Finite Group Databasebeta

← C37×#10174 #22631427

Subgroups — C37×#10174 #22631427

Order 9472.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3226C211yesmaximal, characteristic
2C2×C37×#500C211yesmaximal, characteristic, complemented
3C37×#3152C211yesmaximal, characteristic
4C37×#1966C211yesmaximal, characteristic, complemented
5C37×#1546C211yesmaximal, characteristic, complemented
6C37×#1492C211yesmaximal, characteristic
7C37×#1278C211yesmaximal, characteristic, complemented
8C2×#137×C37—22no—
9C37×#532C2211yescharacteristic
10C2×C4⋊Q8×C37C2211yescharacteristic
11C2×C4⋊Q8×C37C2211yescharacteristic
12C37×#500—22no—
13C37×#500—21no—
14C37×#500—22no—
15C37×#497—22no—
16C37×#494—21no—
17C37×#492—21no—
18C37×#486—21no—
19C37×#480—22no—
20C37×#474—22no—
21C37×#431C2211yescharacteristic
22C2×#107×C37—22no—
23C37×#423C2211yescharacteristic
24C2×#101×C37—22no—
25C4×M16×C37C2211yescharacteristic
26C2×C4×C8×C37C2211yescharacteristic
27C37×#332—22no—
28C37×D4⋊C8—22no—
29C2×#41×C37—21no—
30C22×Q8×C37—21no—
31C22×Q8×C37—21no—
32#137×C37—44no—
33#137×C37—44no—
34C2×Q16×C37—24no—
35C2×SD16×C37—24no—
36C2×M16×C37—21no—
37C2×M16×C37D412yes—
38C22×C8×C37—21no—
39C4⋊Q8×C37—21no—
40C4⋊Q8×C37—21no—
41C4⋊Q8×C37—21no—
42C4⋊Q8×C37—21no—
43C4⋊Q8×C37—21no—
44C4⋊Q8×C37—21no—
45C4⋊Q8×C37—21no—
46C4⋊Q8×C37D412yes—
47C4⋊Q8×C37D411yescharacteristic
48#127×C37D411yescharacteristic
49#124×C37—21no—
50#121×C37—21no—
51#121×C37—21no—
52C4×D4×C37—21no—
53C4×D4×C37—21no—
54C2×C4⋊C4×C37—21no—
55C2×C4⋊C4×C37—21no—
56C2×C4⋊C4×C37—21no—
57C2×C4⋊C4×C37D412yes—
58C2×C42×C37C2311yescharacteristic
59#107×C37—28no—
60#107×C37—24no—
61C4⋊C8×C37—21no—
62C4⋊C8×C37—21no—
63C4⋊C8×C37D412yes—
64#102×C37—24no—
65#101×C37—44no—
66C22⋊C8×C37—21no—
67#97×C37—21no—
68C4×C8×C37—21no—
69C4×C8×C37—21no—
70C4×C8×C37—21no—
71C4×C8×C37D412yes—
72#41×C37—42no—
73#41×C37—42no—
74C2×Q8×C37—21no—
75C2×Q8×C37—22no—
76C2×Q8×C37—41no—
77C2×Q8×C37—41no—
78C2×Q8×C37—41no—
79C2×Q8×C37—42no—
80C2×Q8×C37—42no—
81C2×Q8×C37—42no—
82C2×Q8×C37—42no—
83C2×D4×C37—21no—
84C2×D4×C37—21no—
85C2×D4×C37—21no—
86C2×D4×C37—41no—
87C22×C4×C37—22no—
88C22×C4×C37—41no—
89C22×C4×C37—41no—
90C22×C4×C37C2×D411yescharacteristic
91C22×C4×C37C2×D411yescharacteristic
92C22×C4×C37#4111yescharacteristic
93Q16×C37—48no—
94SD16×C37—48no—
95M16×C37—22no—
96M16×C37—42no—
97M16×C37D814yes—
98C2×C8×C37—24no—
99C2×C8×C37—22no—
100C2×C8×C37—22no—
101C2×C8×C37—22no—
102C2×C8×C37—41no—
103C4⋊C4×C37—22no—
104C4⋊C4×C37—22no—
105C4⋊C4×C37—22no—
106C4⋊C4×C37—21no—
107C4⋊C4×C37—22no—
108C4⋊C4×C37—21no—
109C4⋊C4×C37—41no—
110C4⋊C4×C37—41no—
111C4⋊C4×C37—41no—
112C4⋊C4×C37—41no—
113C22⋊C4×C37—41no—
114C22⋊C4×C37—41no—
115C42×C37C2×D411yescharacteristic
116C42×C37C2×D411yescharacteristic
117C42×C37C2×D411yescharacteristic
118C42×C37C2×D411yescharacteristic
119C23×C37—41no—
120C23×C37#12111yescharacteristic
121Q8×C37—44no—
122Q8×C37—42no—
123Q8×C37—82no—
124Q8×C37—81no—
125Q8×C37—81no—
126Q8×C37—81no—
127Q8×C37—81no—
128D4×C37—42no—
129D4×C37—42no—
130D4×C37—42no—
131D4×C37—82no—
132C2×C4×C37—22no—
133C2×C4×C37—22no—
134C2×C4×C37—41no—
135C2×C4×C37—42no—
136C2×C4×C37—42no—
137C2×C4×C37—44no—
138C2×C4×C37—82no—
139C2×C4×C37—81no—
140C2×C4×C37—81no—
141C2×C4×C37#12011yescharacteristic
142C2×C4×C37#12011yescharacteristic
143C2×C4×C37#12011yescharacteristic
144C2×C4×C37#12111yescharacteristic
145C2×C4×C37#12111yescharacteristic
146C2×C4×C37#12711yescharacteristic
147C2×C4×C37C2×D812yes—
148C2×C4×C37AGL(1,ℤ/8)12yes—
149C8×C37—24no—
150C8×C37—24no—
151C8×C37—42no—
152C8×C37—81no—
153#10174C3711yesmaximal, characteristic, direct product factor, Sylow
154C22×C37—21no—
155C22×C37—21no—
156C22×C37—81no—
157C22×C37—81no—
158C22×C37—81no—
159C22×C37#39211yescharacteristic
160C22×C37#45912yes—
161C4×C37—21no—
162C4×C37—21no—
163C4×C37—82no—
164C4×C37—81no—
165C4×C37—82no—
166C4×C37#44712yes—
167C4×C37#45612yes—
168C4×C37#47312yes—
169C4×C37#49311yescharacteristic
170C4×C37#49611yescharacteristic
171#3226C2×C3711yescharacteristic
172C2×#500C2×C3711yescharacteristic, complemented
173#3152C2×C3711yescharacteristic
174#1966C2×C3711yescharacteristic, complemented
175#1546C2×C3711yescharacteristic, complemented
176#1492C2×C3711yescharacteristic
177#1278C2×C3711yescharacteristic, complemented
178C2×C37—22no—
179C2×C37—161no—
180C2×C37#200511yescharacteristic
181C2×C37#201111yescharacteristic
182C2×C37#202011yescharacteristic
183C2×#137—22no—
184#532C22×C3711yescharacteristic
185C2×C4⋊Q8C22×C3711yescharacteristic
186C2×C4⋊Q8C22×C3711yescharacteristic
187#500—22no—
188#500—21no—
189#500—22no—
190#497—22no—
191#494—21no—
192#492—21no—
193#486—21no—
194#480—22no—
195#474—22no—
196#431C22×C3711yescharacteristic
197C2×#107—22no—
198#423C22×C3711yescharacteristic
199C2×#101—22no—
200C4×M16C22×C3711yescharacteristic
201C2×C4×C8C22×C3711yescharacteristic
202#332—22no—
203D4⋊C8—22no—
204C37#1017411yescharacteristic, direct product factor, Sylow
205C2×#41—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208#137—44no—
209#137—44no—
210C2×Q16—24no—
211C2×SD16—24no—
212C2×M16—21no—
213C2×M16D4×C3712yes—
214C22×C8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8D4×C3712yes—
223C4⋊Q8D4×C3711yescharacteristic
224#127D4×C3711yescharacteristic
225#124—21no—
226#121—21no—
227#121—21no—
228C4×D4—21no—
229C4×D4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4D4×C3712yes—
234C2×C42C23×C3711yescharacteristic
235#107—28no—
236#107—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8D4×C3712yes—
240#102—24no—
241#101—44no—
242C22⋊C8—21no—
243#97—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8D4×C3712yes—
248#41—42no—
249#41—42no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D4×C3711yescharacteristic
267C22×C4C2×D4×C3711yescharacteristic
268C22×C4#41×C3711yescharacteristic
269Q16—48no—
270SD16—48no—
271M16—22no—
272M16—42no—
273M16D8×C3714yes—
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42C2×D4×C3711yescharacteristic
292C42C2×D4×C3711yescharacteristic
293C42C2×D4×C3711yescharacteristic
294C42C2×D4×C3711yescharacteristic
295C23—41no—
296C23#121×C3711yescharacteristic
297Q8—44no—
298Q8—42no—
299Q8—82no—
300Q8—81no—
301Q8—81no—
302Q8—81no—
303Q8—81no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#120×C3711yescharacteristic
318C2×C4#120×C3711yescharacteristic
319C2×C4#120×C3711yescharacteristic
320C2×C4#121×C3711yescharacteristic
321C2×C4#121×C3711yescharacteristic
322C2×C4#127×C3711yescharacteristic
323C2×C4C2×D8×C3712yes—
324C2×C4AGL(1,ℤ/8)×C3712yes—
325C8—24no—
326C8—24no—
327C8—42no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C37×#39211yescharacteristic
335C22C37×#45912yes—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—82no—
341C4C37×#44712yes—
342C4C37×#45612yes—
343C4C37×#47312yes—
344C4C37×#49311yescharacteristic
345C4C37×#49611yescharacteristic
346C2—22no—
347C2—161no—
348C2C37×#200511yescharacteristic
349C2C37×#201111yescharacteristic
350C2C37×#202011yescharacteristic
subgroupG/H
center, socleC22×C37#392
commutator, Frattini, cosocleC2×C42C23×C37
prime radicalC37×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#500C2semidirect
4. C37×#1966C2semidirect
5. C37×#1546C2semidirect
7. C37×#1278C2semidirect
153. #10174C37direct (N × H)
172. C2×#500C2×C37semidirect
174. #1966C2×C37semidirect
175. #1546C2×C37semidirect
177. #1278C2×C37semidirect
204. C37#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32262
C2×C37×#5002
C37×#31522
C37×#19662
C37×#15462
C37×#14922
C37×#12782
#1017437