Finite Group Databasebeta

← #34009918

Subgroups — #34009918

Order 18688.

310 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 84 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#22626293C211yesmaximal, characteristic, complemented
2#22626289C211yesmaximal, characteristic, complemented
3#22624869C212yesmaximal, complemented
4#22624854C212yesmaximal
5#22618574C211yesmaximal, characteristic
6#17989508—21no—
7#17989508—21no—
8C22⋊#11924031C2212yes—
9C22⋊#11924031C2212yes—
10#17989476—21no—
11#17989473—21no—
12#17989055C411yescharacteristic
13#17988647—21no—
14#17988647—21no—
15#17988639—21no—
16#17988639—21no—
17#17988614C2211yescharacteristic
18#17988555C2211yescharacteristic
19#17988553C411yescharacteristic
20M16×Dic73C411yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22#424×C73C411yescharacteristic
23C2×#11924031—22no—
24C2×#11924031—22no—
25C2×#11924028C2×C411yescharacteristic
26C2×#11924012C2×C411yescharacteristic
27C22×#10596831C2311yescharacteristic
28#14556556C811yescharacteristic
29#14556556C811yescharacteristic
30#14556553C811yescharacteristic
31#14556553C811yescharacteristic
32#14556551—21no—
33#14556551—21no—
34#14556548—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36C42×D73—21no—
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39#14556503—21no—
40#14556503—21no—
41#14556503—21no—
42#14556503—21no—
43#14556501—21no—
44C8×Dic73—21no—
45C8×Dic73D411yescharacteristic
46C8×Dic73D411yescharacteristic
47C8×Dic73D411yescharacteristic
48C8×Dic73D411yescharacteristic
49C2×M16×C73C2×C411yescharacteristic
50C22×C8×C73C2×C411yescharacteristic
51#117×C73C2×C411yescharacteristic
52C4⋊C8×C73—21no—
53C4⋊C8×C73—21no—
54#11924031—42no—
55#11924031—42no—
56#11924028—41no—
57#11924028—41no—
58C2×Dic146C2×C811yescharacteristic
59C2×D292C2×C811yescharacteristic
60C2×C22⋊D73—21no—
61C22×Dic73M1611yescharacteristic
62C22×Dic73M1611yescharacteristic
63C2×C4×D73—21no—
64C2×C4×D73—21no—
65C2×C4×D73—21no—
66#11924012—41no—
67C2×#10596831—21no—
68C2×#10596831—41no—
69C2×#10596831C2×D411yescharacteristic
70C2×#10596831C2×D411yescharacteristic
71#11924007—21no—
72#11924007—21no—
73#11924005—21no—
74#11924005—21no—
75C4×Dic73—21no—
76C4×Dic73—21no—
77C4×Dic73C22⋊C411yescharacteristic
78C4×Dic73C22⋊C411yescharacteristic
79C4×Dic73C2×C811yescharacteristic
80C4×Dic73C2×C811yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82M16×C73—41no—
83C2×C8×C73—21no—
84C2×C8×C73—41no—
85C2×C8×C73C22⋊C411yescharacteristic
86C2×C8×C73C22⋊C411yescharacteristic
87C4⋊C4×C73C2×C811yescharacteristic
88C4⋊C4×C73C2×C811yescharacteristic
89C22⋊C4×C73—21no—
90C42×C73—21no—
91C23×C73C2×M1611yescharacteristic
92C22×D73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73C2×C22⋊C411yescharacteristic
98C2×C4×C73C22×C811yescharacteristic
99C22⋊D73—42no—
100C2×Dic73—22no—
101C2×Dic73—21no—
102C2×Dic73—41no—
103C2×Dic73C22⋊C811yescharacteristic
104C2×Dic73C22⋊C811yescharacteristic
105C2×Dic73C22⋊C811yescharacteristic
106C2×Dic73C22⋊C811yescharacteristic
107D292—41no—
108C4×D73—42no—
109C4×D73—41no—
110C4×D73—41no—
111C4×D73—42no—
112Dic146—41no—
113C8×C73—21no—
114C8×C73—21no—
115C8×C73—41no—
116#10596831—21no—
117#10596831—21no—
118#10596831—41no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73C2×C22⋊C811yescharacteristic
123C2×D73—41no—
124C2×D73—41no—
125C2×D73—41no—
126C4×C73—21no—
127C4×C73—41no—
128C4×C73—41no—
129C4×C73#41011yescharacteristic
130C4×C73#41311yescharacteristic
131Dic73—21no—
132Dic73—21no—
133Dic73—21no—
134Dic73—41no—
135Dic73#35012yes—
136#11464—731nomaximal, Sylow
137C2×C73—41no—
138C2×C73#145811yescharacteristic
139C2×C73C2×#35011yescharacteristic
140C2×C73#211711yescharacteristic
141D73—81no—
142#2922—731no—
143#2163—731no—
144#2162—731no—
145#1323—732no—
146#1315—732no—
147C73#1146411yescharacteristic, complemented, Sylow
148C4×#41—731no—
149#433—1461no—
150#433—1461no—
151#431—1461no—
152#431—1461no—
153#424—731no—
154#424#6440811yescharacteristic
155#408—731no—
156#408—731no—
157C4×M16—731no—
158C2×C4×C8—731no—
159C4⋊C16—1461no—
160C4⋊C16—1461no—
161C22⋊C16—732no—
162C22⋊C16—732no—
163#346—1461no—
164C4×C16—1461no—
165C2×#41—731no—
166C2×M16—731no—
167C2×M16C2×#6440811yescharacteristic
168C22×C8—731no—
169C22×C8C2×#6440811yescharacteristic
170C4×Q8—731no—
171C4×Q8—731no—
172C4×D4—731no—
173C4×D4—731no—
174C4×D4—1461no—
175C4×D4—1461no—
176#117—1461no—
177#117C2×#6440811yescharacteristic
178C2×C42—731no—
179C2×C42—1461no—
180C2×C16—1462no—
181C2×C16—1462no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—1461no—
185C4⋊C8—1461no—
186C22⋊C8—1461no—
187C22⋊C8—1461no—
188C22⋊C8—1461no—
189C22⋊C8—1461no—
190#97—1461no—
191C4×C8—731no—
192C4×C8—731no—
193C4×C8—731no—
194C4×C8—731no—
195C4×C8—1461no—
196#41—2921no—
197#41—2921no—
198C2×Q8—731no—
199C2×D4—731no—
200C2×D4—1461no—
201C22×C4—731no—
202C22×C4—731no—
203C22×C4—1461no—
204C22×C4—1461no—
205C22×C4—1461no—
206C22×C4C22×#6440811yescharacteristic
207M16—41no—
208M16—2921no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—2921no—
212C2×C8—731no—
213C2×C8—731no—
214C2×C8—1461no—
215C2×C8C22⋊#6440811yescharacteristic
216C2×C8C22⋊#6440811yescharacteristic
217C4⋊C4—1461no—
218C4⋊C4—1461no—
219C4⋊C4#1192403611yescharacteristic
220C4⋊C4#1192403611yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—1461no—
223C22⋊C4—1461no—
224C42—21no—
225C42—731no—
226C42—731no—
227C42—731no—
228C42—731no—
229C42—1461no—
230C42—1461no—
231C16—2922no—
232C16—2922no—
233C23—1461no—
234C23D73⋊M1611yescharacteristic
235Q8—2921no—
236D4—2921no—
237D4—2922no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—2922no—
243C2×C4—2922no—
244C2×C4—2921no—
245C2×C4—2921no—
246C2×C4—2921no—
247C2×C4—731no—
248C2×C4—731no—
249C2×C4—731no—
250C2×C4—731no—
251C2×C4—1461no—
252C2×C4—1462no—
253C2×C4C2×C22⋊#6440811yescharacteristic
254C2×C4C2×#1192403611yescharacteristic
255C8—21no—
256C8—21no—
257C8—41no—
258C8—2921no—
259C8—1461no—
260C8—1461no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—2921no—
265C22—2921no—
266C22—2921no—
267C22#1798955511yescharacteristic
268C4—21no—
269C4—41no—
270C4—41no—
271C4—2921no—
272C4—732no—
273C4—1461no—
274C4—1461no—
275C4—1461no—
276C4#1798956811yescharacteristic
277C4#1798957411yescharacteristic
278C2—41no—
279C2—5841no—
280C2#2262516511yescharacteristic
281C2#2262617111yescharacteristic
282C2#2262617411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#424×C73C4
socleC22×C73C2×C22⋊C8
prime radicalC2×#11924028C2×C4
cosocleC22×#10596831C23
hypercenter#424#64408
hypocenterC73#11464
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #22626293C2semidirect
2. #22626289C2semidirect
3. #22624869C2semidirect
147. C73#11464semidirect

Maximal subgroups

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

subgroupindex
#226262932
#226262892
#226248692
#226248542
#226185742
#1146473