41no—
109Dic73—82no—
110#11770—731nomaximal, Sylow
111C2×C73—21no—
112C2×C73—162no—
113C2×C73#150511yescharacteristic
114D73—41no—
115#3176—731no—
116#2166—731no—
117#1747—731no—
118#1341—732no—
119#1339—732no—
120C73#1177011yescharacteristic, complemented, Sylow
121#540—1461no—
122#499—731no—
123#493#6440811yescharacteristic, complemented
124#490—1461no—
125#469—732no—
126#466—1461no—
127#442—732no—
128C4×D8—1461no—
129#405—731no—
130#405—1461no—
131#364—1461no—
132#349—732no—
133#347—1461no—
134#338—731no—
135H(3,ℤ/4)—731no—
136H(3,ℤ/4)—1461no—
137C2×D8—21no—
138C2×D8C2×#6440812yes—
139C2×M16—731no—
140C2×M16—1461no—
141C22×C8—731no—
142#127—21no—
143#125—1461no—
144#123—2922no—
145#121—2922no—
146#121—1462no—
147C4×D4—1462no—
148#117—731no—
149#117—1461no—
150M32—1462no—
151#107—1461no—
152#106—732no—
153#102—1462no—
154#102—1462no—
155#97—731no—
156#97—731no—
157#97—1462no—
158C4×C8—731no—
159C4×C8—1462no—
160C4×C8C2×#6440811yescharacteristic
161C2×D4—22no—
162C2×D4—42no—
163C2×D4—2922no—
164C22×C4—2922no—
165C22×C4—731no—
166D8—42no—
167D8—41no—
168D8—41no—
169M16—2921no—
170M16—2921no—
171M16—2921no—
172C2×C8—731no—
173C2×C8—731no—
174C2×C8—1462no—
175C2×C8—1461no—
176C2×C8—1461no—
177C2×C8C22⋊#6440811yescharacteristic
178C2×C8C22×#6440811yescharacteristic
179C4⋊C4—2922no—
180C4⋊C4—2921no—
181C4⋊C4—1461no—
182C4⋊C4—1462no—
183C22⋊C4—2921no—
184C22⋊C4—2922no—
185C22⋊C4—2922no—
186C22⋊C4—1461no—
187C42—731no—
188C42—1462no—
189C42C22⋊#6440811yescharacteristic
190C16—2922no—
191C23—42no—
192C23—731no—
193D4—42no—
194D4—42no—
195D4—162no—
196D4—11682no—
197C2×C4—21no—
198C2×C4—2922no—
199C2×C4—2922no—
200C2×C4—731no—
201C2×C4—11682no—
202C2×C4—1461no—
203C2×C4—1461no—
204C2×C4—1461no—
205C2×C4C2×C22⋊#6440811yescharacteristic
206C8—21no—
207C8—2922no—
208C8—1461no—
209C8—1461no—
210C8#1455665511yescharacteristic
211C8#1455665511yescharacteristic
212C22—82no—
213C22—82no—
214C22—82no—
215C22—2921no—
216C22—1461no—
217C22#1798957911yescharacteristic
218C4—41no—
219C4—2921no—
220C4—5842no—
221C4—5842no—
222C4—1461no—
223C4#1798956711yescharacteristic
224C4C2×#1455665511yescharacteristic
225C2—21no—
226C2—162no—
227C2—2921no—
228C2#2262532011yescharacteristic
subgroupG/H
centerC2#22625320
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#493×C73C4
socleC2×C73#1505
prime radical#22620578C2
cosocleC2×#11924011C23
hypercenter#493#64408
hypocenterC73#11770
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #22626306C2semidirect
2. #22625652C2semidirect
3. #22624905C2semidirect
4. #22624901C2semidirect
14. #17988989C4semidirect
18. #17988699C4semidirect
21. #493×C73C4semidirect
120. C73#11770semidirect
123. #493#64408semidirect

Maximal subgroups

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

subgroupindex
#226263062
#226256522
#226249052
#226249012
#226205782
#1177073