Subgroups — #34061581

Order 19072.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750660	C₂	1	1	yes	maximal, characteristic, complemented
2	#23750659	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750657	C₂	1	2	yes	maximal, complemented
4	#23750654	C₂	1	2	yes	maximal, complemented
5	#23750653	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#11924753	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750646	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750641	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750599	C₂	1	2	yes	maximal, complemented
10	#23750598	C₂	1	2	yes	maximal, complemented
11	#23750170	C₂	1	1	yes	maximal, characteristic
12	C₂×#14558746	C₂²	1	2	yes	—
13	C₂×#14558745	C₂²	1	2	yes	—
14	C₂×D₁₄₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×#11924753	C₂²	1	2	yes	—
16	C₄⋊#11924753	C₂²	1	1	yes	characteristic
17	C₄⋊#11924753	C₂²	1	1	yes	characteristic
18	C₄⋊#11924753	C₂²	1	1	yes	characteristic
19	C₄⋊#11924753	C₂²	1	1	yes	characteristic
20	#18011968	C₂²	1	2	yes	—
21	#18011968	C₂²	1	2	yes	—
22	C₂²⋊#11924753	C₂²	1	2	yes	—
23	C₂²⋊#11924753	C₂²	1	2	yes	—
24	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
25	C₂²⋊#11924753	C₂²	1	2	yes	—
26	C₂²⋊#11924753	C₂²	1	1	yes	characteristic
27	#18011966	C₂²	1	2	yes	—
28	C₄×#11924753	C₂²	1	1	yes	characteristic
29	C₄×#11924753	C₂²	1	1	yes	characteristic
30	C₂×#14558735	C₂²	1	2	yes	—
31	#18011910	C₄	1	1	yes	characteristic
32	#18011907	C₄	1	1	yes	characteristic
33	#18011897	C₄	1	2	yes	—
34	#18011885	C₄	1	1	yes	characteristic
35	#18011883	C₄	1	2	yes	—
36	D₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
37	#18011873	C₂²	1	1	yes	characteristic
38	#18011871	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
40	#121×C₁₄₉	C₄	1	1	yes	characteristic
41	#14558746	—	4	2	no	—
42	#14558745	—	4	2	no	—
43	D₁₄₉⋊C₈	—	4	2	no	—
44	C₂×#11924753	—	4	2	no	—
45	C₂×#11924753	C₂³	1	1	yes	characteristic
46	C₂×#11924753	C₂³	1	1	yes	characteristic
47	C₂×#11924753	C₂³	1	1	yes	characteristic
48	C₂×#11924753	C₂³	1	1	yes	characteristic
49	C₂×#11924753	C₂³	1	2	yes	—
50	C₂×#11924753	C₂³	1	2	yes	—
51	#14558735	—	4	2	no	—
52	#14558735	—	4	2	no	—
53	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
54	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
58	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
61	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
62	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
63	#14558715	C₂×C₄	1	2	yes	—
64	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
65	#14558713	C₂×C₄	1	2	yes	—
66	#14558713	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
70	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
72	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
73	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
76	#11924753	—	2	1	no	—
77	#11924753	—	2	2	no	—
78	#11924753	—	2	1	no	—
79	#11924753	—	2	2	no	—
80	#11924753	—	2	1	no	—
81	#11924753	—	2	1	no	—
82	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
83	Dic₂₉₈	—	4	2	no	—
84	D₅₉₆	—	2	2	no	—
85	C₂²⋊D₁₄₉	—	4	2	no	—
86	C₂²⋊D₁₄₉	—	4	2	no	—
87	C₂²⋊D₁₄₉	—	4	2	no	—
88	C₂×Dic₁₄₉	—	2	1	no	—
89	C₂×Dic₁₄₉	—	2	1	no	—
90	C₂×Dic₁₄₉	—	2	2	no	—
91	C₂×Dic₁₄₉	—	2	2	no	—
92	C₂×Dic₁₄₉	—	4	1	no	—
93	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₄₉	C₂⁴	1	1	yes	characteristic
98	C₄×D₁₄₉	—	2	2	no	—
99	C₄×D₁₄₉	—	2	2	no	—
100	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
102	D₄×C₁₄₉	—	2	2	no	—
103	D₄×C₁₄₉	—	4	2	no	—
104	C₂×C₄×C₁₄₉	—	2	2	no	—
105	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₁₄₉	—	2	1	no	—
109	C₂²×C₁₄₉	—	2	2	no	—
110	C₂²×C₁₄₉	—	2	1	no	—
111	C₂²×C₁₄₉	—	2	2	no	—
112	C₂²×C₁₄₉	—	2	1	no	—
113	C₂²×C₁₄₉	—	2	2	no	—
114	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₄₉	—	2	2	no	—
116	C₂×D₁₄₉	—	2	2	no	—
117	C₂×D₁₄₉	—	2	2	no	—
118	C₄×C₁₄₉	—	2	2	no	—
119	C₄×C₁₄₉	—	2	1	no	—
120	C₄×C₁₄₉	#131	1	2	yes	—
121	Dic₁₄₉	—	2	1	no	—
122	Dic₁₄₉	—	2	2	no	—
123	Dic₁₄₉	—	2	1	no	—
124	Dic₁₄₉	—	2	1	no	—
125	Dic₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
126	Dic₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₁₄₉	—	4	1	no	—
128	C₂×C₁₄₉	—	4	2	no	—
129	C₂×C₁₄₉	#518	1	1	yes	characteristic
130	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
131	C₂×C₁₄₉	#568	1	1	yes	characteristic
132	D₁₄₉	—	4	2	no	—
133	C₁₄₉	#2981	1	1	yes	characteristic, complemented, Sylow
134	#2981	—	149	1	no	maximal, Sylow
135	#532	—	149	1	no	—
136	#436	—	149	1	no	—
137	#436	—	149	1	no	—
138	#435	—	149	2	no	—
139	#435	—	149	2	no	—
140	C₈×D₄	—	149	1	no	—
141	C₈×D₄	—	149	1	no	—
142	#432	—	149	1	no	—
143	#431	—	149	1	no	—
144	#408	—	149	2	no	—
145	#408	—	149	2	no	—
146	C₂×#41	—	149	2	no	—
147	C₂×M₁₆	—	149	2	no	—
148	C₂×M₁₆	—	149	2	no	—
149	C₂²×C₈	—	149	2	no	—
150	C₂²×C₈	—	149	2	no	—
151	C₄⋊Q₈	—	149	1	no	—
152	#127	—	149	1	no	—
153	#124	—	149	2	no	—
154	#121	—	149	2	no	—
155	#121	—	149	1	no	—
156	#121	#10597010	1	1	yes	characteristic
157	C₄×D₄	—	149	1	no	—
158	C₄×D₄	—	149	1	no	—
159	C₄×D₄	—	149	2	no	—
160	C₂×C₄²	—	149	1	no	—
161	C₄⋊C₈	—	149	1	no	—
162	C₄⋊C₈	—	149	1	no	—
163	C₄⋊C₈	—	149	1	no	—
164	C₄⋊C₈	—	149	1	no	—
165	C₂²⋊C₈	—	149	1	no	—
166	C₂²⋊C₈	—	149	1	no	—
167	C₂²⋊C₈	—	149	2	no	—
168	C₂²⋊C₈	—	149	2	no	—
169	C₂²⋊C₈	—	149	2	no	—
170	C₂²⋊C₈	—	149	2	no	—
171	C₂²⋊C₈	—	149	2	no	—
172	#97	—	149	2	no	—
173	C₄×C₈	—	149	1	no	—
174	C₄×C₈	—	149	1	no	—
175	#41	—	596	2	no	—
176	#41	—	596	2	no	—
177	C₂×Q₈	—	149	2	no	—
178	C₂×D₄	—	149	2	no	—
179	C₂×D₄	—	149	2	no	—
180	C₂×D₄	—	149	2	no	—
181	C₂×D₄	—	149	1	no	—
182	C₂×D₄	C₂×#10597010	1	1	yes	characteristic
183	C₂×D₄	C₂×#10597010	1	2	yes	—
184	C₂²×C₄	—	149	1	no	—
185	C₂²×C₄	—	149	2	no	—
186	C₂²×C₄	—	149	2	no	—
187	C₂²×C₄	—	149	1	no	—
188	C₂²×C₄	C₂×#10597010	1	1	yes	characteristic
189	M₁₆	—	596	2	no	—
190	M₁₆	—	596	2	no	—
191	C₂×C₈	—	596	2	no	—
192	C₂×C₈	—	596	2	no	—
193	C₂×C₈	—	149	1	no	—
194	C₂×C₈	—	149	1	no	—
195	C₂×C₈	—	149	1	no	—
196	C₂×C₈	—	149	1	no	—
197	C₂×C₈	—	149	2	no	—
198	C₂×C₈	—	149	2	no	—
199	C₄⋊C₄	—	149	2	no	—
200	C₄⋊C₄	—	149	1	no	—
201	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
202	C₂²⋊C₄	—	149	2	no	—
203	C₂²⋊C₄	—	149	2	no	—
204	C₂²⋊C₄	—	149	2	no	—
205	C₂²⋊C₄	C₂×#10597010	1	2	yes	—
206	C₄²	—	149	1	no	—
207	C₄²	—	149	1	no	—
208	C₄²	—	149	1	no	—
209	C₄²	—	149	1	no	—
210	C₂³	—	149	2	no	—
211	C₂³	C₂²×#10597010	1	2	yes	—
212	C₂³	C₂²×#10597010	1	1	yes	characteristic
213	Q₈	—	596	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	4	2	no	—
216	D₄	—	298	2	no	—
217	D₄	—	596	2	no	—
218	D₄	—	596	2	no	—
219	D₄	—	596	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	298	2	no	—
222	C₂×C₄	—	298	2	no	—
223	C₂×C₄	—	298	2	no	—
224	C₂×C₄	—	298	2	no	—
225	C₂×C₄	—	298	1	no	—
226	C₂×C₄	—	298	1	no	—
227	C₂×C₄	—	596	1	no	—
228	C₂×C₄	—	149	1	no	—
229	C₂×C₄	—	149	1	no	—
230	C₂×C₄	—	149	1	no	—
231	C₂×C₄	—	149	2	no	—
232	C₂×C₄	—	149	1	no	—
233	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
234	C₂×C₄	C₂²×#10597010	1	2	yes	—
235	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
236	C₈	—	298	1	no	—
237	C₈	—	298	1	no	—
238	C₈	—	298	2	no	—
239	C₈	—	298	2	no	—
240	C₈	—	298	1	no	—
241	C₈	—	298	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	298	2	no	—
249	C₂²	—	298	2	no	—
250	C₂²	—	298	2	no	—
251	C₂²	C₂³×#10597010	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	298	1	no	—
255	C₄	—	298	2	no	—
256	C₄	—	298	1	no	—
257	C₄	—	298	1	no	—
258	C₄	—	149	1	no	—
259	C₄	—	149	1	no	—
260	C₄	#18011998	1	2	yes	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	596	2	no	—
264	C₂	#23750692	1	1	yes	characteristic
265	C₂	C₂×#18011998	1	1	yes	characteristic
266	C₂	#23750704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#121×C₁₄₉	C₄
prime radical	#23750170	C₂
cosocle	C₂×Dic₁₄₉	C₂⁴
hypercenter	#121	#10597010
hypocenter	C₁₄₉	#2981
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #23750660	C₂	semidirect
2. #23750659	C₂	semidirect
3. #23750657	C₂	semidirect
4. #23750654	C₂	semidirect
5. #23750653	C₂	semidirect
6. D₄×#11924753	C₂	semidirect
7. #23750646	C₂	semidirect
8. #23750641	C₂	semidirect
9. #23750599	C₂	semidirect
10. #23750598	C₂	semidirect
133. C₁₄₉	#2981	semidirect

Maximal subgroups

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

subgroup	index
#23750660	2
#23750659	2
#23750657	2
#23750654	2
#23750653	2
D₄×#11924753	2
#23750646	2
#23750641	2
#23750599	2
#23750598	2
#23750170	2
#2981	149