Finite Group Databasebeta

← C137×#9974 #46859360

Subgroups — C137×#9974 #46859360

Order 35072.

768 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 102 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#3186×C137C211yesmaximal, characteristic, complemented
2#3006×C137C212yesmaximal, complemented
3#1780×C137C211yesmaximal, characteristic, complemented
4#1471×C137C212yesmaximal, complemented
5#1293×C137C211yesmaximal, characteristic, complemented
6C22×#41×C137C2211yescharacteristic
7C2×AGL(1,ℤ/8)×C137—22no—
8C2×#121×C137C2212yes—
9#480×C137—22no—
10#466×C137—22no—
11#449×C137—24no—
12#448×C137—22no—
13#447×C137—22no—
14#447×C137—22no—
15#417×C137—22no—
16#407×C137C2212yes—
17C2×C22⋊C8×C137—22no—
18C2×C22⋊C8×C137C2211yescharacteristic
19C2×C22⋊C8×C137C2211yescharacteristic
20#323×C137—24no—
21C2×#41×C137—22no—
22C2×#41×C137—22no—
23C2×#41×C137—44no—
24C22×Q8×C137D411yescharacteristic
25C22×D4×C137—21no—
26C22×D4×C137—22no—
27C22×D4×C137D411yescharacteristic
28C22×D4×C137D412yes—
29C23×C4×C137—21no—
30C23×C4×C137C2311yescharacteristic
31AGL(1,ℤ/8)×C137—48no—
32C2×SD16×C137—24no—
33C2×D8×C137—24no—
34C2×D8×C137—22no—
35C2×M16×C137—22no—
36C22×C8×C137—21no—
37C22×C8×C137D412yes—
38#121×C137—22no—
39#121×C137—22no—
40#121×C137—22no—
41#121×C137—22no—
42C2×C4⋊C4×C137D412yes—
43C2×C22⋊C4×C137—22no—
44#107×C137—22no—
45#103×C137—24no—
46#102×C137—24no—
47#102×C137—24no—
48C22⋊C8×C137—22no—
49C22⋊C8×C137—22no—
50C22⋊C8×C137—22no—
51C22⋊C8×C137—44no—
52C22⋊C8×C137D414yes—
53C24×C137—21no—
54C24×C137—22no—
55C24×C137#4111yescharacteristic
56#41×C137—28no—
57#41×C137—44no—
58#41×C137—44no—
59#41×C137—44no—
60C2×Q8×C137—22no—
61C2×Q8×C137—42no—
62C2×D4×C137—22no—
63C2×D4×C137—22no—
64C2×D4×C137—22no—
65C2×D4×C137—42no—
66C2×D4×C137—44no—
67C2×D4×C137—42no—
68C2×D4×C137—42no—
69C2×D4×C137—42no—
70C2×D4×C137—42no—
71C2×D4×C137—42no—
72C2×D4×C137—84no—
73C22×C4×C137—24no—
74C22×C4×C137—21no—
75C22×C4×C137—22no—
76C22×C4×C137—42no—
77C22×C4×C137—42no—
78C22×C4×C137—41no—
79C22×C4×C137—41no—
80C22×C4×C137C2×D412yes—
81C22×C4×C137C2×D411yescharacteristic
82C22×C4×C137C2×D411yescharacteristic
83C22×C4×C137C2×D412yes—
84SD16×C137—48no—
85D8×C137—48no—
86D8×C137—82no—
87M16×C137—44no—
88C2×C8×C137—22no—
89C2×C8×C137—24no—
90C2×C8×C137—42no—
91C2×C8×C137—42no—
92C2×C8×C137—41no—
93C4⋊C4×C137—22no—
94C4⋊C4×C137—22no—
95C22⋊C4×C137—42no—
96C22⋊C4×C137—42no—
97C23×C137—22no—
98C23×C137—22no—
99C23×C137—41no—
100C23×C137—41no—
101C23×C137—41no—
102C23×C137—42no—
103C23×C137—44no—
104C23×C137—41no—
105C23×C137—42no—
106C23×C137—42no—
107C23×C137—82no—
108C23×C137—82no—
109C23×C137#12112yes—
110C23×C137#12111yescharacteristic
111Q8×C137—24no—
112Q8×C137—42no—
113D4×C137—24no—
114D4×C137—42no—
115D4×C137—44no—
116D4×C137—48no—
117D4×C137—44no—
118D4×C137—84no—
119D4×C137—84no—
120C2×C4×C137—22no—
121C2×C4×C137—24no—
122C2×C4×C137—22no—
123C2×C4×C137—41no—
124C2×C4×C137—42no—
125C2×C4×C137—44no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—44no—
129C2×C4×C137—42no—
130C2×C4×C137—82no—
131C2×C4×C137#12012yes—
132C2×C4×C137#12011yescharacteristic
133C2×C4×C137#12711yescharacteristic
134C8×C137—44no—
135C8×C137—42no—
136C8×C137—81no—
137C22×C137—21no—
138C22×C137—21no—
139C22×C137—22no—
140C22×C137—22no—
141C22×C137—22no—
142C22×C137—44no—
143C22×C137—42no—
144C22×C137—82no—
145C22×C137—82no—
146C22×C137—81no—
147C22×C137—81no—
148C22×C137—82no—
149C22×C137—82no—
150C22×C137—82no—
151C22×C137—164no—
152C22×C137#39211yescharacteristic
153C22×C137#47212yes—
154C4×C137—22no—
155C4×C137—21no—
156C4×C137—42no—
157C4×C137—82no—
158C4×C137B(4,𝔽2)12yes—
159C2×C137—22no—
160C2×C137—42no—
161C2×C137—82no—
162C2×C137—162no—
163C2×C137#201711yescharacteristic
164C2×C137#201811yescharacteristic
165C2×C137#202111yescharacteristic
166#9974C13711yesmaximal, characteristic, direct product factor, Sylow
167C137#997411yescharacteristic, direct product factor, Sylow
168#3186C2×C13711yescharacteristic, complemented
169#3006C2×C13712yescomplemented
170#1780C2×C13711yescharacteristic, complemented
171#1471C2×C13712yescomplemented
172#1293C2×C13711yescharacteristic, complemented
173C22×#41C22×C13711yescharacteristic
174C2×AGL(1,ℤ/8)—22no—
175C2×#121C22×C13712yes—
176#480—22no—
177#466—22no—
178#449—24no—
179#448—22no—
180#447—22no—
181#447—22no—
182#417—22no—
183#407C22×C13712yes—
184C2×C22⋊C8—22no—
185C2×C22⋊C8C22×C13711yescharacteristic
186C2×C22⋊C8C22×C13711yescharacteristic
187#323—24no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—44no—
191C22×Q8D4×C13711yescharacteristic
192C22×D4—21no—
193C22×D4—22no—
194C22×D4D4×C13712yes—
195C22×D4D4×C13711yescharacteristic
196C23×C4—21no—
197C23×C4C23×C13711yescharacteristic
198AGL(1,ℤ/8)—48no—
199C2×SD16—24no—
200C2×D8—24no—
201C2×D8—22no—
202C2×M16—22no—
203C22×C8—21no—
204C22×C8D4×C13712yes—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209C2×C4⋊C4D4×C13712yes—
210C2×C22⋊C4—22no—
211#107—22no—
212#103—24no—
213#102—24no—
214#102—24no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—44no—
219C22⋊C8D4×C13714yes—
220C24—21no—
221C24—22no—
222C24#41×C13711yescharacteristic
223#41—28no—
224#41—44no—
225#41—44no—
226#41—44no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—84no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4C2×D4×C13712yes—
248C22×C4C2×D4×C13711yescharacteristic
249C22×C4C2×D4×C13711yescharacteristic
250C22×C4C2×D4×C13712yes—
251SD16—48no—
252D8—48no—
253D8—82no—
254M16—44no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—41no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—44no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—82no—
275C23—82no—
276C23#121×C13712yes—
277C23#121×C13711yescharacteristic
278Q8—24no—
279Q8—42no—
280D4—24no—
281D4—42no—
282D4—44no—
283D4—48no—
284D4—44no—
285D4—84no—
286D4—84no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4#120×C13712yes—
299C2×C4#120×C13711yescharacteristic
300C2×C4#127×C13711yescharacteristic
301C8—44no—
302C8—42no—
303C8—81no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—44no—
310C22—42no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22—81no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—164no—
319C22#392×C13711yescharacteristic
320C22#472×C13712yes—
321C4—22no—
322C4—21no—
323C4—42no—
324C4—82no—
325C4B(4,𝔽2)×C13712yes—
326C2—22no—
327C2—42no—
328C2—82no—
329C2—162no—
330C2#2017×C13711yescharacteristic
331C2#2018×C13711yescharacteristic
332C2#2021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#392
commutator, Frattini, cosocleC23×C4C23×C137
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3186×C137C2semidirect
2. #3006×C137C2semidirect
3. #1780×C137C2semidirect
4. #1471×C137C2semidirect
5. #1293×C137C2semidirect
166. #9974C137direct (N × H)
167. C137#9974direct (N × H)
168. #3186C2×C137semidirect
169. #3006C2×C137semidirect
170. #1780C2×C137semidirect
171. #1471C2×C137semidirect
172. #1293C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3186×C1372
#3006×C1372
#1780×C1372
#1471×C1372
#1293×C1372
#9974137