Subgroups — #34047698

Order 19072.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011891	C₂	1	1	yes	maximal, characteristic
2	C₂×#18011876	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊C₄×D₁₄₉	C₂	1	1	yes	maximal, characteristic
4	#23749531	C₂	1	1	yes	maximal, characteristic, complemented
5	#23749524	C₂	1	1	yes	maximal, characteristic, complemented
6	#23749523	C₂	1	2	yes	maximal, complemented
7	#23749510	C₂	1	2	yes	maximal, complemented
8	#23749499	C₂	1	2	yes	maximal
9	C₄×#14558713	C₂	1	2	yes	maximal
10	C₄×#14558713	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
15	#18011891	C₄	1	8	yes	complemented
16	#18011876	—	2	4	no	—
17	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
20	C₂×#14558715	C₂²	1	2	yes	—
21	C₂×#14558715	C₂²	1	2	yes	—
22	C₄⋊C₄×D₁₄₉	—	2	4	no	—
23	C₂×#14558713	C₂²	1	1	yes	characteristic
24	C₂×#14558713	C₂²	1	2	yes	—
25	C₂×#14558713	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
28	#18011818	C₂²	1	1	yes	characteristic
29	#18011818	C₂²	1	2	yes	—
30	#18011818	C₂²	1	1	yes	characteristic
31	#18011818	C₂²	1	2	yes	—
32	#18011818	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₄₉	C₂²	1	2	yes	—
37	#95×C₁₄₉	C₂²	1	2	yes	—
38	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₂₉₈	—	2	2	no	—
40	C₂×Dic₂₉₈	—	2	2	no	—
41	C₂×Dic₂₉₈	C₂×C₄	1	4	yes	complemented
42	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
43	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
46	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₄₉	—	2	4	no	—
48	C₂×C₄×D₁₄₉	—	2	2	no	—
49	C₂×C₄×D₁₄₉	—	2	4	no	—
50	C₂×C₄×D₁₄₉	—	2	2	no	—
51	C₂×C₄×D₁₄₉	—	4	2	no	—
52	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	—
53	C₄⋊Dic₁₄₉	—	2	2	no	—
54	C₄⋊Dic₁₄₉	—	4	2	no	—
55	C₄⋊Dic₁₄₉	C₂×C₄	1	8	yes	complemented
56	#14558715	—	2	4	no	—
57	#14558715	—	2	4	no	—
58	#14558713	—	2	2	no	—
59	#14558713	C₂×C₄	1	8	yes	—
60	#14558713	Q₈	1	4	yes	complemented
61	C₄×Dic₁₄₉	—	2	4	no	—
62	C₄×Dic₁₄₉	—	2	2	no	—
63	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
69	C₄⋊C₄×C₁₄₉	D₄	1	4	yes	—
70	C₄²×C₁₄₉	—	2	2	no	—
71	C₄²×C₁₄₉	—	4	2	no	—
72	C₂²×D₁₄₉	—	2	1	no	—
73	C₂²×D₁₄₉	—	2	1	no	—
74	C₂²×D₁₄₉	—	2	1	no	—
75	C₂²×D₁₄₉	—	2	1	no	—
76	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₄₉	C₂×Q₈	1	2	yes	—
78	C₂²×D₁₄₉	#41	1	2	yes	—
79	Dic₂₉₈	—	2	8	no	—
80	C₂×Dic₁₄₉	—	2	1	no	—
81	C₂×Dic₁₄₉	—	2	2	no	—
82	C₂×Dic₁₄₉	—	2	1	no	—
83	C₂×Dic₁₄₉	—	2	1	no	—
84	C₂×Dic₁₄₉	—	2	2	no	—
85	C₂×Dic₁₄₉	—	2	1	no	—
86	C₂×Dic₁₄₉	—	2	2	no	—
87	C₂×Dic₁₄₉	—	2	1	no	—
88	C₂×Dic₁₄₉	—	2	2	no	—
89	C₂×Dic₁₄₉	—	2	1	no	—
90	C₂×Dic₁₄₉	—	2	2	no	—
91	C₂×Dic₁₄₉	—	2	1	no	—
92	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
94	C₂×Dic₁₄₉	C₂×Q₈	1	2	yes	—
95	C₄×D₁₄₉	—	2	8	no	—
96	C₄×D₁₄₉	—	4	8	no	—
97	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₄₉	—	2	2	no	—
99	C₂×C₄×C₁₄₉	—	2	1	no	—
100	C₂×C₄×C₁₄₉	—	2	2	no	—
101	C₂×C₄×C₁₄₉	—	2	1	no	—
102	C₂×C₄×C₁₄₉	—	2	2	no	—
103	C₂×C₄×C₁₄₉	—	2	2	no	—
104	C₂×C₄×C₁₄₉	—	2	1	no	—
105	C₂×C₄×C₁₄₉	—	2	1	no	—
106	C₂×C₄×C₁₄₉	—	2	1	no	—
107	C₂×C₄×C₁₄₉	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₁₄₉	#41	1	2	yes	—
113	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₁₄₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂×D₁₄₉	—	2	2	no	—
121	C₂×D₁₄₉	—	2	2	no	—
122	C₂×D₁₄₉	—	2	2	no	—
123	C₂×D₁₄₉	—	2	2	no	—
124	C₂×D₁₄₉	—	2	2	no	—
125	C₂×D₁₄₉	—	2	2	no	—
126	C₂×D₁₄₉	C₄×Q₈	1	4	yes	complemented
127	C₄×C₁₄₉	—	2	4	no	—
128	C₄×C₁₄₉	—	2	4	no	—
129	C₄×C₁₄₉	—	2	2	no	—
130	C₄×C₁₄₉	—	4	1	no	—
131	C₄×C₁₄₉	C₄×D₄	1	4	yes	—
132	Dic₁₄₉	—	2	2	no	—
133	Dic₁₄₉	—	2	4	no	—
134	Dic₁₄₉	—	2	2	no	—
135	Dic₁₄₉	—	4	2	no	—
136	Dic₁₄₉	—	4	1	no	—
137	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₁₄₉	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₂×C₁₄₉	#520	1	1	yes	characteristic
140	C₂×C₁₄₉	#546	1	1	yes	characteristic
141	C₂×C₁₄₉	D₄×Q₈	1	1	yes	characteristic
142	C₂×C₁₄₉	#552	1	1	yes	characteristic
143	C₂×C₁₄₉	#554	1	1	yes	characteristic
144	D₁₄₉	—	2	4	no	—
145	C₁₄₉	#2360	1	1	yes	characteristic, complemented, Sylow
146	#2360	—	149	1	no	maximal, Sylow
147	C₂×C₂²⋊Q₈	—	149	1	no	—
148	C₂×C₄×Q₈	—	149	1	no	—
149	C₂²×C₄⋊C₄	—	149	1	no	—
150	#391	—	149	1	no	—
151	#389	—	149	1	no	—
152	#389	—	149	2	no	—
153	#387	—	149	2	no	—
154	#385	—	149	2	no	—
155	C₄×C₄⋊C₄	D₁₄₉	1	1	yes	characteristic, complemented
156	C₄×C₂²⋊C₄	—	149	1	no	—
157	C₄×C₂²⋊C₄	—	149	2	no	—
158	C₂²×Q₈	—	149	1	no	—
159	C₂³×C₄	—	149	2	no	—
160	C₂³×C₄	—	149	1	no	—
161	C₂²⋊Q₈	—	149	8	no	—
162	C₄×Q₈	—	298	4	no	—
163	C₂×C₄⋊C₄	—	298	4	no	—
164	C₂×C₄⋊C₄	—	149	2	no	—
165	C₂×C₄⋊C₄	—	149	1	no	—
166	C₂×C₄⋊C₄	—	149	2	no	—
167	C₂×C₄⋊C₄	—	149	2	no	—
168	C₂×C₄⋊C₄	—	149	2	no	—
169	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	149	1	no	—
172	C₂×C₂²⋊C₄	—	149	2	no	—
173	C₂×C₂²⋊C₄	—	149	1	no	—
174	C₂×C₄²	—	149	2	no	—
175	C₂×C₄²	—	149	1	no	—
176	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
177	C₂×C₄²	C₂×D₁₄₉	1	2	yes	—
178	#95	—	149	1	no	—
179	#95	—	149	2	no	—
180	#95	—	149	1	no	—
181	#95	—	149	2	no	—
182	#95	—	149	2	no	—
183	#95	C₂×D₁₄₉	1	2	yes	—
184	C₂⁴	—	149	1	no	—
185	C₂×Q₈	—	298	2	no	—
186	C₂×Q₈	—	298	2	no	—
187	C₂×Q₈	—	149	4	no	—
188	C₂²×C₄	—	298	4	no	—
189	C₂²×C₄	—	298	2	no	—
190	C₂²×C₄	—	298	2	no	—
191	C₂²×C₄	—	298	4	no	—
192	C₂²×C₄	—	596	2	no	—
193	C₂²×C₄	—	149	2	no	—
194	C₂²×C₄	—	149	1	no	—
195	C₂²×C₄	—	149	1	no	—
196	C₂²×C₄	—	149	2	no	—
197	C₂²×C₄	—	149	1	no	—
198	C₂²×C₄	—	149	4	no	—
199	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
204	C₄⋊C₄	—	298	4	no	—
205	C₄⋊C₄	—	298	2	no	—
206	C₄⋊C₄	—	298	4	no	—
207	C₄⋊C₄	—	596	2	no	—
208	C₄⋊C₄	—	149	8	no	—
209	C₄⋊C₄	C₄×D₁₄₉	1	4	yes	complemented
210	C₄⋊C₄	D₅₉₆	1	4	yes	—
211	C₂²⋊C₄	—	298	2	no	—
212	C₂²⋊C₄	—	149	8	no	—
213	C₂²⋊C₄	—	149	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	2	no	—
216	C₄²	—	298	4	no	—
217	C₄²	—	298	2	no	—
218	C₂³	—	298	1	no	—
219	C₂³	—	298	1	no	—
220	C₂³	—	298	1	no	—
221	C₂³	—	298	1	no	—
222	C₂³	—	149	2	no	—
223	C₂³	—	149	2	no	—
224	C₂³	—	149	2	no	—
225	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
226	Q₈	—	298	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	298	1	no	—
237	C₂×C₄	—	298	2	no	—
238	C₂×C₄	—	298	1	no	—
239	C₂×C₄	—	298	1	no	—
240	C₂×C₄	—	298	2	no	—
241	C₂×C₄	—	298	1	no	—
242	C₂×C₄	—	298	8	no	—
243	C₂×C₄	—	298	1	no	—
244	C₂×C₄	—	298	2	no	—
245	C₂×C₄	—	298	1	no	—
246	C₂×C₄	—	298	2	no	—
247	C₂×C₄	—	298	1	no	—
248	C₂×C₄	—	298	2	no	—
249	C₂×C₄	—	596	8	no	—
250	C₂×C₄	—	149	4	no	—
251	C₂×C₄	—	149	2	no	—
252	C₂×C₄	—	149	2	no	—
253	C₂×C₄	C₂×C₄×D₁₄₉	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
255	C₂×C₄	C₂×D₅₉₆	1	4	yes	—
256	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
257	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
258	C₂×C₄	#14558736	1	2	yes	—
259	C₂²	—	298	2	no	—
260	C₂²	—	298	2	no	—
261	C₂²	—	298	2	no	—
262	C₂²	—	298	2	no	—
263	C₂²	—	298	2	no	—
264	C₂²	—	298	2	no	—
265	C₂²	—	149	4	no	—
266	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
267	C₂²	C₂²×D₅₉₆	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₁₄₉	1	1	yes	characteristic
269	C₂²	#41×D₁₄₉	1	1	yes	characteristic
270	C₂²	C₂×#14558736	1	1	yes	characteristic
271	C₂²	#18011956	1	1	yes	characteristic
272	C₂²	#18011958	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	4	1	no	—
277	C₄	—	298	2	no	—
278	C₄	—	298	2	no	—
279	C₄	—	298	4	no	—
280	C₄	—	596	2	no	—
281	C₄	—	596	1	no	—
282	C₄	C₄×D₅₉₆	1	4	yes	—
283	C₂	—	298	4	no	—
284	C₂	C₂×C₄×D₅₉₆	1	1	yes	characteristic
285	C₂	C₄×Q₈×D₁₄₉	1	1	yes	characteristic
286	C₂	#23750104	1	1	yes	characteristic
287	C₂	#23750270	1	1	yes	characteristic
288	C₂	Q₈×D₅₉₆	1	1	yes	characteristic
289	C₂	#23750314	1	1	yes	characteristic
290	C₂	#23750330	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂³×D₁₄₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₄₉
hypocenter	C₁₄₉	#2360
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#18011876	C₂	semidirect
4. #23749531	C₂	semidirect
5. #23749524	C₂	semidirect
6. #23749523	C₂	semidirect
7. #23749510	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₄₉	C₂	semidirect
15. #18011891	C₄	semidirect
41. C₂×Dic₂₉₈	C₂×C₄	semidirect
55. C₄⋊Dic₁₄₉	C₂×C₄	semidirect
60. #14558713	Q₈	semidirect
68. C₄⋊C₄×C₁₄₉	C₂×C₄	semidirect
126. C₂×D₁₄₉	C₄×Q₈	semidirect
145. C₁₄₉	#2360	semidirect
155. C₄×C₄⋊C₄	D₁₄₉	semidirect
209. C₄⋊C₄	C₄×D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011891	2
C₂×#18011876	2
C₂×C₄⋊C₄×D₁₄₉	2
#23749531	2
#23749524	2
#23749523	2
#23749510	2
#23749499	2
C₄×#14558713	2
C₄×C₄⋊C₄×C₁₄₉	2
#2360	149