Finite Group Databasebeta

← #34012045

Subgroups — #34012045

Order 18688.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#22625570C211yesmaximal, characteristic, complemented
2#22625563C211yesmaximal, characteristic
3#22624962C211yesmaximal, characteristic, complemented
4#22624888C211yesmaximal, characteristic, complemented
5#22624886C211yesmaximal, characteristic
6#22624842C211yesmaximal, characteristic, complemented
7#22611782C211yesmaximal, characteristic
8#17989637—21no—
9#17989621—21no—
10#17989473C2211yescharacteristic
11#17989450C2211yescharacteristic
12#17989450C2211yescharacteristic
13#17989437—21no—
14#17989435—21no—
15C8×#10596833C2211yescharacteristic
16C2×#14556536C411yescharacteristic
17#17988681C2211yescharacteristic
18#17988669C2211yescharacteristic
19#17988390C411yescharacteristic
20#17988387C411yescharacteristic
21C16×Dic73C2211yescharacteristic
22#367×C73C411yescharacteristic
23C2×#11924038—41no—
24C2×#11924031—21no—
25C4⋊#10596833—21no—
26C4⋊#10596833—21no—
27C4⋊#10596833—41no—
28#14556648—41no—
29#14556646—41no—
30C4×#10596833D411yescharacteristic
31C4×#10596833D411yescharacteristic
32C2×Dic292C2×C411yescharacteristic
33C2×D584C2×C411yescharacteristic
34#14556556—21no—
35#14556553—21no—
36#14556536—22no—
37C2×#11924003C2×C411yescharacteristic
38#14556514—21no—
39#14556511—21no—
40C8×Dic73C2311yescharacteristic
41C2×C16×C73C2×C411yescharacteristic
42#107×C73C2×C411yescharacteristic
43#107×C73C2×C411yescharacteristic
44#11924038—82no—
45C2×#10596833—21no—
46C2×#10596833—41no—
47C2×#10596833—41no—
48C2×#10596833D811yescharacteristic
49C2×#10596833D811yescharacteristic
50#11924031—22no—
51C2×Dic146—21no—
52C2×D292—21no—
53C2×C4×D73—41no—
54Dic292—21no—
55Dic292—21no—
56D584—21no—
57D584—21no—
58C2×#10596831#4111yescharacteristic
59#11924007—41no—
60#11924005—41no—
61C4×Dic73—41no—
62C4×Dic73C2×D411yescharacteristic
63#11924003—21no—
64C2×C8×C73C22×C411yescharacteristic
65C4⋊C4×C73—21no—
66C4⋊C4×C73—21no—
67C16×C73M1612yescomplemented
68C22×D73—41no—
69C2×C4×C73—41no—
70C2×C4×C73—41no—
71C2×C4×C73C4×D411yescharacteristic
72C2×Dic73—41no—
73C2×Dic73C2×D811yescharacteristic
74C2×Dic73#13511yescharacteristic
75D292—41no—
76D292—41no—
77C4×D73—81no—
78C4×D73—81no—
79Dic146—41no—
80Dic146—41no—
81#10596833—21no—
82#10596833—21no—
83#10596833—41no—
84#10596833—81no—
85#10596833—81no—
86C8×C73C2×M1611yescharacteristic
87C8×C73#13111yescharacteristic
88#10596831—21no—
89C22×C73C4×D811yescharacteristic
90C2×D73—81no—
91C2×D73—81no—
92C2×D73—81no—
93C4×C73—81no—
94C4×C73—81no—
95C4×C73#43611yescharacteristic
96C4×C73#44311yescharacteristic
97Dic73—21no—
98Dic73—81no—
99Dic73#50911yescharacteristic
100Dic73#51011yescharacteristic
101#12084—731nomaximal, Sylow
102C2×C73#158911yescharacteristic
103C2×C73#217511yescharacteristic
104C2×C73#217911yescharacteristic
105D73—161no—
106C4×SD32—731no—
107#1591—731no—
108#1589—731no—
109#1371—731no—
110#1332—731no—
111#1331—731no—
112#1311—731no—
113C73#1208411yescharacteristic, complemented, Sylow
114C2×SD32—731no—
115#446—1461no—
116C4×Q16—731no—
117C4×D8—731no—
118#436—1461no—
119#367#6440811yescharacteristic
120#358—731no—
121#357—731no—
122C4×C16—731no—
123C4×C16—731no—
124#335—731no—
125#335—731no—
126Q8⋊C8—1461no—
127D4⋊C8—1461no—
128C82—731no—
129C2×Q16—731no—
130C2×D8—731no—
131C2×M16—2921no—
132C4×Q8—1461no—
133C4×D4—1461no—
134SD32—1462no—
135C2×C16—731no—
136C2×C16—1461no—
137C2×C16C2×#6440811yescharacteristic
138#107C2×#6440811yescharacteristic
139#107C2×#6440811yescharacteristic
140C4⋊C8—2921no—
141C4⋊C8—1461no—
142C4⋊C8—1461no—
143#103—1461no—
144#102—1461no—
145C22⋊C8—2921no—
146#97—2921no—
147C4×C8—731no—
148C4×C8—731no—
149C4×C8—731no—
150C2×Q8—1461no—
151C2×D4—1461no—
152C22×C4—2921no—
153Q16—1461no—
154Q16—1461no—
155D8—1461no—
156D8—1461no—
157M16—5842no—
158C2×C8—2921no—
159C2×C8—2921no—
160C2×C8—731no—
161C2×C8—731no—
162C2×C8—731no—
163C2×C8—1461no—
164C2×C8C22×#6440811yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—2921no—
168C22⋊C4—2921no—
169C42—2921no—
170C42—731no—
171C16—1462no—
172C16—1461no—
173C16#1192403812yescomplemented
174C23—2921no—
175Q8—2921no—
176Q8—2921no—
177D4—2921no—
178D4—2921no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—2921no—
182C2×C4—5841no—
183C2×C4—5841no—
184C2×C4—731no—
185C2×C4—731no—
186C2×C4D4×#6440811yescharacteristic
187C8—2921no—
188C8—5841no—
189C8—5841no—
190C8—1461no—
191C8—1461no—
192C8—1461no—
193C8C2×#1192403811yescharacteristic
194C8#1455667911yescharacteristic
195C22—5841no—
196C22—5841no—
197C22—5841no—
198C22D8×#6440811yescharacteristic
199C4—81no—
200C4—81no—
201C4—5841no—
202C4—731no—
203C4—731no—
204C4—1461no—
205C4#1798962111yescharacteristic
206C4#1798963111yescharacteristic
207C2—11681no—
208C2#2262556311yescharacteristic
209C2#2262632711yescharacteristic
210C2#2262633311yescharacteristic
subgroupG/H
centerC22D8×#64408
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#367×C73C4
socleC22×C73C4×D8
prime radicalC2×D584C2×C4
cosocleC8×Dic73C23
hypercenter#367#64408
hypocenterC73#12084
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #367

Complements

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

NH ≅ G/Ntype
1. #22625570C2semidirect
3. #22624962C2semidirect
4. #22624888C2semidirect
6. #22624842C2semidirect
67. C16×C73M16semidirect
113. C73#12084semidirect
173. C16#11924038semidirect

Maximal subgroups

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

subgroupindex
#226255702
#226255632
#226249622
#226248882
#226248862
#226248422
#226117822
#1208473