Finite Group Databasebeta

← C37×#10286 #22631539

Subgroups — C37×#10286 #22631539

Order 9472.

542 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3227C211yesmaximal, characteristic
2C37×#3226C211yesmaximal, characteristic
3C37×#2862C211yesmaximal, characteristic, complemented
4C37×#1962C211yesmaximal, characteristic
5C37×#1549C211yesmaximal, characteristic, complemented
6C37×#1546C211yesmaximal, characteristic, complemented
7C37×#1276C211yesmaximal, characteristic, complemented
8C37×#532C2211yescharacteristic
9C2×C4⋊Q8×C37C2211yescharacteristic
10C2×C4⋊Q8×C37C2211yescharacteristic
11C37×#500—21no—
12C37×#498—21no—
13C37×#486—21no—
14C37×#486—21no—
15C37×#481—22no—
16C37×#480—22no—
17C37×#476—22no—
18C37×#474—22no—
19C37×#431C2211yescharacteristic
20C37×#431C2211yescharacteristic
21C37×#423C2211yescharacteristic
22C2×#104×C37—22no—
23C37×#391—22no—
24C37×#391—22no—
25C37×#382—22no—
26C43×C37C2211yescharacteristic
27C37×#332—22no—
28C37×#332—22no—
29C2×#41×C37—21no—
30C22×Q8×C37—21no—
31C22×Q8×C37—21no—
32C2×M16×C37—21no—
33C4⋊Q8×C37—21no—
34C4⋊Q8×C37—21no—
35C4⋊Q8×C37—21no—
36C4⋊Q8×C37—21no—
37C4⋊Q8×C37—21no—
38C4⋊Q8×C37—21no—
39C4⋊Q8×C37—21no—
40C4⋊Q8×C37—21no—
41C4⋊Q8×C37D411yescharacteristic
42#127×C37D411yescharacteristic
43#124×C37—21no—
44#121×C37—21no—
45#121×C37—21no—
46C4×D4×C37—21no—
47C4×D4×C37—21no—
48C2×C4⋊C4×C37—21no—
49C2×C4⋊C4×C37—21no—
50C2×C4⋊C4×C37D412yes—
51C2×C4⋊C4×C37D412yes—
52C2×C42×C37—21no—
53C2×C42×C37—21no—
54C2×C42×C37D412yes—
55C2×C42×C37C2311yescharacteristic
56#107×C37—24no—
57#106×C37—24no—
58C4⋊C8×C37—21no—
59C4⋊C8×C37—21no—
60C4⋊C8×C37D412yes—
61C4⋊C8×C37D412yes—
62#104×C37—44no—
63#102×C37—24no—
64#102×C37—24no—
65C22⋊C8×C37—21no—
66C22⋊C8×C37—21no—
67C4×C8×C37—21no—
68C4×C8×C37—21no—
69#95×C37—28no—
70#41×C37—42no—
71#41×C37—42no—
72C2×Q8×C37—21no—
73C2×Q8×C37—44no—
74C2×Q8×C37—41no—
75C2×Q8×C37—41no—
76C2×Q8×C37—41no—
77C2×Q8×C37—41no—
78C2×Q8×C37—42no—
79C2×Q8×C37—42no—
80C2×D4×C37—21no—
81C2×D4×C37—21no—
82C2×D4×C37—21no—
83C2×D4×C37—41no—
84C22×C4×C37—22no—
85C22×C4×C37—22no—
86C22×C4×C37—22no—
87C22×C4×C37—41no—
88C22×C4×C37C2×D411yescharacteristic
89C22×C4×C37C2×D411yescharacteristic
90C22×C4×C37C2×D411yescharacteristic
91M16×C37—42no—
92C2×C8×C37—22no—
93C2×C8×C37—22no—
94C2×C8×C37—41no—
95C4⋊C4×C37—22no—
96C4⋊C4×C37—22no—
97C4⋊C4×C37—21no—
98C4⋊C4×C37—21no—
99C4⋊C4×C37—22no—
100C4⋊C4×C37—22no—
101C4⋊C4×C37—41no—
102C4⋊C4×C37—41no—
103C4⋊C4×C37—41no—
104C4⋊C4×C37—41no—
105C22⋊C4×C37—41no—
106C22⋊C4×C37—41no—
107C42×C37—24no—
108C42×C37—22no—
109C42×C37—22no—
110C42×C37—42no—
111C42×C37C2×D411yescharacteristic
112C42×C37C2×D411yescharacteristic
113C42×C37C2×D411yescharacteristic
114C42×C37#4111yescharacteristic
115C23×C37—41no—
116C23×C37#12011yescharacteristic
117Q8×C37—42no—
118Q8×C37—82no—
119Q8×C37—82no—
120Q8×C37—82no—
121Q8×C37—81no—
122Q8×C37—81no—
123D4×C37—42no—
124D4×C37—42no—
125D4×C37—42no—
126D4×C37—82no—
127C2×C4×C37—22no—
128C2×C4×C37—22no—
129C2×C4×C37—24no—
130C2×C4×C37—42no—
131C2×C4×C37—42no—
132C2×C4×C37—44no—
133C2×C4×C37—41no—
134C2×C4×C37—42no—
135C2×C4×C37—42no—
136C2×C4×C37—44no—
137C2×C4×C37—81no—
138C2×C4×C37—81no—
139C2×C4×C37#12011yescharacteristic
140C2×C4×C37#12011yescharacteristic
141C2×C4×C37#12111yescharacteristic
142C2×C4×C37#12111yescharacteristic
143C2×C4×C37#12111yescharacteristic
144C2×C4×C37#12711yescharacteristic
145C2×C4×C37#13512yes—
146C2×C4×C37AGL(1,ℤ/8)12yes—
147C8×C37—42no—
148C8×C37—42no—
149C8×C37—81no—
150#10286C3711yesmaximal, characteristic, direct product factor, Sylow
151C22×C37—21no—
152C22×C37—21no—
153C22×C37—81no—
154C22×C37—81no—
155C22×C37—81no—
156C22×C37#39211yescharacteristic
157C22×C37#44912yes—
158C4×C37—21no—
159C4×C37—21no—
160C4×C37—44no—
161C4×C37—82no—
162C4×C37—82no—
163C4×C37—81no—
164C4×C37#45412yes—
165C4×C37#45612yes—
166C4×C37#46312yes—
167C4×C37#49511yescharacteristic
168C4×C37#49611yescharacteristic
169#3227C2×C3711yescharacteristic
170#3226C2×C3711yescharacteristic
171#2862C2×C3711yescharacteristic, complemented
172#1962C2×C3711yescharacteristic
173#1549C2×C3711yescharacteristic, complemented
174#1546C2×C3711yescharacteristic, complemented
175#1276C2×C3711yescharacteristic, complemented
176C2×C37—22no—
177C2×C37—161no—
178C2×C37#200311yescharacteristic
179C2×C37#200511yescharacteristic
180C2×C37#201211yescharacteristic
181#532C22×C3711yescharacteristic
182C2×C4⋊Q8C22×C3711yescharacteristic
183C2×C4⋊Q8C22×C3711yescharacteristic
184#500—21no—
185#498—21no—
186#486—21no—
187#486—21no—
188#481—22no—
189#480—22no—
190#476—22no—
191#474—22no—
192#431C22×C3711yescharacteristic
193#431C22×C3711yescharacteristic
194#423C22×C3711yescharacteristic
195C2×#104—22no—
196#391—22no—
197#391—22no—
198#382—22no—
199C43C22×C3711yescharacteristic
200#332—22no—
201#332—22no—
202C37#1028611yescharacteristic, direct product factor, Sylow
203C2×#41—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C2×M16—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8D4×C3711yescharacteristic
216#127D4×C3711yescharacteristic
217#124—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4D4×C3712yes—
225C2×C4⋊C4D4×C3712yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42D4×C3712yes—
229C2×C42C23×C3711yescharacteristic
230#107—24no—
231#106—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8D4×C3712yes—
235C4⋊C8D4×C3712yes—
236#104—44no—
237#102—24no—
238#102—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—28no—
244#41—42no—
245#41—42no—
246C2×Q8—21no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×D4×C3711yescharacteristic
263C22×C4C2×D4×C3711yescharacteristic
264C22×C4C2×D4×C3711yescharacteristic
265M16—42no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42C2×D4×C3711yescharacteristic
286C42C2×D4×C3711yescharacteristic
287C42C2×D4×C3711yescharacteristic
288C42#41×C3711yescharacteristic
289C23—41no—
290C23#120×C3711yescharacteristic
291Q8—42no—
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#120×C3711yescharacteristic
314C2×C4#120×C3711yescharacteristic
315C2×C4#121×C3711yescharacteristic
316C2×C4#121×C3711yescharacteristic
317C2×C4#121×C3711yescharacteristic
318C2×C4#127×C3711yescharacteristic
319C2×C4#135×C3712yes—
320C2×C4AGL(1,ℤ/8)×C3712yes—
321C8—42no—
322C8—42no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C37×#39211yescharacteristic
330C22C37×#44912yes—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—82no—
335C4—82no—
336C4—81no—
337C4C37×#45412yes—
338C4C37×#45612yes—
339C4C37×#46312yes—
340C4C37×#49511yescharacteristic
341C4C37×#49611yescharacteristic
342C2—22no—
343C2—161no—
344C2C37×#200311yescharacteristic
345C2C37×#200511yescharacteristic
346C2C37×#201211yescharacteristic
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
3. C37×#2862C2semidirect
5. C37×#1549C2semidirect
6. C37×#1546C2semidirect
7. C37×#1276C2semidirect
150. #10286C37direct (N × H)
171. #2862C2×C37semidirect
173. #1549C2×C37semidirect
174. #1546C2×C37semidirect
175. #1276C2×C37semidirect
202. C37#10286direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32272
C37×#32262
C37×#28622
C37×#19622
C37×#15492
C37×#15462
C37×#12762
#1028637