Subgroups — #34061464

Order 19072.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011999	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₄₉⋊#131	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	#23750624	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750622	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750591	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	1	yes	characteristic, complemented
16	#18011999	C₂²	1	2	yes	complemented
17	#18011999	C₂²	1	1	yes	characteristic, complemented
18	#18011998	—	2	1	no	—
19	#18011998	—	2	1	no	—
20	#18011998	C₂²	1	1	yes	characteristic, complemented
21	#18011998	C₂²	1	1	yes	characteristic, complemented
22	C₂×#14558746	—	2	1	no	—
23	C₂×#14558746	C₂²	1	1	yes	characteristic
24	D₁₄₉⋊M₁₆	—	2	1	no	—
25	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×D₁₄₉⋊C₈	—	2	1	no	—
27	C₂×D₁₄₉⋊C₈	—	2	1	no	—
28	C₂×D₁₄₉⋊C₈	C₂²	1	1	yes	characteristic
29	#18011990	C₂²	1	1	yes	characteristic
30	#18011982	C₂²	1	2	yes	complemented
31	#18011981	C₂²	1	2	yes	complemented
32	#18011976	C₂²	1	2	yes	complemented
33	#18011976	C₂²	1	2	yes	complemented
34	#18011976	C₂²	1	2	yes	complemented
35	#18011975	C₂²	1	2	yes	complemented
36	#18011965	C₂²	1	2	yes	complemented
37	C₈×#10597010	C₂²	1	2	yes	complemented
38	C₂×#14558737	C₂²	1	1	yes	characteristic
39	#41×D₁₄₉	C₂²	1	1	yes	characteristic
40	#18011943	C₄	1	2	yes	complemented
41	#18011942	C₄	1	1	yes	characteristic, complemented
42	#18011939	C₄	1	1	yes	characteristic, complemented
43	#18011938	C₄	1	2	yes	complemented
44	#18011936	C₄	1	1	yes	characteristic, complemented
45	#18011934	C₂²	1	2	yes	—
46	#18011932	C₂²	1	2	yes	—
47	M₁₆×D₁₄₉	C₂²	1	1	yes	characteristic
48	#137×C₁₄₉	C₄	1	1	yes	characteristic, complemented
49	#14558746	—	2	1	no	—
50	#14558746	—	2	1	no	—
51	#14558746	—	2	2	no	—
52	#14558746	—	2	2	no	—
53	#14558746	—	2	1	no	—
54	#14558746	C₂³	1	2	yes	—
55	#14558746	C₂³	1	1	yes	characteristic
56	#14558746	C₂³	1	1	yes	characteristic
57	#14558745	—	2	1	no	—
58	#14558745	—	2	1	no	—
59	#14558745	—	2	1	no	—
60	D₁₄₉⋊C₈	—	2	1	no	—
61	D₁₄₉⋊C₈	—	2	1	no	—
62	D₁₄₉⋊C₈	—	2	2	no	—
63	D₁₄₉⋊C₈	—	2	2	no	—
64	D₁₄₉⋊C₈	—	2	1	no	—
65	D₁₄₉⋊C₈	D₄	1	1	yes	characteristic
66	D₁₄₉⋊C₈	D₄	1	1	yes	characteristic
67	D₁₄₉⋊C₈	C₂³	1	1	yes	characteristic
68	D₁₄₉⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×#11924753	—	2	1	no	—
70	C₂×#11924753	—	2	1	no	—
71	C₂×#11924753	—	2	1	no	—
72	C₂×#11924753	D₄	1	1	yes	characteristic
73	C₂×#11924753	D₄	1	1	yes	characteristic
74	C₄⋊#10597010	—	2	1	no	—
75	C₂²⋊#10597010	—	2	1	no	—
76	C₄×#10597010	C₂³	1	2	yes	—
77	#14558737	—	2	1	no	—
78	#14558737	—	2	1	no	—
79	#14558737	C₂×C₄	1	2	yes	—
80	#14558737	C₂³	1	2	yes	—
81	#14558737	C₂³	1	1	yes	characteristic
82	#14558736	—	2	1	no	—
83	#14558736	C₂×C₄	1	1	yes	characteristic
84	#14558735	—	2	1	no	—
85	#14558735	C₂³	1	1	yes	characteristic
86	Q₈×D₁₄₉	C₂×C₄	1	1	yes	characteristic
87	C₂×D₅₉₆	—	2	1	no	—
88	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
89	D₄×D₁₄₉	—	2	1	no	—
90	D₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄×D₁₄₉	—	2	1	no	—
92	C₂×C₄×D₁₄₉	—	2	1	no	—
93	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
94	#14558726	C₂×C₄	1	2	yes	—
95	#14558725	C₂×C₄	1	2	yes	—
96	#14558723	C₂×C₄	1	2	yes	—
97	#14558723	C₂×C₄	1	2	yes	—
98	D₁₁₉₂	C₂×C₄	1	2	yes	—
99	#14558721	C₂×C₄	1	2	yes	—
100	#14558720	C₂×C₄	1	1	yes	characteristic
101	#14558719	C₂×C₄	1	2	yes	—
102	C₈×D₁₄₉	C₂³	1	2	yes	—
103	#41×C₁₄₉	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
105	Q₁₆×C₁₄₉	C₂×C₄	1	2	yes	—
106	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₄₉	C₂×C₄	1	1	yes	characteristic
108	C₂×#10597010	—	2	2	no	—
109	#11924753	—	2	1	no	—
110	#11924753	—	2	1	no	—
111	#11924753	—	2	2	no	—
112	#11924753	C₂×D₄	1	1	yes	characteristic
113	#11924753	C₂×D₄	1	1	yes	characteristic
114	#11924753	C₂×D₄	1	1	yes	characteristic
115	#11924753	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₁₄₉	—	2	1	no	—
117	C₂²×D₁₄₉	—	2	1	no	—
118	C₂²×D₁₄₉	#41	1	1	yes	characteristic
119	Dic₂₉₈	—	2	1	no	—
120	Dic₂₉₈	C₂²×C₄	1	1	yes	characteristic
121	D₅₉₆	—	2	2	no	—
122	D₅₉₆	—	2	1	no	—
123	D₅₉₆	—	2	2	no	—
124	D₅₉₆	—	2	1	no	—
125	D₅₉₆	C₂²×C₄	1	2	yes	—
126	D₅₉₆	C₂²×C₄	1	1	yes	characteristic
127	C₂²⋊D₁₄₉	—	2	1	no	—
128	C₂²⋊D₁₄₉	—	2	1	no	—
129	C₂²⋊D₁₄₉	—	2	1	no	—
130	C₂×Dic₁₄₉	—	2	1	no	—
131	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₄₉	—	2	1	no	—
133	C₄×D₁₄₉	—	2	1	no	—
134	C₄×D₁₄₉	—	2	2	no	—
135	C₄×D₁₄₉	—	2	1	no	—
136	C₄×D₁₄₉	—	2	2	no	—
137	C₄×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
138	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
139	C₄×D₁₄₉	#41	1	1	yes	characteristic
140	C₄×D₁₄₉	C₂⁴	1	1	yes	characteristic
141	#11924746	C₂²×C₄	1	2	yes	—
142	Q₈×C₁₄₉	—	2	1	no	—
143	Q₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
144	Q₈×C₁₄₉	C₂²×C₄	1	2	yes	—
145	D₄×C₁₄₉	—	2	1	no	—
146	D₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₄₉	—	2	1	no	—
148	C₂×C₄×C₁₄₉	—	2	1	no	—
149	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
150	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₄₉	—	2	1	no	—
152	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₄₉	—	2	1	no	—
154	C₂×D₁₄₉	—	2	2	no	—
155	C₂×D₁₄₉	—	4	1	no	—
156	C₂×D₁₄₉	—	4	2	no	—
157	C₂×D₁₄₉	—	4	2	no	—
158	C₂×D₁₄₉	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₄₉	C₂×#41	1	1	yes	characteristic
160	#10597010	—	4	2	no	—
161	C₄×C₁₄₉	—	2	1	no	—
162	C₄×C₁₄₉	—	2	2	no	—
163	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
164	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
165	Dic₁₄₉	—	2	1	no	—
166	Dic₁₄₉	C₄×D₄	1	1	yes	characteristic
167	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₄₉	—	2	1	no	—
169	C₂×C₁₄₉	—	4	1	no	—
170	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
171	D₁₄₉	—	2	1	no	—
172	D₁₄₉	—	2	1	no	—
173	D₁₄₉	—	4	1	no	—
174	D₁₄₉	—	4	2	no	—
175	C₁₄₉	#2955	1	1	yes	characteristic, complemented, Sylow
176	#2955	—	149	1	no	maximal, Sylow
177	#575	—	149	1	no	—
178	C₂×#131	—	149	1	no	—
179	C₂×#131	—	149	1	no	—
180	#444	—	149	2	no	—
181	#444	—	149	2	no	—
182	#443	—	149	2	no	—
183	#443	—	149	2	no	—
184	#430	—	149	1	no	—
185	#421	—	149	1	no	—
186	#421	—	149	1	no	—
187	#405	—	149	1	no	—
188	C₂×#41	—	149	1	no	—
189	C₂×#41	—	149	1	no	—
190	#137	—	149	1	no	—
191	#137	—	149	2	no	—
192	#137	#10597010	1	1	yes	characteristic, complemented
193	AGL(1,ℤ/8)	—	149	1	no	—
194	AGL(1,ℤ/8)	—	149	2	no	—
195	AGL(1,ℤ/8)	—	149	1	no	—
196	#135	—	149	2	no	—
197	#135	—	149	2	no	—
198	#131	—	298	1	no	—
199	#131	—	298	1	no	—
200	#131	—	298	1	no	—
201	#131	—	298	1	no	—
202	#131	—	149	1	no	—
203	#131	—	149	1	no	—
204	#131	—	149	1	no	—
205	#131	—	149	2	no	—
206	#131	—	149	1	no	—
207	#131	—	149	2	no	—
208	C₂×M₁₆	—	298	1	no	—
209	C₂×M₁₆	—	298	1	no	—
210	C₂×M₁₆	—	149	1	no	—
211	C₂×M₁₆	—	149	1	no	—
212	C₂×M₁₆	—	149	1	no	—
213	C₂²×C₈	—	298	1	no	—
214	C₂²×C₈	—	298	1	no	—
215	C₂²×C₈	—	149	1	no	—
216	#117	—	149	1	no	—
217	#108	—	149	2	no	—
218	#108	—	149	2	no	—
219	#104	—	149	2	no	—
220	#104	—	149	2	no	—
221	#104	—	149	2	no	—
222	#104	—	149	2	no	—
223	#97	—	149	2	no	—
224	C₄×C₈	—	149	2	no	—
225	#41	—	298	1	no	—
226	#41	—	298	1	no	—
227	#41	—	298	1	no	—
228	#41	—	298	1	no	—
229	#41	—	149	2	no	—
230	#41	—	149	1	no	—
231	#41	—	149	2	no	—
232	#41	—	149	1	no	—
233	#41	—	149	1	no	—
234	#41	C₂×#10597010	1	1	yes	characteristic
235	C₂×Q₈	—	149	1	no	—
236	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic
237	C₂×D₄	—	298	1	no	—
238	C₂×D₄	—	298	1	no	—
239	C₂×D₄	—	149	1	no	—
240	C₂×D₄	—	149	1	no	—
241	C₂²×C₄	—	298	1	no	—
242	C₂²×C₄	—	298	1	no	—
243	C₂²×C₄	—	149	1	no	—
244	Q₁₆	—	149	2	no	—
245	Q₁₆	C₂×#10597010	1	2	yes	—
246	SD₁₆	—	149	2	no	—
247	SD₁₆	—	149	2	no	—
248	SD₁₆	—	149	2	no	—
249	SD₁₆	C₂×#10597010	1	2	yes	—
250	D₈	—	149	2	no	—
251	D₈	—	149	2	no	—
252	M₁₆	—	298	1	no	—
253	M₁₆	—	298	1	no	—
254	M₁₆	—	298	1	no	—
255	M₁₆	—	298	1	no	—
256	M₁₆	—	298	1	no	—
257	M₁₆	—	298	2	no	—
258	M₁₆	—	298	1	no	—
259	M₁₆	—	298	2	no	—
260	M₁₆	—	149	2	no	—
261	M₁₆	—	149	1	no	—
262	M₁₆	—	149	1	no	—
263	M₁₆	—	149	1	no	—
264	M₁₆	—	149	2	no	—
265	M₁₆	C₂×#10597010	1	1	yes	characteristic
266	C₂×C₈	—	298	1	no	—
267	C₂×C₈	—	298	1	no	—
268	C₂×C₈	—	298	1	no	—
269	C₂×C₈	—	298	1	no	—
270	C₂×C₈	—	298	1	no	—
271	C₂×C₈	—	298	2	no	—
272	C₂×C₈	—	298	1	no	—
273	C₂×C₈	—	298	2	no	—
274	C₂×C₈	—	149	1	no	—
275	C₂×C₈	—	149	1	no	—
276	C₂×C₈	—	149	1	no	—
277	C₂×C₈	—	149	1	no	—
278	C₂×C₈	—	149	1	no	—
279	C₂×C₈	—	149	1	no	—
280	C₂×C₈	—	149	2	no	—
281	C₄⋊C₄	—	298	1	no	—
282	C₂²⋊C₄	—	298	1	no	—
283	C₄²	—	149	2	no	—
284	C₂³	—	298	1	no	—
285	C₂³	—	298	1	no	—
286	C₂³	—	149	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	298	1	no	—
289	Q₈	—	149	1	no	—
290	Q₈	C₂²×#10597010	1	1	yes	characteristic
291	Q₈	C₂²×#10597010	1	2	yes	—
292	D₄	—	2	1	no	—
293	D₄	—	298	1	no	—
294	D₄	—	298	2	no	—
295	D₄	—	298	2	no	—
296	D₄	—	298	1	no	—
297	D₄	—	298	1	no	—
298	D₄	—	298	1	no	—
299	D₄	—	298	1	no	—
300	D₄	—	149	2	no	—
301	D₄	—	149	1	no	—
302	D₄	C₂²×#10597010	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	298	1	no	—
306	C₂×C₄	—	298	1	no	—
307	C₂×C₄	—	298	2	no	—
308	C₂×C₄	—	298	2	no	—
309	C₂×C₄	—	298	1	no	—
310	C₂×C₄	—	298	1	no	—
311	C₂×C₄	—	298	2	no	—
312	C₂×C₄	—	149	1	no	—
313	C₂×C₄	—	149	1	no	—
314	C₂×C₄	—	149	1	no	—
315	C₂×C₄	—	149	1	no	—
316	C₂×C₄	—	149	1	no	—
317	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
318	C₈	—	298	1	no	—
319	C₈	—	298	1	no	—
320	C₈	—	298	2	no	—
321	C₈	—	149	1	no	—
322	C₈	—	149	1	no	—
323	C₈	—	149	1	no	—
324	C₈	—	149	1	no	—
325	C₈	—	149	2	no	—
326	C₈	C₂²×#10597010	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	298	2	no	—
329	C₂²	—	298	1	no	—
330	C₂²	—	596	1	no	—
331	C₂²	—	596	2	no	—
332	C₂²	—	596	2	no	—
333	C₂²	—	149	1	no	—
334	C₂²	—	149	1	no	—
335	C₂²	D₄×#10597010	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	298	1	no	—
339	C₄	—	596	2	no	—
340	C₄	—	149	1	no	—
341	C₄	—	149	1	no	—
342	C₄	D₄×#10597010	1	1	yes	characteristic
343	C₄	C₂³×#10597010	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	1	no	—
346	C₂	—	298	1	no	—
347	C₂	—	298	1	no	—
348	C₂	—	596	2	no	—
349	C₂	—	596	1	no	—
350	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	#2955
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. C₂×#18011999	C₂	semidirect
2. D₁₄₉⋊#131	C₂	semidirect
3. #23750671	C₂	semidirect
4. #23750670	C₂	semidirect
5. #23750669	C₂	semidirect
6. #23750668	C₂	semidirect
7. #23750640	C₂	semidirect
8. #23750624	C₂	semidirect
9. #23750622	C₂	semidirect
10. #23750591	C₂	semidirect
14. #18011999	C₂²	semidirect
15. #18011999	C₂²	semidirect
16. #18011999	C₂²	semidirect
17. #18011999	C₂²	semidirect
20. #18011998	C₂²	semidirect
21. #18011998	C₂²	semidirect
30. #18011982	C₂²	semidirect
31. #18011981	C₂²	semidirect
32. #18011976	C₂²	semidirect
33. #18011976	C₂²	semidirect
34. #18011976	C₂²	semidirect
35. #18011975	C₂²	semidirect
36. #18011965	C₂²	semidirect
37. C₈×#10597010	C₂²	semidirect
40. #18011943	C₄	semidirect
41. #18011942	C₄	semidirect
42. #18011939	C₄	semidirect
43. #18011938	C₄	semidirect
44. #18011936	C₄	semidirect
48. #137×C₁₄₉	C₄	semidirect
175. C₁₄₉	#2955	semidirect
192. #137	#10597010	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011999	2
D₁₄₉⋊#131	2
#23750671	2
#23750670	2
#23750669	2
#23750668	2
#23750640	2
#23750624	2
#23750622	2
#23750591	2
#23750424	2
#2955	149