Finite Group Databasebeta

← C137×#12094 #46861480

Subgroups — C137×#12094 #46861480

Order 35072.

570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 74 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#2815×C137C211yesmaximal, characteristic, complemented
2#2022×C137C211yesmaximal, characteristic, complemented
3#1595×C137C212yesmaximal, complemented
4#1343×C137C212yesmaximal, complemented
5#1317×C137C211yesmaximal, characteristic, complemented
6#534×C137C2211yescharacteristic
7C2×#123×C137—21no—
8C2×#121×C137—21no—
9#447×C137—22no—
10C2×#99×C137—21no—
11#397×C137—22no—
12#392×C137—22no—
13#392×C137—21no—
14#392×C137C2212yes—
15#381×C137C2211yescharacteristic
16#342×C137C2211yescharacteristic
17#327×C137—22no—
18#323×C137C2212yes—
19C22×D4×C137—21no—
20C22×D4×C137—21no—
21C22×D4×C137—22no—
22C22×D4×C137—21no—
23C22×D4×C137D412yes—
24C23×C4×C137D411yescharacteristic
25C2×D8×C137—42no—
26#123×C137—42no—
27#123×C137—41no—
28#123×C137—42no—
29#121×C137—22no—
30#121×C137—42no—
31#121×C137—42no—
32#120×C137—41no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—41no—
35C2×C22⋊C4×C137—21no—
36C2×C22⋊C4×C137—22no—
37C2×C22⋊C4×C137—22no—
38C2×C22⋊C4×C137—22no—
39C2×C22⋊C4×C137—41no—
40C2×C22⋊C4×C137D411yescharacteristic
41C2×C22⋊C4×C137C2311yescharacteristic
42#102×C137—42no—
43#99×C137—42no—
44C22⋊C8×C137—22no—
45#95×C137—21no—
46#95×C137—22no—
47#95×C137D412yes—
48C24×C137—21no—
49C24×C137—22no—
50C24×C137C2×D411yescharacteristic
51C2×D4×C137—22no—
52C2×D4×C137—42no—
53C2×D4×C137—42no—
54C2×D4×C137—41no—
55C2×D4×C137—41no—
56C2×D4×C137—41no—
57C2×D4×C137—41no—
58C2×D4×C137—42no—
59C2×D4×C137—41no—
60C2×D4×C137—42no—
61C2×D4×C137—41no—
62C2×D4×C137—82no—
63C2×D4×C137—81no—
64C2×D4×C137—81no—
65C2×D4×C137—82no—
66C2×D4×C137—82no—
67C2×D4×C137—82no—
68C2×D4×C137—81no—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—21no—
73C22×C4×C137—22no—
74C22×C4×C137—41no—
75C22×C4×C137—41no—
76C22×C4×C137—41no—
77C22×C4×C137D812yescomplemented
78C22×C4×C137C2×D412yes—
79D8×C137—84no—
80C2×C8×C137—42no—
81C4⋊C4×C137—22no—
82C4⋊C4×C137—42no—
83C4⋊C4×C137—44no—
84C22⋊C4×C137—22no—
85C22⋊C4×C137—41no—
86C22⋊C4×C137—44no—
87C22⋊C4×C137—42no—
88C22⋊C4×C137—44no—
89C22⋊C4×C137—41no—
90C22⋊C4×C137—41no—
91C22⋊C4×C137—82no—
92C22⋊C4×C137—82no—
93C23×C137—21no—
94C23×C137—21no—
95C23×C137—42no—
96C23×C137—41no—
97C23×C137—42no—
98C23×C137—42no—
99C23×C137—42no—
100C23×C137—42no—
101C23×C137—41no—
102C23×C137—42no—
103C23×C137—42no—
104C23×C137—41no—
105C23×C137—42no—
106C23×C137—82no—
107C23×C137#12011yescharacteristic
108C23×C137C2×D811yescharacteristic
109C23×C137AGL(1,ℤ/8)11yescharacteristic
110D4×C137—44no—
111D4×C137—82no—
112D4×C137—82no—
113D4×C137—84no—
114D4×C137—81no—
115D4×C137—81no—
116D4×C137—84no—
117D4×C137—84no—
118D4×C137—84no—
119D4×C137—81no—
120D4×C137—81no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—41no—
124C2×C4×C137—41no—
125C2×C4×C137—42no—
126C2×C4×C137—42no—
127C2×C4×C137—42no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—42no—
134C2×C4×C137—41no—
135C2×C4×C137—41no—
136C2×C4×C137—81no—
137C2×C4×C137—82no—
138C8×C137—82no—
139C22×C137—22no—
140C22×C137—21no—
141C22×C137—21no—
142C22×C137—41no—
143C22×C137—41no—
144C22×C137—42no—
145C22×C137—81no—
146C22×C137—82no—
147C22×C137—82no—
148C22×C137—82no—
149C22×C137—82no—
150C22×C137—82no—
151C22×C137—82no—
152C22×C137—81no—
153C22×C137—82no—
154C22×C137—82no—
155C22×C137—82no—
156C22×C137—161no—
157C22×C137#44711yescharacteristic
158C22×C137#45311yescharacteristic
159C22×C137B(4,𝔽2)11yescharacteristic
160C4×C137—42no—
161C4×C137—42no—
162C4×C137—82no—
163C4×C137—81no—
164C4×C137—82no—
165C4×C137—81no—
166C2×C137—21no—
167C2×C137—21no—
168C2×C137—41no—
169C2×C137—41no—
170C2×C137—162no—
171C2×C137—161no—
172C2×C137#159511yescharacteristic
173C2×C137#220012yes—
174#12094C13711yesmaximal, characteristic, direct product factor, Sylow
175C137#1209411yescharacteristic, direct product factor, Sylow
176#2815C2×C13711yescharacteristic, complemented
177#2022C2×C13711yescharacteristic, complemented
178#1595C2×C13712yescomplemented
179#1343C2×C13712yescomplemented
180#1317C2×C13711yescharacteristic, complemented
181#534C22×C13711yescharacteristic
182C2×#123—21no—
183C2×#121—21no—
184#447—22no—
185C2×#99—21no—
186#397—22no—
187#392—22no—
188#392—21no—
189#392C22×C13712yes—
190#381C22×C13711yescharacteristic
191#342C22×C13711yescharacteristic
192#327—22no—
193#323C22×C13712yes—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—21no—
198C22×D4D4×C13712yes—
199C23×C4D4×C13711yescharacteristic
200C2×D8—42no—
201#123—42no—
202#123—41no—
203#123—42no—
204#121—22no—
205#121—42no—
206#121—42no—
207#120—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4D4×C13711yescharacteristic
216C2×C22⋊C4C23×C13711yescharacteristic
217#102—42no—
218#99—42no—
219C22⋊C8—22no—
220#95—21no—
221#95—22no—
222#95D4×C13712yes—
223C24—21no—
224C24—22no—
225C24C2×D4×C13711yescharacteristic
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4D8×C13712yescomplemented
253C22×C4C2×D4×C13712yes—
254D8—84no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23—82no—
282C23#120×C13711yescharacteristic
283C23C2×D8×C13711yescharacteristic
284C23AGL(1,ℤ/8)×C13711yescharacteristic
285D4—44no—
286D4—82no—
287D4—82no—
288D4—84no—
289D4—81no—
290D4—81no—
291D4—84no—
292D4—84no—
293D4—84no—
294D4—81no—
295D4—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C8—82no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—81no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—161no—
332C22#447×C13711yescharacteristic
333C22#453×C13711yescharacteristic
334C22B(4,𝔽2)×C13711yescharacteristic
335C4—42no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—82no—
340C4—81no—
341C2—21no—
342C2—21no—
343C2—41no—
344C2—41no—
345C2—162no—
346C2—161no—
347C2#1595×C13711yescharacteristic
348C2#2200×C13712yes—
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C137
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C22⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2815×C137C2semidirect
2. #2022×C137C2semidirect
3. #1595×C137C2semidirect
4. #1343×C137C2semidirect
5. #1317×C137C2semidirect
77. C22×C4×C137D8semidirect
174. #12094C137direct (N × H)
175. C137#12094direct (N × H)
176. #2815C2×C137semidirect
177. #2022C2×C137semidirect
178. #1595C2×C137semidirect
179. #1343C2×C137semidirect
180. #1317C2×C137semidirect
252. C22×C4D8×C137semidirect

Maximal subgroups

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

subgroupindex
#2815×C1372
#2022×C1372
#1595×C1372
#1343×C1372
#1317×C1372
#12094137