Finite Group Databasebeta

← C37×#13227 #22634480

Subgroups — C37×#13227 #22634480

Order 9472.

472 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C37×#2546C211yesmaximal, characteristic
2C2×C37×#356C214yesmaximal
3C37×#1801C211yesmaximal, characteristic
4C4×#101×C37C211yesmaximal, characteristic
5C2×C4⋊Q8×C37C411yescharacteristic
6C2×C4⋊Q8×C37C2212yes—
7C2×C4×Q8×C37C2211yescharacteristic
8C37×#423—21no—
9C2×#101×C37C2212yes—
10C2×#101×C37C2212yes—
11C4×M16×C37—21no—
12C37×#398—21no—
13C37×#398—21no—
14C37×#391C412yes—
15C37×#389—21no—
16C37×#389—21no—
17C4×C4⋊C4×C37C411yescharacteristic
18C37×#356—28no—
19C37×#343—21no—
20C37×#343—21no—
21C22×Q8×C37C2×C412yes—
22C22×Q8×C37C2311yescharacteristic
23C2×M16×C37—22no—
24C2×M16×C37—22no—
25C4⋊Q8×C37—24no—
26C4⋊Q8×C37—24no—
27C4⋊Q8×C37—41no—
28C4⋊Q8×C37—41no—
29C4×Q8×C37—41no—
30C4×Q8×C37—41no—
31C2×C4⋊C4×C37—21no—
32C2×C4⋊C4×C37—21no—
33C2×C4⋊C4×C37—21no—
34C2×C4⋊C4×C37—22no—
35C2×C4⋊C4×C37—22no—
36C2×C4⋊C4×C37C2×C411yescharacteristic
37C2×C4⋊C4×C37D411yescharacteristic
38C2×C4⋊C4×C37D411yescharacteristic
39C2×C42×C37C2×C412yes—
40C2×C42×C37C2×C411yescharacteristic
41C2×C42×C37D411yescharacteristic
42C2×C42×C37D411yescharacteristic
43C4⋊C8×C37—41no—
44C4⋊C8×C37—41no—
45#101×C37—22no—
46#101×C37—24no—
47#101×C37D414yes—
48#97×C37—41no—
49C4×C8×C37—41no—
50#95×C37—21no—
51#95×C37—22no—
52#95×C37—22no—
53C2×Q8×C37—24no—
54C2×Q8×C37—41no—
55C2×Q8×C37—41no—
56C2×Q8×C37—42no—
57C2×Q8×C37—42no—
58C2×Q8×C37C2×D412yes—
59C2×Q8×C37#4112yes—
60C22×C4×C37—22no—
61C22×C4×C37—22no—
62C22×C4×C37C22⋊C411yescharacteristic
63C22×C4×C37C22⋊C411yescharacteristic
64C22×C4×C37C22⋊C411yescharacteristic
65C22×C4×C37C22⋊C411yescharacteristic
66C22×C4×C37C22×C411yescharacteristic
67C22×C4×C37C2×D411yescharacteristic
68C22×C4×C37C2×D411yescharacteristic
69M16×C37—24no—
70M16×C37—42no—
71M16×C37—42no—
72M16×C37—42no—
73C2×C8×C37—42no—
74C2×C8×C37—42no—
75C4⋊C4×C37—24no—
76C4⋊C4×C37—41no—
77C4⋊C4×C37—41no—
78C4⋊C4×C37—41no—
79C4⋊C4×C37—42no—
80C4⋊C4×C37—41no—
81C4⋊C4×C37—41no—
82C4⋊C4×C37—44no—
83C4⋊C4×C37—42no—
84C4⋊C4×C37—42no—
85C42×C37—21no—
86C42×C37—21no—
87C42×C37—21no—
88C42×C37—21no—
89C42×C37—21no—
90C42×C37—21no—
91C42×C37—24no—
92C23×C37C2×C22⋊C411yescharacteristic
93Q8×C37—42no—
94Q8×C37—42no—
95Q8×C37—84no—
96C2×C4×C37—21no—
97C2×C4×C37—21no—
98C2×C4×C37—21no—
99C2×C4×C37—21no—
100C2×C4×C37—21no—
101C2×C4×C37—21no—
102C2×C4×C37—22no—
103C2×C4×C37—24no—
104C2×C4×C37—21no—
105C2×C4×C37—21no—
106C2×C4×C37—21no—
107C2×C4×C37—42no—
108C2×C4×C37—42no—
109C2×C4×C37—41no—
110C2×C4×C37—41no—
111C2×C4×C37—41no—
112C2×C4×C37—42no—
113C2×C4×C37—42no—
114C2×C4×C37—42no—
115C2×C4×C37#9911yescharacteristic
116C2×C4×C37#9911yescharacteristic
117C2×C4×C37C4×D412yes—
118C2×C4×C37#12112yes—
119C2×C4×C37#12411yescharacteristic
120C2×C4×C37#12711yescharacteristic
121C8×C37—44no—
122C8×C37—82no—
123#13227C3711yesmaximal, characteristic, direct product factor, Sylow
124C22×C37—21no—
125C22×C37—21no—
126C22×C37#39011yescharacteristic
127C22×C37C2×#9911yescharacteristic
128C22×C37#41011yescharacteristic
129C4×C37—21no—
130C4×C37—22no—
131C4×C37—21no—
132C4×C37—22no—
133C4×C37—21no—
134C4×C37—44no—
135C4×C37—41no—
136C4×C37—41no—
137C4×C37—41no—
138C4×C37—82no—
139C4×C37#35612yescomplemented
140#2546C2×C3711yescharacteristic
141C2×#356C2×C3714yes—
142#1801C2×C3711yescharacteristic
143C4×#101C2×C3711yescharacteristic
144C2×C37—21no—
145C2×C37—21no—
146C2×C37#196411yescharacteristic
147C2×C37C2×#35611yescharacteristic
148C2×C37#213511yescharacteristic
149C2×C4⋊Q8C4×C3711yescharacteristic
150C2×C4⋊Q8C22×C3712yes—
151C2×C4×Q8C22×C3711yescharacteristic
152#423—21no—
153C2×#101C22×C3712yes—
154C2×#101C22×C3712yes—
155C4×M16—21no—
156#398—21no—
157#398—21no—
158#391C4×C3712yes—
159#389—21no—
160#389—21no—
161C4×C4⋊C4C4×C3711yescharacteristic
162#356—28no—
163#343—21no—
164#343—21no—
165C37#1322711yescharacteristic, direct product factor, Sylow
166C22×Q8C2×C4×C3712yes—
167C22×Q8C23×C3711yescharacteristic
168C2×M16—22no—
169C2×M16—22no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×C4×C3711yescharacteristic
182C2×C4⋊C4D4×C3711yescharacteristic
183C2×C4⋊C4D4×C3711yescharacteristic
184C2×C42C2×C4×C3712yes—
185C2×C42C2×C4×C3711yescharacteristic
186C2×C42D4×C3711yescharacteristic
187C2×C42D4×C3711yescharacteristic
188C4⋊C8—41no—
189C4⋊C8—41no—
190#101—22no—
191#101—24no—
192#101D4×C3714yes—
193#97—41no—
194C4×C8—41no—
195#95—21no—
196#95—22no—
197#95—22no—
198C2×Q8—24no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8C2×D4×C3712yes—
204C2×Q8#41×C3712yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C22⋊C4×C3711yescharacteristic
208C22×C4C22⋊C4×C3711yescharacteristic
209C22×C4C22⋊C4×C3711yescharacteristic
210C22×C4C22⋊C4×C3711yescharacteristic
211C22×C4C22×C4×C3711yescharacteristic
212C22×C4C2×D4×C3711yescharacteristic
213C22×C4C2×D4×C3711yescharacteristic
214M16—24no—
215M16—42no—
216M16—42no—
217M16—42no—
218C2×C8—42no—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—24no—
237C23C2×C22⋊C4×C3711yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#99×C3711yescharacteristic
261C2×C4#99×C3711yescharacteristic
262C2×C4C4×D4×C3712yes—
263C2×C4#121×C3712yes—
264C2×C4#124×C3711yescharacteristic
265C2×C4#127×C3711yescharacteristic
266C8—44no—
267C8—82no—
268C22—21no—
269C22—21no—
270C22C37×#39011yescharacteristic
271C22C2×#99×C3711yescharacteristic
272C22C37×#41011yescharacteristic
273C4—21no—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4C37×#35612yescomplemented
284C2—21no—
285C2—21no—
286C2C37×#196411yescharacteristic
287C2C2×C37×#35611yescharacteristic
288C2C37×#213511yescharacteristic
subgroupG/H
center, socleC22×C37C2×#99
commutatorC22×C4C22×C4×C37
Frattini, cosocleC22×Q8C23×C37
prime radicalC23×C37C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C2×C4×Q8×C37 ◁ G

Complements

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

NH ≅ G/Ntype
123. #13227C37direct (N × H)
139. C4×C37#356semidirect
165. C37#13227direct (N × H)
283. C4C37×#356semidirect

Maximal subgroups

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

subgroupindex
C37×#25462
C2×C37×#3562
C37×#18012
C4×#101×C372
#1322737