Subgroups — #34061469

Order 19072.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#23750703	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750670	C₂	1	4	yes	maximal, complemented
4	#23750668	C₂	1	4	yes	maximal, complemented
5	D₁₄₉⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750624	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750623	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750588	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#18011932	C₂	1	1	yes	maximal, characteristic
10	#18011999	—	2	2	no	—
11	#18011999	C₂²	1	4	yes	complemented
12	#18011998	—	2	2	no	—
13	#18011998	C₂²	1	4	yes	complemented
14	D₁₄₉⋊M₁₆	—	2	2	no	—
15	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
16	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
17	D₁₄₉⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#14558745	—	2	2	no	—
19	#18011990	C₂²	1	1	yes	characteristic
20	#18011982	C₂²	1	2	yes	complemented
21	#18011981	C₂²	1	2	yes	complemented
22	#18011976	C₂²	1	4	yes	complemented
23	#18011975	C₂²	1	4	yes	complemented
24	#18011965	C₂²	1	2	yes	complemented
25	C₈×#10597010	C₂²	1	2	yes	complemented
26	C₂×#14558737	C₂²	1	1	yes	characteristic
27	C₂×#14558735	C₂²	1	1	yes	characteristic
28	#18011932	C₄	1	2	yes	complemented
29	#18011932	C₄	1	2	yes	complemented
30	#18011932	C₂²	1	2	yes	—
31	#18011932	C₂²	1	2	yes	—
32	C₂×#14558726	C₄	1	1	yes	characteristic, complemented
33	C₂×#14558725	C₄	1	1	yes	characteristic, complemented
34	C₂×#14558721	C₄	1	1	yes	characteristic, complemented
35	C₂×C₈×D₁₄₉	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₄₉	C₄	1	1	yes	characteristic, complemented
37	#14558746	—	2	4	no	—
38	#14558746	D₄	1	2	yes	—
39	#14558746	C₂³	1	2	yes	—
40	#14558746	C₂³	1	2	yes	—
41	#14558745	—	2	4	no	—
42	#14558745	—	2	1	no	—
43	#14558745	—	2	1	no	—
44	#14558745	D₄	1	2	yes	—
45	D₁₄₉⋊C₈	—	2	4	no	—
46	D₁₄₉⋊C₈	—	2	1	no	—
47	D₁₄₉⋊C₈	—	2	1	no	—
48	D₁₄₉⋊C₈	C₂³	1	2	yes	—
49	C₂×#11924753	—	2	4	no	—
50	C₄⋊#10597010	—	2	1	no	—
51	C₂²⋊#10597010	—	2	1	no	—
52	C₄×#10597010	C₂³	1	2	yes	—
53	#14558737	—	2	1	no	—
54	#14558737	—	2	1	no	—
55	#14558737	C₂×C₄	1	2	yes	—
56	#14558737	C₂³	1	2	yes	—
57	#14558735	—	2	1	no	—
58	#14558735	—	2	1	no	—
59	#14558735	C₂×C₄	1	2	yes	—
60	#14558735	C₂³	1	2	yes	—
61	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
62	C₂×D₅₉₆	—	2	1	no	—
63	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₄₉	—	2	1	no	—
65	C₂²×Dic₁₄₉	—	2	1	no	—
66	C₂×C₄×D₁₄₉	—	2	1	no	—
67	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
68	#14558726	C₂×C₄	1	2	yes	—
69	#14558726	C₂×C₄	1	2	yes	—
70	#14558725	C₂×C₄	1	2	yes	—
71	#14558725	C₂×C₄	1	2	yes	—
72	#14558721	C₂×C₄	1	2	yes	—
73	#14558721	C₂×C₄	1	2	yes	—
74	C₂×#11924746	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₄₉	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₄₉	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₄₉	C₂³	1	1	yes	characteristic
78	C₈×D₁₄₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	—
82	SD₁₆×C₁₄₉	C₂×C₄	1	2	yes	—
83	C₂×C₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
84	C₂×#10597010	—	2	2	no	—
85	#11924753	—	2	2	no	—
86	#11924753	—	2	2	no	—
87	#11924753	C₂×D₄	1	4	yes	—
88	C₂²×D₁₄₉	—	2	1	no	—
89	C₂²×D₁₄₉	#41	1	1	yes	characteristic
90	Dic₂₉₈	—	2	1	no	—
91	Dic₂₉₈	C₂²×C₄	1	2	yes	—
92	D₅₉₆	—	2	2	no	—
93	D₅₉₆	—	2	2	no	—
94	D₅₉₆	—	2	1	no	—
95	D₅₉₆	C₂²×C₄	1	2	yes	—
96	C₂²⋊D₁₄₉	—	2	2	no	—
97	C₂²⋊D₁₄₉	—	2	2	no	—
98	C₂×Dic₁₄₉	—	2	1	no	—
99	C₂×Dic₁₄₉	—	2	2	no	—
100	C₂×Dic₁₄₉	—	2	2	no	—
101	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₄₉	—	2	2	no	—
103	C₄×D₁₄₉	—	2	2	no	—
104	C₄×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₄₉	C₂×D₄	1	1	yes	characteristic
106	C₄×D₁₄₉	#41	1	1	yes	characteristic
107	C₄×D₁₄₉	C₂⁴	1	1	yes	characteristic
108	#11924746	C₂²×C₄	1	1	yes	characteristic
109	#11924746	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₁₄₉	—	2	1	no	—
111	Q₈×C₁₄₉	—	2	1	no	—
112	Q₈×C₁₄₉	C₂²×C₄	1	2	yes	—
113	D₄×C₁₄₉	—	2	1	no	—
114	D₄×C₁₄₉	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₄₉	—	2	1	no	—
116	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₁₄₉	—	2	2	no	—
120	C₂²×C₁₄₉	—	8	1	no	—
121	C₂²×C₁₄₉	C₄×D₄	1	1	yes	characteristic
122	C₂×D₁₄₉	—	2	2	no	—
123	C₂×D₁₄₉	—	4	1	no	—
124	C₂×D₁₄₉	—	8	1	no	—
125	C₂×D₁₄₉	C₄×D₄	1	1	yes	characteristic
126	C₂×D₁₄₉	C₂×#41	1	1	yes	characteristic
127	#10597010	—	4	2	no	—
128	C₄×C₁₄₉	—	2	2	no	—
129	C₄×C₁₄₉	C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₄₉	—	2	2	no	—
132	Dic₁₄₉	C₄×D₄	1	1	yes	characteristic
133	Dic₁₄₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₁₄₉	—	2	1	no	—
135	C₂×C₁₄₉	—	4	2	no	—
136	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
137	D₁₄₉	—	2	1	no	—
138	D₁₄₉	—	2	1	no	—
139	D₁₄₉	—	4	2	no	—
140	C₁₄₉	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	149	1	no	maximal, Sylow
142	C₂×#135	—	149	1	no	—
143	#568	—	149	1	no	—
144	#568	—	149	1	no	—
145	#444	—	149	4	no	—
146	#443	—	149	4	no	—
147	C₂×#108	—	149	1	no	—
148	#421	—	149	1	no	—
149	#421	—	149	1	no	—
150	#405	—	149	1	no	—
151	C₂×#41	—	149	1	no	—
152	C₂×#41	—	149	1	no	—
153	#135	—	149	2	no	—
154	#135	—	149	2	no	—
155	#135	—	149	2	no	—
156	#135	—	149	2	no	—
157	C₂×Q₁₆	—	149	1	no	—
158	C₂×SD₁₆	—	149	1	no	—
159	C₂×SD₁₆	#10597010	1	1	yes	characteristic, complemented
160	C₂×D₈	—	149	1	no	—
161	#131	—	298	2	no	—
162	#131	—	298	2	no	—
163	#131	—	149	4	no	—
164	#131	—	149	4	no	—
165	C₂×M₁₆	—	298	2	no	—
166	C₂×M₁₆	—	298	2	no	—
167	C₂×M₁₆	—	149	1	no	—
168	C₂×M₁₆	—	149	1	no	—
169	C₂×M₁₆	—	149	1	no	—
170	C₂²×C₈	—	149	1	no	—
171	#117	—	149	1	no	—
172	#108	—	149	2	no	—
173	#108	—	149	2	no	—
174	#104	—	149	4	no	—
175	#104	—	149	4	no	—
176	#97	—	149	2	no	—
177	C₄×C₈	—	149	2	no	—
178	#41	—	298	1	no	—
179	#41	—	298	1	no	—
180	#41	—	298	1	no	—
181	#41	—	298	1	no	—
182	#41	—	149	2	no	—
183	#41	—	149	2	no	—
184	#41	—	149	2	no	—
185	#41	—	149	2	no	—
186	C₂×Q₈	—	149	1	no	—
187	C₂×Q₈	C₂×#10597010	1	1	yes	characteristic
188	C₂×D₄	—	298	1	no	—
189	C₂×D₄	—	298	1	no	—
190	C₂×D₄	—	149	1	no	—
191	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
192	C₂²×C₄	—	298	1	no	—
193	C₂²×C₄	—	298	1	no	—
194	C₂²×C₄	—	149	1	no	—
195	Q₁₆	—	149	2	no	—
196	Q₁₆	—	149	2	no	—
197	SD₁₆	—	149	2	no	—
198	SD₁₆	—	149	2	no	—
199	SD₁₆	C₂×#10597010	1	2	yes	—
200	SD₁₆	C₂×#10597010	1	2	yes	—
201	D₈	—	149	2	no	—
202	D₈	—	149	2	no	—
203	M₁₆	—	298	1	no	—
204	M₁₆	—	298	1	no	—
205	M₁₆	—	298	4	no	—
206	M₁₆	—	298	4	no	—
207	M₁₆	—	149	2	no	—
208	M₁₆	—	149	2	no	—
209	M₁₆	—	149	2	no	—
210	M₁₆	—	149	2	no	—
211	C₂×C₈	—	298	1	no	—
212	C₂×C₈	—	298	1	no	—
213	C₂×C₈	—	298	4	no	—
214	C₂×C₈	—	298	4	no	—
215	C₂×C₈	—	149	2	no	—
216	C₂×C₈	—	149	1	no	—
217	C₂×C₈	—	149	1	no	—
218	C₂×C₈	—	149	1	no	—
219	C₂×C₈	—	149	1	no	—
220	C₂×C₈	—	149	1	no	—
221	C₂×C₈	C₂×#10597010	1	1	yes	characteristic
222	C₄⋊C₄	—	298	1	no	—
223	C₂²⋊C₄	—	298	1	no	—
224	C₄²	—	149	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	298	1	no	—
227	C₂³	—	149	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	298	1	no	—
230	Q₈	—	149	2	no	—
231	Q₈	C₂²×#10597010	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	298	2	no	—
234	D₄	—	298	2	no	—
235	D₄	—	298	2	no	—
236	D₄	—	298	2	no	—
237	D₄	—	298	1	no	—
238	D₄	—	149	2	no	—
239	D₄	C₂²×#10597010	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	298	2	no	—
242	C₂×C₄	—	298	2	no	—
243	C₂×C₄	—	298	1	no	—
244	C₂×C₄	—	298	2	no	—
245	C₂×C₄	—	298	2	no	—
246	C₂×C₄	—	298	2	no	—
247	C₂×C₄	—	149	1	no	—
248	C₂×C₄	—	149	1	no	—
249	C₂×C₄	—	149	1	no	—
250	C₂×C₄	—	149	1	no	—
251	C₂×C₄	—	149	1	no	—
252	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
253	C₈	—	298	2	no	—
254	C₈	—	298	2	no	—
255	C₈	—	149	4	no	—
256	C₈	—	149	1	no	—
257	C₈	—	149	1	no	—
258	C₈	C₂²×#10597010	1	1	yes	characteristic
259	C₈	C₂²×#10597010	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	298	2	no	—
263	C₂²	—	596	1	no	—
264	C₂²	—	149	1	no	—
265	C₂²	—	149	1	no	—
266	C₂²	—	1192	1	no	—
267	C₂²	D₄×#10597010	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	298	2	no	—
270	C₄	—	596	2	no	—
271	C₄	—	149	1	no	—
272	C₄	—	149	1	no	—
273	C₄	D₄×#10597010	1	1	yes	characteristic
274	C₄	C₂³×#10597010	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	298	1	no	—
278	C₂	—	298	1	no	—
279	C₂	—	596	2	no	—
280	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	C₂×SD₁₆×C₁₄₉	C₄
socle	C₂×C₁₄₉	C₂×C₄×D₄
prime radical	C₂×#18011932	C₂
cosocle	C₄×D₁₄₉	C₂⁴
hypercenter	C₂×SD₁₆	#10597010
hypocenter	C₁₄₉	#2957
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #23750705	C₂	semidirect
2. #23750703	C₂	semidirect
3. #23750670	C₂	semidirect
4. #23750668	C₂	semidirect
5. D₁₄₉⋊#108	C₂	semidirect
6. #23750624	C₂	semidirect
7. #23750623	C₂	semidirect
8. #23750588	C₂	semidirect
11. #18011999	C₂²	semidirect
13. #18011998	C₂²	semidirect
20. #18011982	C₂²	semidirect
21. #18011981	C₂²	semidirect
22. #18011976	C₂²	semidirect
23. #18011975	C₂²	semidirect
24. #18011965	C₂²	semidirect
25. C₈×#10597010	C₂²	semidirect
28. #18011932	C₄	semidirect
29. #18011932	C₄	semidirect
32. C₂×#14558726	C₄	semidirect
33. C₂×#14558725	C₄	semidirect
34. C₂×#14558721	C₄	semidirect
36. C₂×SD₁₆×C₁₄₉	C₄	semidirect
140. C₁₄₉	#2957	semidirect
159. C₂×SD₁₆	#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
#23750703	2
#23750670	2
#23750668	2
D₁₄₉⋊#108	2
#23750624	2
#23750623	2
#23750588	2
C₂×#18011932	2
#2957	149