Finite Group Databasebeta

← #33006405

Subgroups — #33006405

Order 18688.

304 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 75 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#22611626C211yesmaximal, characteristic
2#22611620C211yesmaximal, characteristic
3#22608104C212yesmaximal, complemented
4#22608100C212yesmaximal, complemented
5C73×#3223C211yesmaximal, characteristic
6#17988571—21no—
7C2×#14556510—21no—
8#17988386—21no—
9#17988386C2212yes—
10#17988386C2212yes—
11#17988384—21no—
12#541×C73C411yescharacteristic
13#536×C73C2211yescharacteristic
14#497×C73C411yescharacteristic
15#494×C73C2211yescharacteristic
16#491×C73C2211yescharacteristic
17#488×C73C411yescharacteristic
18#464×C73—21no—
19#462×C73—21no—
20#452×C73—21no—
21#451×C73—21no—
22#433×C73C411yescharacteristic
23C2×#11924012—41no—
24C2×#11924012—41no—
25#14556510—22no—
26#14556510—22no—
27#14556510—41no—
28#14556508—41no—
29C2×#41×C73D411yescharacteristic
30C22×Q8×C73D411yescharacteristic
31C2×Q16×C73—21no—
32C2×Q16×C73—21no—
33C2×Q16×C73—21no—
34C2×SD16×C73—21no—
35C4⋊Q8×C73C2×C411yescharacteristic
36C4⋊Q8×C73D411yescharacteristic
37C4⋊Q8×C73C2311yescharacteristic
38#126×C73—41no—
39#125×C73—41no—
40#125×C73C2×C411yescharacteristic
41#124×C73C2×C411yescharacteristic
42#124×C73D411yescharacteristic
43#123×C73—41no—
44C22⋊Q8×C73—21no—
45C22⋊Q8×C73—41no—
46C4×Q8×C73—21no—
47#117×C73C2×C411yescharacteristic
48C4⋊C8×C73—21no—
49#103×C73—21no—
50#103×C73—21no—
51#103×C73—21no—
52#102×C73—21no—
53C22⋊C8×C73—21no—
54#97×C73C2×C411yescharacteristic
55C4×C8×C73C2×C411yescharacteristic
56#11924012—42no—
57#11924012—42no—
58#11924012—81no—
59#11924012—81no—
60C2×#10596831—81no—
61C2×#10596831—81no—
62#41×C73—81no—
63C2×Q8×C73—21no—
64C2×Q8×C73—21no—
65C2×Q8×C73—81no—
66C2×Q8×C73C22⋊C411yescharacteristic
67C2×Q8×C73C2×D411yescharacteristic
68C2×Q8×C73C2×D411yescharacteristic
69C2×D4×C73—21no—
70C2×D4×C73C22⋊C411yescharacteristic
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73C22⋊C411yescharacteristic
74Q16×C73—81no—
75Q16×C73—81no—
76Q16×C73—81no—
77SD16×C73—81no—
78C2×C8×C73—21no—
79C2×C8×C73—21no—
80C4⋊C4×C73—21no—
81C4⋊C4×C73—21no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C4⋊C4×C73—41no—
88C22⋊C4×C73—21no—
89C22⋊C4×C73—41no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C42×C73—41no—
93C42×C73C22⋊C411yescharacteristic
94C42×C73C22×C411yescharacteristic
95C23×C73—21no—
96C23×C73#9911yescharacteristic
97Q8×C73—41no—
98Q8×C73—41no—
99Q8×C73—41no—
100Q8×C73—41no—
101Q8×C73—41no—
102Q8×C73—81no—
103D4×C73—41no—
104D4×C73—81no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73#9911yescharacteristic
118C2×C4×C73C2×C22⋊C411yescharacteristic
119C8×C73—41no—
120C8×C73—41no—
121#10596831—82no—
122#10596831—82no—
123C22×C73—21no—
124C22×C73—41no—
125C22×C73—41no—
126C22×C73—81no—
127C22×C73C2×#9911yescharacteristic
128C4×C73—21no—
129C4×C73—21no—
130C4×C73—41no—
131C4×C73—41no—
132C4×C73—41no—
133C4×C73—41no—
134C4×C73—41no—
135C4×C73—81no—
136C4×C73—81no—
137#13612—731nomaximal, Sylow
138C2×C73—21no—
139C2×C73—41no—
140C2×C73—81no—
141C2×C73#212711yescharacteristic
142#3223D7311yescharacteristic
143#2135—731no—
144#2133—731no—
145#1413—732no—
146#1411—732no—
147C73#1361211yescharacteristic, complemented, Sylow
148#541Dic7311yescharacteristic
149#536C2×D7311yescharacteristic
150#497Dic7311yescharacteristic
151#494C2×D7311yescharacteristic
152#491C2×D7311yescharacteristic
153#488Dic7311yescharacteristic
154#464—21no—
155#462—21no—
156#452—21no—
157#451—21no—
158#433Dic7311yescharacteristic
159#413—1461no—
160C2×#101—1461no—
161#356—732no—
162#356—732no—
163#356—1461no—
164#355—1461no—
165C2×#41C22⋊D7311yescharacteristic
166C22×Q8C22⋊D7311yescharacteristic
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×SD16—21no—
171C2×M16—2921no—
172C2×M16—2921no—
173C4⋊Q8C2×Dic7311yescharacteristic
174C4⋊Q8C22⋊D7311yescharacteristic
175C4⋊Q8C22×D7311yescharacteristic
176#126—41no—
177#125—41no—
178#125C2×Dic7311yescharacteristic
179#124C2×Dic7311yescharacteristic
180#124C22⋊D7311yescharacteristic
181#123—41no—
182C22⋊Q8—21no—
183C22⋊Q8—41no—
184C4×Q8—21no—
185#117C2×Dic7311yescharacteristic
186C4⋊C8—21no—
187#103—21no—
188#103—21no—
189#103—21no—
190#102—21no—
191#101—2921no—
192#101—1462no—
193#101—1462no—
194#100—2921no—
195C22⋊C8—21no—
196#97C2×Dic7311yescharacteristic
197C4×C8C2×Dic7311yescharacteristic
198#41—81no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—81no—
202C2×Q8C22⋊Dic7311yescharacteristic
203C2×Q8C2×C22⋊D7311yescharacteristic
204C2×Q8C2×C22⋊D7311yescharacteristic
205C2×D4—21no—
206C2×D4C22⋊Dic7311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4C22⋊Dic7311yescharacteristic
210Q16—81no—
211Q16—81no—
212Q16—81no—
213SD16—81no—
214M16—2922no—
215M16—2922no—
216M16—5841no—
217M16—5841no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—5841no—
221C2×C8—5841no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—21no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C42—41no—
235C42C22⋊Dic7311yescharacteristic
236C42C22×Dic7311yescharacteristic
237C23—21no—
238C23#1455650611yescharacteristic
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—81no—
245D4—41no—
246D4—81no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4#1455650611yescharacteristic
260C2×C4C2×C22⋊Dic7311yescharacteristic
261C8—41no—
262C8—41no—
263C8—5842no—
264C8—5842no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22C2×#1455650611yescharacteristic
270C4—21no—
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—81no—
278C4—81no—
279C2—21no—
280C2—41no—
281C2—81no—
282C2#2261160011yescharacteristic
subgroupG/H
centerC2#22611600
commutatorC42×C73C22×C4
FrattiniC4⋊Q8C22×D73
FittingC73×#3223C2
socleC2×C73#2127
prime radicalC2×#41×C73D4
cosocleC4⋊Q8×C73C23
hypercenter#3223D73
hypocenterC73#13612
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #3223

Complements

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

NH ≅ G/Ntype
3. #22608104C2semidirect
4. #22608100C2semidirect
147. C73#13612semidirect

Maximal subgroups

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

subgroupindex
#226116262
#226116202
#226081042
#226081002
C73×#32232
#1361273