Finite Group Databasebeta

← #30824283

Subgroups — #30824283

Order 17152.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569320C211yesmaximal, characteristic, complemented
2#22568777C211yesmaximal, characteristic, complemented
3#22558886C213yesmaximal, complemented
4#22558881C213yesmaximal, complemented
5#22558858C213yesmaximal, complemented
6C2×#16863142C213yesmaximal, complemented
7C67×#3499C211yesmaximal, characteristic
8C2×#14524050—21no—
9Q8×C67⋊C8—21no—
10D4×C67⋊C8—23no—
11#16863449C2213yes—
12#16863423—23no—
13#16863420C2213yes—
14C2×C4⋊(C67⋊C8)C2213yes—
15C2×C4⋊(C67⋊C8)C2213yes—
16#16863142C2213yes—
17#16863142C2216yes—
18#16863142C2213yes—
19#16863140C2211yescharacteristic
20#16863140C2213yes—
21C2×2−1+4×C67—21no—
22D4×Q8×C67—21no—
23D4×Q8×C67C412yes—
24#547×C67—23no—
25#547×C67C416yes—
26#536×C67—23no—
27#532×C67C2213yes—
28C2×C4⋊Q8×C67C2213yes—
29C4×#41×C67C2211yescharacteristic
30#14524050—44no—
31C2×#11761612—23no—
32C22×C67⋊C8—23no—
33C22×C67⋊C8—23no—
34C4⋊(C67⋊C8)—23no—
35C4⋊(C67⋊C8)—23no—
36C4⋊(C67⋊C8)—23no—
37C4⋊(C67⋊C8)C2311yescharacteristic
38C4⋊(C67⋊C8)C2313yes—
39C4⋊(C67⋊C8)C2313yes—
40C4⋊(C67⋊C8)C2311yescharacteristic
41C22⋊(C67⋊C8)—23no—
42C4×C67⋊C8—23no—
432−1+4×C67—44no—
442−1+4×C67—44no—
45C2×#41×C67—26no—
46C2×#41×C67—23no—
47C2×#41×C67D411yescharacteristic
48C22×Q8×C67—22no—
49C22×Q8×C67—23no—
50C4⋊Q8×C67—23no—
51C4⋊Q8×C67—23no—
52C4⋊Q8×C67C2×C416yes—
53C4⋊Q8×C67C2313yes—
54#127×C67C2311yescharacteristic
55#124×C67—23no—
56#123×C67—212no—
57C22⋊Q8×C67—26no—
58C22⋊Q8×C67—26no—
59#121×C67—23no—
60#121×C67C2×C416yes—
61C4×Q8×C67C2×C411yescharacteristic
62C4×Q8×C67D411yescharacteristic
63C4×D4×C67—23no—
64C4×D4×C67C2×C416yes—
65C4×D4×C67C2×C413yes—
66C4×D4×C67D413yes—
67#117×C67D413yes—
68C2×C4⋊C4×C67—23no—
69C2×C4⋊C4×C67C2×C416yes—
70C2×C42×C67C2313yes—
71#11761612—46no—
72C2×C67⋊C8—21no—
73C2×C67⋊C8—23no—
74C2×C67⋊C8—21no—
75C2×C67⋊C8—23no—
76C2×C67⋊C8—46no—
77C2×C67⋊C8—83no—
78#41×C67—424no—
79#41×C67—46no—
80#41×C67—46no—
81#41×C67D814yes—
82#41×C67SD1614yes—
83C2×Q8×C67—26no—
84C2×Q8×C67—23no—
85C2×Q8×C67—48no—
86C2×Q8×C67—46no—
87C2×Q8×C67—46no—
88C2×Q8×C67C22⋊C411yescharacteristic
89C2×D4×C67—26no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67C22⋊C413yes—
93C2×D4×C67C22×C412yes—
94C22×C4×C67—26no—
95C22×C4×C67—26no—
96C22×C4×C67C22×C413yes—
97C22×C4×C67C2×D413yes—
98C22×C4×C67C2×D413yes—
99C4⋊C4×C67—26no—
100C4⋊C4×C67—26no—
101C4⋊C4×C67—23no—
102C4⋊C4×C67C22⋊C413yes—
103C4⋊C4×C67C22⋊C413yes—
104C4⋊C4×C67C22×C416yes—
105C22⋊C4×C67—26no—
106C22⋊C4×C67—26no—
107C22⋊C4×C67C22⋊C416yes—
108C42×C67C22×C413yes—
109C42×C67C2×D413yes—
110C42×C67C2×D413yes—
111C42×C67C2411yescharacteristic
112C23×C67—22no—
113C23×C67C2×C22⋊C413yes—
114Q8×C67—43no—
115Q8×C67—43no—
116Q8×C67—412no—
117Q8×C67#10214yes—
118D4×C67—42no—
119D4×C67—412no—
120D4×C67—41no—
121D4×C67—41no—
122D4×C67—41no—
123D4×C67—41no—
124D4×C67#102112yes—
125C2×C4×C67—23no—
126C2×C4×C67—23no—
127C2×C4×C67—26no—
128C2×C4×C67—43no—
129C2×C4×C67—412no—
130C2×C4×C67—412no—
131C2×C4×C67C2×C22⋊C413yes—
132C2×C4×C67C2×C22⋊C413yes—
133C2×C4×C67C2×C22⋊C413yes—
134C2×C4×C67C2×D816yes—
135C2×C4×C67C2×SD1616yes—
136C2×C4×C67C23×C411yescharacteristic
137C2×C4×C67C22×D411yescharacteristic
138C2×C4×C67C22×D411yescharacteristic
139C67⋊C8—41no—
140C67⋊C8—43no—
141C67⋊C8—43no—
142C67⋊C8—41no—
143C22×C67—23no—
144C22×C67—23no—
145C22×C67—42no—
146C22×C67—42no—
147C22×C67—42no—
148C22×C67C2×#10216yes—
149C22×C67C22×C22⋊C411yescharacteristic
150C4×C67—21no—
151C4×C67—23no—
152C4×C67—23no—
153C4×C67—46no—
154C4×C67C2×#10216yes—
155C4×C67C22×D811yescharacteristic
156C4×C67C22×SD1611yescharacteristic
157C4×C67#57711yescharacteristic
158C4×C67#57811yescharacteristic
159#17721—671nomaximal, Sylow
160C2×C67—26no—
161C2×C67—82no—
162C2×C67#288911yescharacteristic
163C2×C67C22×#10211yescharacteristic
164C2×C67#289811yescharacteristic
165#3499D6711yescharacteristic
166#2988—671no—
167#2906—671no—
168#1547—673no—
169#1546—673no—
170#1540—673no—
171C2×#332—673no—
172C67#1772111yescharacteristic, complemented, Sylow
173C2×2−1+4—21no—
174C2×#131—1341no—
175D4×Q8—21no—
176D4×Q8Dic6712yes—
177#547—23no—
178#547Dic6716yes—
179#536—23no—
180#532C2×D6713yes—
181C2×C4⋊Q8C2×D6713yes—
182C4×#41C2×D6711yescharacteristic
183C8×Q8—1341no—
184C8×D4—1343no—
185#431—673no—
186#424—1343no—
187#423—673no—
188C2×C4⋊C8—673no—
189C2×C4⋊C8—673no—
190#332—673no—
191#332—676no—
192#332—673no—
193#331—671no—
194#331—673no—
1952−1+4—44no—
1962−1+4—44no—
197C2×#41—26no—
198C2×#41—23no—
199C2×#41C22⋊D6711yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202#131—2684no—
203C2×M16—1343no—
204C22×C8—1343no—
205C22×C8—1343no—
206C4⋊Q8—23no—
207C4⋊Q8—23no—
208C4⋊Q8C2×Dic6716yes—
209C4⋊Q8C22×D6713yes—
210#127C22×D6711yescharacteristic
211#124—23no—
212#123—212no—
213C22⋊Q8—26no—
214C22⋊Q8—26no—
215#121—23no—
216#121C2×Dic6716yes—
217C4×Q8C2×Dic6711yescharacteristic
218C4×Q8C22⋊D6711yescharacteristic
219C4×D4—23no—
220C4×D4C2×Dic6716yes—
221C4×D4C2×Dic6713yes—
222C4×D4C22⋊D6713yes—
223#117C22⋊D6713yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4C2×Dic6716yes—
226C2×C42C22×D6713yes—
227C4⋊C8—671no—
228C4⋊C8—673no—
229C4⋊C8—673no—
230C4⋊C8—671no—
231C4⋊C8—1343no—
232C4⋊C8—1343no—
233C4⋊C8—1343no—
234C22⋊C8—1343no—
235C4×C8—1343no—
236#41—424no—
237#41—46no—
238#41—46no—
239#41#1176161314yes—
240#41#1176161614yes—
241C2×Q8—26no—
242C2×Q8—23no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic6711yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic6713yes—
251C2×D4C22×Dic6712yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic6713yes—
255C22×C4C2×C22⋊D6713yes—
256C22×C4C2×C22⋊D6713yes—
257M16—2686no—
258C2×C8—2686no—
259C2×C8—5363no—
260C2×C8—1341no—
261C2×C8—1343no—
262C2×C8—1341no—
263C2×C8—1343no—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—23no—
267C4⋊C4C22⋊Dic6713yes—
268C4⋊C4C22⋊Dic6713yes—
269C4⋊C4C22×Dic6716yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic6716yes—
273C42C22×Dic6713yes—
274C42C2×C22⋊D6713yes—
275C42C2×C22⋊D6713yes—
276C42C23×D6711yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic6713yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1452396414yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#14523964112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×C22⋊Dic6713yes—
297C2×C4C2×C22⋊Dic6713yes—
298C2×C4C2×C22⋊Dic6713yes—
299C2×C4C2×#1176161316yes—
300C2×C4C2×#1176161616yes—
301C2×C4C23×Dic6711yescharacteristic
302C2×C4C22×C22⋊D6711yescharacteristic
303C2×C4C22×C22⋊D6711yescharacteristic
304C8—2681no—
305C8—2683no—
306C8—2683no—
307C8—2681no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1452396416yes—
314C22C22×C22⋊Dic6711yescharacteristic
315C4—23no—
316C4—21no—
317C4—23no—
318C4—46no—
319C4C2×#1452396416yes—
320C4C22×#1176161311yescharacteristic
321C4C22×#1176161611yescharacteristic
322C4#1686420411yescharacteristic
323C4#1686421111yescharacteristic
324C2—26no—
325C2—82no—
326C2#2256869211yescharacteristic
327C2C22×#1452396411yescharacteristic
328C2#2256874211yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
Fitting, prime radicalC67×#3499C2
socleC22×C67C22×C22⋊C4
cosocleC42×C67C24
hypercenter#3499D67
hypocenterC67#17721
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #22569320C2semidirect
2. #22568777C2semidirect
3. #22558886C2semidirect
4. #22558881C2semidirect
5. #22558858C2semidirect
6. C2×#16863142C2semidirect
172. C67#17721semidirect

Maximal subgroups

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

subgroupindex
#225693202
#225687772
#225588862
#225588812
#225588582
C2×#168631422
C67×#34992
#1772167