Finite Group Databasebeta

← #34010040

Subgroups — #34010040

Order 18688.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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
2#22626289C211yesmaximal, characteristic
3#22624870C212yesmaximal
4#22624869C212yesmaximal
5#22618863C211yesmaximal, characteristic
6#17989508—21no—
7#17989508—21no—
8C22⋊#11924031C2212yes—
9C22⋊#11924031C2212yes—
10#17989476—21no—
11#17989473—21no—
12#17989140C411yescharacteristic
13#17988715—21no—
14#17988715—21no—
15C8×Dic146—21no—
16C8×Dic146—21no—
17#17988615C2211yescharacteristic
18#17988614C2211yescharacteristic
19#17988610C411yescharacteristic
20#17988542C411yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22#405×C73C411yescharacteristic
23C2×#11924031—22no—
24C2×#11924031—22no—
25C2×#11924012C2×C411yescharacteristic
26C22×#10596831C2311yescharacteristic
27#14556590C811yescharacteristic
28#14556590C811yescharacteristic
29#14556582C811yescharacteristic
30#14556582C811yescharacteristic
31#14556568—21no—
32#14556568—21no—
33C4×Dic146—21no—
34C4×Dic146—21no—
35#14556550C2×C411yescharacteristic
36C2×C4×Dic73C2×C411yescharacteristic
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39#14556520—21no—
40#14556520—21no—
41#14556520—21no—
42#14556520—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
52#97×C73—21no—
53C4×C8×C73—21no—
54#11924031—42no—
55#11924031—42no—
56C2×Dic146—21no—
57C2×Dic146—21no—
58C22×Dic73M1611yescharacteristic
59C22×Dic73M1611yescharacteristic
60#11924012—41no—
61C2×#10596831—21no—
62C2×#10596831—41no—
63C2×#10596831C2×D411yescharacteristic
64C2×#10596831C2×D411yescharacteristic
65C4⋊Dic73—21no—
66C4⋊Dic73—21no—
67C4⋊Dic73C2×C811yescharacteristic
68C4⋊Dic73C2×C811yescharacteristic
69#11924007—21no—
70#11924007—21no—
71#11924007—21no—
72#11924007—21no—
73C22⋊Dic73—21no—
74C4×Dic73—21no—
75C4×Dic73C22⋊C411yescharacteristic
76C4×Dic73C22⋊C411yescharacteristic
77C4×Dic73C2×C811yescharacteristic
78C4×Dic73C2×C811yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80M16×C73—41no—
81C2×C8×C73—22no—
82C2×C8×C73—21no—
83C2×C8×C73C22⋊C411yescharacteristic
84C2×C8×C73C22⋊C411yescharacteristic
85C4⋊C4×C73C2×C811yescharacteristic
86C4⋊C4×C73C2×C811yescharacteristic
87C22⋊C4×C73—21no—
88C42×C73—21no—
89C23×C73C2×M1611yescharacteristic
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73C2×C22⋊C411yescharacteristic
95C2×C4×C73C22×C811yescharacteristic
96C2×Dic73—22no—
97C2×Dic73—21no—
98C2×Dic73—21no—
99C2×Dic73—41no—
100C2×Dic73C22⋊C811yescharacteristic
101C2×Dic73C22⋊C811yescharacteristic
102C2×Dic73C22⋊C811yescharacteristic
103C2×Dic73C22⋊C811yescharacteristic
104Dic146—42no—
105Dic146—42no—
106C8×C73—21no—
107C8×C73—41no—
108C8×C73#10112yes—
109#10596831—21no—
110#10596831—21no—
111#10596831—41no—
112C22×C73—21no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73C2×C22⋊C811yescharacteristic
116C4×C73—21no—
117C4×C73—41no—
118C4×C73—41no—
119C4×C73#41011yescharacteristic
120C4×C73C2×#10111yescharacteristic
121Dic73—21no—
122Dic73—21no—
123Dic73—21no—
124Dic73—41no—
125Dic73—41no—
126Dic73#35012yes—
127#11498—731nomaximal, Sylow
128C2×C73—41no—
129C2×C73#146111yescharacteristic
130C2×C73C2×#35011yescharacteristic
131C2×C73#211711yescharacteristic
132#2960—731no—
133#2163—731no—
134#2162—731no—
135#1323—732no—
136#1323—732no—
137C73#1149811yescharacteristic, complemented, Sylow
138#533—731no—
139#446—1461no—
140#446—1461no—
141C8×Q8—1461no—
142C8×Q8—1461no—
143#424—731no—
144#424—731no—
145#423—731no—
146#405—731no—
147#405#6440811yescharacteristic
148C2×C4×C8—731no—
149C4⋊C16—1461no—
150C4⋊C16—1461no—
151C22⋊C16—732no—
152C22⋊C16—732no—
153#346—1461no—
154C4×C16—1461no—
155C2×M16—731no—
156C2×M16C2×#6440811yescharacteristic
157C22×C8—731no—
158C22×C8C2×#6440811yescharacteristic
159C4⋊Q8—731no—
160C4⋊Q8—731no—
161#125—731no—
162#125—731no—
163C22⋊Q8—1461no—
164C22⋊Q8—1461no—
165C4×Q8—1461no—
166C4×Q8—1461no—
167#117—731no—
168#117C2×#6440811yescharacteristic
169C2×C42—731no—
170C2×C16—1462no—
171C2×C16—1462no—
172C4⋊C8—1461no—
173C4⋊C8—1461no—
174C4⋊C8—1461no—
175C4⋊C8—1461no—
176C4⋊C8—1461no—
177C4⋊C8—1461no—
178#97—21no—
179#97—1461no—
180C4×C8—21no—
181C4×C8—731no—
182C4×C8—731no—
183C4×C8—731no—
184C4×C8—731no—
185C4×C8—1461no—
186C2×Q8—1461no—
187C2×Q8—1461no—
188C22×C4—731no—
189C22×C4—731no—
190C22×C4C22×#6440811yescharacteristic
191M16—41no—
192M16—2921no—
193C2×C8—22no—
194C2×C8—21no—
195C2×C8—2921no—
196C2×C8—731no—
197C2×C8—731no—
198C2×C8—1461no—
199C2×C8C22⋊#6440811yescharacteristic
200C2×C8C22⋊#6440811yescharacteristic
201C4⋊C4—731no—
202C4⋊C4—731no—
203C4⋊C4—1461no—
204C4⋊C4—1461no—
205C4⋊C4—1461no—
206C4⋊C4—1461no—
207C4⋊C4—1461no—
208C4⋊C4—1461no—
209C4⋊C4#1192403611yescharacteristic
210C4⋊C4#1192403611yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—1461no—
213C42—21no—
214C42—731no—
215C42—731no—
216C42—731no—
217C42—731no—
218C42—1461no—
219C16—2922no—
220C16—2922no—
221C23D73⋊M1611yescharacteristic
222Q8—2922no—
223Q8—2922no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—2921no—
229C2×C4—731no—
230C2×C4—731no—
231C2×C4—731no—
232C2×C4—731no—
233C2×C4—1461no—
234C2×C4—1461no—
235C2×C4—1462no—
236C2×C4C2×C22⋊#6440811yescharacteristic
237C2×C4C2×#1192403611yescharacteristic
238C8—21no—
239C8—41no—
240C8—2921no—
241C8—1461no—
242C8—1461no—
243C8#1455665212yes—
244C22—21no—
245C22—21no—
246C22—21no—
247C22#1798955511yescharacteristic
248C4—21no—
249C4—41no—
250C4—41no—
251C4—2921no—
252C4—2921no—
253C4—732no—
254C4—1461no—
255C4—1461no—
256C4—1461no—
257C4#1798956611yescharacteristic
258C4C2×#1455665211yescharacteristic
259C2—41no—
260C2#2262517311yescharacteristic
261C2#2262617111yescharacteristic
262C2#2262617411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#405×C73C4
socleC22×C73C2×C22⋊C8
prime radicalC23×C73C2×M16
cosocleC22×#10596831C23
hypercenter#405#64408
hypocenterC73#11498
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
137. C73#11498semidirect

Maximal subgroups

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

subgroupindex
#226262932
#226262892
#226248702
#226248692
#226188632
#1149873