Subgroups — #34061466

Order 19072.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750705	C₂	1	1	yes	maximal, characteristic, complemented
2	#23750704	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750671	C₂	1	2	yes	maximal, complemented
4	#23750670	C₂	1	2	yes	maximal, complemented
5	#23750669	C₂	1	2	yes	maximal, complemented
6	#23750668	C₂	1	2	yes	maximal, complemented
7	#23750640	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#18011976	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₄₉⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#10597010	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750424	C₂	1	1	yes	maximal, characteristic
12	#18011999	—	2	1	no	—
13	#18011999	—	2	1	no	—
14	#18011999	C₂²	1	2	yes	complemented
15	#18011999	C₂²	1	2	yes	complemented
16	#18011999	C₂²	1	2	yes	complemented
17	#18011998	—	2	2	no	—
18	#18011998	C₂²	1	2	yes	complemented
19	C₂×#14558746	—	2	1	no	—
20	C₂×#14558746	C₂²	1	1	yes	characteristic
21	D₁₄₉⋊M₁₆	—	2	1	no	—
22	D₁₄₉⋊M₁₆	—	2	2	no	—
23	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
24	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10597010	C₂²	1	1	yes	characteristic
26	#18011982	C₂²	1	2	yes	complemented
27	#18011981	C₂²	1	2	yes	complemented
28	#18011976	C₂²	1	2	yes	complemented
29	#18011976	C₂²	1	1	yes	characteristic, complemented
30	#18011976	C₂²	1	2	yes	complemented
31	#18011976	C₂²	1	1	yes	characteristic, complemented
32	#18011975	C₂²	1	1	yes	characteristic, complemented
33	#18011975	C₂²	1	1	yes	characteristic, complemented
34	#18011965	C₂²	1	2	yes	complemented
35	C₈×#10597010	C₂²	1	2	yes	complemented
36	C₂×#14558737	C₂²	1	1	yes	characteristic
37	#41×D₁₄₉	C₂²	1	1	yes	characteristic
38	#18011943	C₄	1	2	yes	complemented
39	#18011942	C₄	1	1	yes	characteristic, complemented
40	#18011939	C₄	1	1	yes	characteristic, complemented
41	#18011938	C₄	1	2	yes	complemented
42	#18011936	C₄	1	1	yes	characteristic, complemented
43	#18011934	C₂²	1	2	yes	—
44	#18011932	C₂²	1	2	yes	—
45	M₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
46	#137×C₁₄₉	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10597010	—	2	1	no	—
48	#14558746	—	2	2	no	—
49	#14558746	—	2	2	no	—
50	#14558746	—	2	2	no	—
51	#14558746	—	2	1	no	—
52	#14558746	D₄	1	1	yes	characteristic
53	#14558746	D₄	1	1	yes	characteristic
54	#14558746	C₂³	1	1	yes	characteristic
55	#14558746	C₂³	1	1	yes	characteristic
56	#14558746	C₂³	1	2	yes	—
57	#14558745	—	2	2	no	—
58	#14558745	—	2	1	no	—
59	#14558745	D₄	1	2	yes	—
60	D₁₄₉⋊C₈	—	2	2	no	—
61	D₁₄₉⋊C₈	—	2	2	no	—
62	D₁₄₉⋊C₈	—	2	2	no	—
63	D₁₄₉⋊C₈	—	2	1	no	—
64	D₁₄₉⋊C₈	C₂³	1	2	yes	—
65	C₂×#11924753	—	2	2	no	—
66	C₂×#11924753	—	2	1	no	—
67	C₄×#10597010	—	2	1	no	—
68	C₄×#10597010	C₂³	1	1	yes	characteristic
69	C₄×#10597010	C₂³	1	1	yes	characteristic
70	#14558737	—	2	1	no	—
71	#14558737	—	2	1	no	—
72	#14558737	C₂×C₄	1	2	yes	—
73	#14558737	C₂³	1	2	yes	—
74	#14558737	C₂³	1	1	yes	characteristic
75	#14558736	—	2	1	no	—
76	#14558736	C₂×C₄	1	1	yes	characteristic
77	#14558735	—	2	1	no	—
78	#14558735	C₂³	1	1	yes	characteristic
79	Q₈×D₁₄₉	C₂×C₄	1	1	yes	characteristic
80	C₂×D₅₉₆	—	2	1	no	—
81	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
82	D₄×D₁₄₉	—	2	1	no	—
83	D₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₄₉	—	2	1	no	—
85	C₂×C₄×D₁₄₉	—	2	1	no	—
86	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
87	#14558726	C₂×C₄	1	2	yes	complemented
88	#14558725	C₂×C₄	1	2	yes	complemented
89	#14558723	C₂×C₄	1	2	yes	complemented
90	#14558723	C₂×C₄	1	2	yes	complemented
91	D₁₁₉₂	C₂×C₄	1	2	yes	complemented
92	#14558721	C₂×C₄	1	2	yes	complemented
93	#14558720	C₂×C₄	1	1	yes	characteristic
94	#14558719	C₂×C₄	1	2	yes	—
95	C₈×D₁₄₉	C₂³	1	2	yes	—
96	#41×C₁₄₉	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
98	Q₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic
101	C₂×#10597010	—	2	1	no	—
102	C₂×#10597010	—	2	1	no	—
103	C₂×#10597010	—	4	2	no	—
104	#11924753	—	2	2	no	—
105	#11924753	—	2	1	no	—
106	#11924753	—	2	1	no	—
107	#11924753	C₂×D₄	1	2	yes	—
108	#11924753	C₂×D₄	1	2	yes	—
109	C₂²×D₁₄₉	—	2	1	no	—
110	C₂²×D₁₄₉	—	2	1	no	—
111	C₂²×D₁₄₉	#41	1	1	yes	characteristic
112	Dic₂₉₈	—	2	1	no	—
113	Dic₂₉₈	C₂²×C₄	1	1	yes	characteristic
114	D₅₉₆	—	2	2	no	—
115	D₅₉₆	—	2	1	no	—
116	D₅₉₆	—	2	2	no	—
117	D₅₉₆	—	2	1	no	—
118	D₅₉₆	C₂²×C₄	1	2	yes	—
119	D₅₉₆	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₄₉	—	2	1	no	—
121	C₂²⋊D₁₄₉	—	2	1	no	—
122	C₂²⋊D₁₄₉	—	2	1	no	—
123	C₂×Dic₁₄₉	—	2	1	no	—
124	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₄₉	—	2	1	no	—
126	C₄×D₁₄₉	—	2	1	no	—
127	C₄×D₁₄₉	—	2	2	no	—
128	C₄×D₁₄₉	—	2	1	no	—
129	C₄×D₁₄₉	—	2	2	no	—
130	C₄×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₄₉	#41	1	1	yes	characteristic
133	C₄×D₁₄₉	C₂⁴	1	1	yes	characteristic
134	#11924746	C₂²×C₄	1	2	yes	—
135	Q₈×C₁₄₉	—	2	1	no	—
136	Q₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
137	Q₈×C₁₄₉	C₂²×C₄	1	2	yes	—
138	D₄×C₁₄₉	—	2	1	no	—
139	D₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₄₉	—	2	1	no	—
141	C₂×C₄×C₁₄₉	—	2	1	no	—
142	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
143	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₄₉	—	2	1	no	—
145	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
146	C₂×D₁₄₉	—	2	1	no	—
147	C₂×D₁₄₉	—	2	2	no	—
148	C₂×D₁₄₉	—	4	1	no	—
149	C₂×D₁₄₉	—	8	1	no	—
150	C₂×D₁₄₉	—	8	1	no	—
151	C₂×D₁₄₉	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₄₉	C₂×#41	1	1	yes	characteristic
153	#10597010	—	2	2	no	—
154	#10597010	—	4	1	no	—
155	C₄×C₁₄₉	—	2	1	no	—
156	C₄×C₁₄₉	—	2	2	no	—
157	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
158	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
159	Dic₁₄₉	—	2	1	no	—
160	Dic₁₄₉	C₄×D₄	1	1	yes	characteristic
161	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₄₉	—	2	1	no	—
163	C₂×C₁₄₉	—	4	1	no	—
164	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
165	D₁₄₉	—	2	1	no	—
166	D₁₄₉	—	2	1	no	—
167	D₁₄₉	—	4	1	no	—
168	D₁₄₉	—	4	2	no	—
169	C₁₄₉	#2956	1	1	yes	characteristic, complemented, Sylow
170	#2956	—	149	1	no	maximal, Sylow
171	#575	—	149	1	no	—
172	#568	—	149	1	no	—
173	#568	—	149	1	no	—
174	#444	—	149	2	no	—
175	#444	—	149	2	no	—
176	#443	—	149	2	no	—
177	#443	—	149	2	no	—
178	#430	—	149	1	no	—
179	C₂×#104	—	149	1	no	—
180	C₂×#104	—	149	1	no	—
181	C₄×M₁₆	—	149	1	no	—
182	C₂×#41	—	149	1	no	—
183	C₂×#41	—	149	1	no	—
184	#137	—	149	1	no	—
185	#137	—	149	2	no	—
186	#137	#10597010	1	1	yes	characteristic, complemented
187	AGL(1,ℤ/8)	—	149	1	no	—
188	AGL(1,ℤ/8)	—	149	2	no	—
189	AGL(1,ℤ/8)	—	149	1	no	—
190	#135	—	149	2	no	—
191	#135	—	149	2	no	—
192	#131	—	298	2	no	—
193	#131	—	298	1	no	—
194	#131	—	298	1	no	—
195	#131	—	149	2	no	—
196	#131	—	149	2	no	—
197	#131	—	149	2	no	—
198	#131	—	149	2	no	—
199	C₂×M₁₆	—	298	2	no	—
200	C₂×M₁₆	—	298	1	no	—
201	C₂×M₁₆	—	298	1	no	—
202	C₂×M₁₆	—	149	1	no	—
203	C₂×M₁₆	—	149	1	no	—
204	C₂×M₁₆	—	149	1	no	—
205	C₂×M₁₆	—	149	1	no	—
206	C₂×C₄²	—	149	1	no	—
207	#108	—	149	2	no	—
208	#108	—	149	2	no	—
209	#104	—	149	1	no	—
210	#104	—	149	1	no	—
211	#104	—	149	2	no	—
212	#104	—	149	1	no	—
213	#104	—	149	2	no	—
214	#104	—	149	1	no	—
215	#97	—	149	2	no	—
216	C₄×C₈	—	149	2	no	—
217	#41	—	298	1	no	—
218	#41	—	298	1	no	—
219	#41	—	298	1	no	—
220	#41	—	298	1	no	—
221	#41	—	149	2	no	—
222	#41	—	149	1	no	—
223	#41	—	149	2	no	—
224	#41	—	149	1	no	—
225	#41	—	149	1	no	—
226	#41	C₂×#10597010	1	1	yes	characteristic
227	C₂×Q₈	—	149	1	no	—
228	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic
229	C₂×D₄	—	298	1	no	—
230	C₂×D₄	—	298	1	no	—
231	C₂×D₄	—	149	1	no	—
232	C₂×D₄	—	149	1	no	—
233	C₂²×C₄	—	298	1	no	—
234	C₂²×C₄	—	298	1	no	—
235	C₂²×C₄	—	298	1	no	—
236	C₂²×C₄	—	149	1	no	—
237	Q₁₆	—	149	2	no	—
238	Q₁₆	C₂×#10597010	1	2	yes	complemented
239	SD₁₆	—	149	2	no	—
240	SD₁₆	—	149	2	no	—
241	SD₁₆	—	149	2	no	—
242	SD₁₆	C₂×#10597010	1	2	yes	complemented
243	D₈	—	149	2	no	—
244	D₈	—	149	2	no	—
245	M₁₆	—	298	1	no	—
246	M₁₆	—	298	1	no	—
247	M₁₆	—	298	2	no	—
248	M₁₆	—	298	2	no	—
249	M₁₆	—	298	2	no	—
250	M₁₆	—	298	2	no	—
251	M₁₆	—	149	1	no	—
252	M₁₆	—	149	1	no	—
253	M₁₆	—	149	2	no	—
254	M₁₆	—	149	2	no	—
255	M₁₆	—	149	1	no	—
256	M₁₆	—	149	1	no	—
257	M₁₆	—	149	1	no	—
258	M₁₆	—	149	2	no	—
259	M₁₆	C₂×#10597010	1	1	yes	characteristic
260	C₂×C₈	—	298	1	no	—
261	C₂×C₈	—	298	1	no	—
262	C₂×C₈	—	298	2	no	—
263	C₂×C₈	—	298	2	no	—
264	C₂×C₈	—	298	2	no	—
265	C₂×C₈	—	298	2	no	—
266	C₂×C₈	—	149	2	no	—
267	C₂×C₈	—	149	2	no	—
268	C₄²	—	298	1	no	—
269	C₄²	—	149	1	no	—
270	C₄²	—	149	1	no	—
271	C₂³	—	298	1	no	—
272	C₂³	—	298	1	no	—
273	C₂³	—	149	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	298	1	no	—
276	Q₈	—	149	1	no	—
277	Q₈	C₂²×#10597010	1	1	yes	characteristic
278	Q₈	C₂²×#10597010	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	298	1	no	—
281	D₄	—	298	2	no	—
282	D₄	—	298	2	no	—
283	D₄	—	298	1	no	—
284	D₄	—	298	1	no	—
285	D₄	—	298	1	no	—
286	D₄	—	298	1	no	—
287	D₄	—	149	2	no	—
288	D₄	—	149	1	no	—
289	D₄	C₂²×#10597010	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	298	1	no	—
293	C₂×C₄	—	298	1	no	—
294	C₂×C₄	—	298	2	no	—
295	C₂×C₄	—	298	2	no	—
296	C₂×C₄	—	298	1	no	—
297	C₂×C₄	—	298	1	no	—
298	C₂×C₄	—	298	1	no	—
299	C₂×C₄	—	298	1	no	—
300	C₂×C₄	—	596	2	no	—
301	C₂×C₄	—	149	1	no	—
302	C₂×C₄	—	149	1	no	—
303	C₂×C₄	—	149	1	no	—
304	C₂×C₄	—	149	1	no	—
305	C₂×C₄	—	149	1	no	—
306	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
307	C₈	—	298	2	no	—
308	C₈	—	298	1	no	—
309	C₈	—	298	1	no	—
310	C₈	—	149	2	no	—
311	C₈	—	149	2	no	—
312	C₈	—	149	2	no	—
313	C₈	C₂²×#10597010	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	298	2	no	—
316	C₂²	—	298	1	no	—
317	C₂²	—	596	1	no	—
318	C₂²	—	149	1	no	—
319	C₂²	—	149	1	no	—
320	C₂²	—	1192	1	no	—
321	C₂²	—	1192	1	no	—
322	C₂²	D₄×#10597010	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	298	2	no	—
326	C₄	—	298	1	no	—
327	C₄	—	596	1	no	—
328	C₄	—	149	1	no	—
329	C₄	—	149	1	no	—
330	C₄	D₄×#10597010	1	1	yes	characteristic
331	C₄	C₂³×#10597010	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	298	1	no	—
335	C₂	—	298	1	no	—
336	C₂	—	596	2	no	—
337	C₂	—	596	1	no	—
338	C₂	C₂×D₄×#10597010	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10597010
commutator	C₄×C₁₄₉	C₂³×C₄
Frattini	C₄	C₂³×#10597010
Fitting	#137×C₁₄₉	C₄
socle	C₂×C₁₄₉	C₂×C₄×D₄
prime radical	#23750424	C₂
cosocle	C₄×D₁₄₉	C₂⁴
hypercenter	#137	#10597010
hypocenter	C₁₄₉	#2956
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #23750705	C₂	semidirect
2. #23750704	C₂	semidirect
3. #23750671	C₂	semidirect
4. #23750670	C₂	semidirect
5. #23750669	C₂	semidirect
6. #23750668	C₂	semidirect
7. #23750640	C₂	semidirect
8. C₂×#18011976	C₂	semidirect
9. D₁₄₉⋊#104	C₂	semidirect
10. M₁₆×#10597010	C₂	semidirect
14. #18011999	C₂²	semidirect
15. #18011999	C₂²	semidirect
16. #18011999	C₂²	semidirect
18. #18011998	C₂²	semidirect
26. #18011982	C₂²	semidirect
27. #18011981	C₂²	semidirect
28. #18011976	C₂²	semidirect
29. #18011976	C₂²	semidirect
30. #18011976	C₂²	semidirect
31. #18011976	C₂²	semidirect
32. #18011975	C₂²	semidirect
33. #18011975	C₂²	semidirect
34. #18011965	C₂²	semidirect
35. C₈×#10597010	C₂²	semidirect
38. #18011943	C₄	semidirect
39. #18011942	C₄	semidirect
40. #18011939	C₄	semidirect
41. #18011938	C₄	semidirect
42. #18011936	C₄	semidirect
46. #137×C₁₄₉	C₄	semidirect
87. #14558726	C₂×C₄	semidirect
88. #14558725	C₂×C₄	semidirect
89. #14558723	C₂×C₄	semidirect
90. #14558723	C₂×C₄	semidirect
91. D₁₁₉₂	C₂×C₄	semidirect
92. #14558721	C₂×C₄	semidirect
98. Q₁₆×C₁₄₉	C₂×C₄	semidirect
99. SD₁₆×C₁₄₉	C₂×C₄	semidirect
169. C₁₄₉	#2956	semidirect
186. #137	#10597010	semidirect
238. Q₁₆	C₂×#10597010	semidirect
242. SD₁₆	C₂×#10597010	semidirect

Maximal subgroups

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

subgroup	index
#23750705	2
#23750704	2
#23750671	2
#23750670	2
#23750669	2
#23750668	2
#23750640	2
C₂×#18011976	2
D₁₄₉⋊#104	2
M₁₆×#10597010	2
#23750424	2
#2956	149