Subgroups — #34058335

Order 19072.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011956	C₂	1	1	yes	maximal, characteristic, complemented
2	#23750274	C₂	1	8	yes	maximal, complemented
3	#23750263	C₂	1	8	yes	maximal, complemented
4	#23750199	C₂	1	2	yes	maximal, complemented
5	#23750184	C₂	1	4	yes	maximal, complemented
6	#23750136	C₂	1	2	yes	maximal, complemented
7	C₂×#18011894	C₂	1	4	yes	maximal, complemented
8	#23750100	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#18011956	—	2	8	no	—
11	C₂×#14558736	C₂²	1	2	yes	—
12	C₂×#14558735	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₁₄₉	C₂²	1	4	yes	—
14	C₂×D₄×D₁₄₉	C₂²	1	4	yes	complemented
15	#18011909	C₂²	1	4	yes	complemented
16	#18011900	C₂²	1	4	yes	complemented
17	#18011894	C₂²	1	8	yes	complemented
18	#18011894	C₂²	1	16	yes	complemented
19	#18011894	C₂²	1	4	yes	complemented
20	#18011892	C₂²	1	8	yes	complemented
21	#18011890	C₂²	1	4	yes	complemented
22	#18011887	C₂²	1	8	yes	complemented
23	#18011885	C₂²	1	4	yes	complemented
24	#18011883	C₂²	1	16	yes	complemented
25	#18011880	C₂²	1	4	yes	complemented
26	#18011879	C₂²	1	8	yes	complemented
27	D₄×Dic₁₄₉	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
29	#18011870	C₂²	1	2	yes	complemented
30	C₂×#14558715	C₂²	1	8	yes	complemented
31	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	complemented
32	C₂×#14558713	C₂²	1	8	yes	complemented
33	C₂×#41×C₁₄₉	—	2	8	no	—
34	C₂×#41×C₁₄₉	C₂²	1	8	yes	complemented
35	C₂²×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
36	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₁₄₉	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	complemented
39	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₁₄₉	C₂³	1	4	yes	—
41	#14558736	—	2	8	no	—
42	#14558735	—	2	16	no	—
43	C₂×Dic₂₉₈	C₂³	1	2	yes	—
44	C₂×D₅₉₆	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₁₄₉	—	2	16	no	—
46	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
47	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
48	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
49	D₄×D₁₄₉	—	2	16	no	—
50	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
51	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
52	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
53	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
54	#14558715	C₂³	1	16	yes	—
55	#14558715	C₂³	1	8	yes	—
56	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
57	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
58	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
59	#14558713	C₂³	1	8	yes	—
60	#14558713	C₂³	1	16	yes	—
61	C₄×Dic₁₄₉	C₂³	1	4	yes	—
62	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
63	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
64	#41×C₁₄₉	—	2	32	no	—
65	C₂×Q₈×C₁₄₉	—	2	4	no	—
66	C₂×Q₈×C₁₄₉	C₂³	1	4	yes	—
67	C₂×D₄×C₁₄₉	—	2	8	no	—
68	C₂×D₄×C₁₄₉	D₄	1	8	yes	complemented
69	C₂×D₄×C₁₄₉	C₂³	1	8	yes	—
70	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₄₉	—	2	8	no	—
72	C₂²×C₄×C₁₄₉	—	2	4	no	—
73	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₄₉	C₂³	1	8	yes	—
76	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
78	C₂²×D₁₄₉	—	2	8	no	—
79	C₂²×D₁₄₉	—	4	8	no	—
80	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
81	Dic₂₉₈	—	2	4	no	—
82	D₅₉₆	—	2	4	no	—
83	C₂²⋊D₁₄₉	—	2	16	no	—
84	C₂²⋊D₁₄₉	—	2	8	no	—
85	C₂²⋊D₁₄₉	—	4	8	no	—
86	C₂×Dic₁₄₉	—	2	8	no	—
87	C₂×Dic₁₄₉	—	4	8	no	—
88	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
89	C₂×Dic₁₄₉	C₂⁴	1	8	yes	—
90	C₄×D₁₄₉	—	2	8	no	—
91	C₂³×C₁₄₉	—	2	2	no	—
92	C₂³×C₁₄₉	—	4	4	no	—
93	C₂³×C₁₄₉	—	4	2	no	—
94	C₂³×C₁₄₉	C₂×D₄	1	8	yes	—
95	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
96	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
97	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
98	Q₈×C₁₄₉	—	2	8	no	—
99	D₄×C₁₄₉	—	2	16	no	—
100	D₄×C₁₄₉	C₂×D₄	1	16	yes	—
101	C₂×C₄×C₁₄₉	—	2	2	no	—
102	C₂×C₄×C₁₄₉	—	2	4	no	—
103	C₂×C₄×C₁₄₉	—	2	16	no	—
104	C₂×C₄×C₁₄₉	—	2	8	no	—
105	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
109	C₂²×C₁₄₉	—	2	2	no	—
110	C₂²×C₁₄₉	—	2	8	no	—
111	C₂²×C₁₄₉	—	2	2	no	—
112	C₂²×C₁₄₉	—	2	4	no	—
113	C₂²×C₁₄₉	—	4	8	no	—
114	C₂²×C₁₄₉	—	4	4	no	—
115	C₂²×C₁₄₉	C₂²×D₄	1	2	yes	—
116	C₂²×C₁₄₉	C₂²×D₄	1	8	yes	—
117	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
118	C₂×D₁₄₉	—	2	4	no	—
119	C₂×D₁₄₉	—	2	8	no	—
120	C₂×D₁₄₉	—	8	8	no	—
121	C₄×C₁₄₉	—	2	2	no	—
122	C₄×C₁₄₉	—	2	4	no	—
123	C₄×C₁₄₉	C₂²×D₄	1	4	yes	—
124	Dic₁₄₉	—	2	4	no	—
125	Dic₁₄₉	—	2	8	no	—
126	C₂×C₁₄₉	—	2	8	no	—
127	C₂×C₁₄₉	—	2	2	no	—
128	C₂×C₁₄₉	—	4	2	no	—
129	C₂×C₁₄₉	C₂³×D₄	1	1	yes	characteristic
130	C₂×C₁₄₉	#585	1	2	yes	—
131	D₁₄₉	—	4	4	no	—
132	C₁₄₉	#3485	1	1	yes	characteristic, complemented, Sylow
133	#3485	—	149	1	no	maximal, Sylow
134	C₂×H(5,𝔽₂)	—	149	1	no	—
135	C₂²×#41	D₁₄₉	1	1	yes	characteristic, complemented
136	#547	—	149	8	no	—
137	#546	—	149	8	no	—
138	#536	—	149	2	no	—
139	#534	—	149	4	no	—
140	#525	—	149	2	no	—
141	C₂×#123	—	149	4	no	—
142	#518	—	149	1	no	—
143	H(5,𝔽₂)	—	298	8	no	—
144	C₂×#41	—	2	8	no	—
145	C₂×#41	—	149	4	no	—
146	C₂×#41	—	149	2	no	—
147	C₂×#41	C₂×D₁₄₉	1	8	yes	complemented
148	C₂²×Q₈	C₂×D₁₄₉	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	149	4	no	—
150	C₂²×D₄	—	149	4	no	—
151	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₁₄₉	1	2	yes	complemented
153	C₂³×C₄	C₂×D₁₄₉	1	2	yes	complemented
154	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic, complemented
155	C₄⋊Q₈	—	149	4	no	—
156	#124	—	149	4	no	—
157	#123	—	149	8	no	—
158	#123	—	149	4	no	—
159	#123	—	149	8	no	—
160	#123	—	149	16	no	—
161	C₂²⋊Q₈	—	149	8	no	—
162	C₂²⋊Q₈	—	149	4	no	—
163	#121	—	149	16	no	—
164	#121	—	149	4	no	—
165	#120	—	149	4	no	—
166	#120	—	149	8	no	—
167	C₄×D₄	—	149	8	no	—
168	C₄×D₄	—	149	8	no	—
169	#117	—	149	2	no	—
170	C₂×C₄⋊C₄	—	149	8	no	—
171	C₂×C₂²⋊C₄	—	149	4	no	—
172	C₂×C₂²⋊C₄	—	149	8	no	—
173	C₂⁴	—	149	4	no	—
174	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
175	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
176	#41	—	2	32	no	—
177	#41	—	298	16	no	—
178	#41	—	298	8	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	149	2	no	—
181	C₂×Q₈	C₂²×D₁₄₉	1	4	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	298	16	no	—
184	C₂×D₄	—	298	16	no	—
185	C₂×D₄	—	149	4	no	—
186	C₂×D₄	—	149	8	no	—
187	C₂×D₄	—	149	8	no	—
188	C₂×D₄	—	149	2	no	—
189	C₂×D₄	C₂²⋊D₁₄₉	1	8	yes	complemented
190	C₂×D₄	C₂²×D₁₄₉	1	8	yes	—
191	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	149	4	no	—
195	C₂²×C₄	—	149	4	no	—
196	C₂²×C₄	—	149	8	no	—
197	C₂²×C₄	C₂²×D₁₄₉	1	8	yes	—
198	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
199	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
202	C₄⋊C₄	—	149	8	no	—
203	C₄⋊C₄	—	149	16	no	—
204	C₄⋊C₄	—	149	4	no	—
205	C₂²⋊C₄	—	149	4	no	—
206	C₂²⋊C₄	—	149	8	no	—
207	C₂²⋊C₄	—	149	8	no	—
208	C₂²⋊C₄	—	149	8	no	—
209	C₂²⋊C₄	—	149	16	no	—
210	C₄²	—	149	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	298	8	no	—
215	C₂³	—	596	8	no	—
216	C₂³	—	149	4	no	—
217	C₂³	C₂×C₂²⋊D₁₄₉	1	8	yes	—
218	C₂³	C₂³×D₁₄₉	1	2	yes	—
219	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
220	C₂³	C₂³×D₁₄₉	1	4	yes	—
221	Q₈	—	2	8	no	—
222	Q₈	—	298	4	no	—
223	D₄	—	2	16	no	—
224	D₄	—	298	8	no	—
225	D₄	—	298	16	no	—
226	D₄	—	298	4	no	—
227	D₄	—	596	8	no	—
228	D₄	C₂×C₂²⋊D₁₄₉	1	16	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	298	8	no	—
234	C₂×C₄	—	298	8	no	—
235	C₂×C₄	—	596	8	no	—
236	C₂×C₄	—	149	8	no	—
237	C₂×C₄	—	149	4	no	—
238	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	4	yes	—
239	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
240	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
241	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	298	4	no	—
249	C₂²	—	298	8	no	—
250	C₂²	—	1192	8	no	—
251	C₂²	C₂²×C₂²⋊D₁₄₉	1	2	yes	—
252	C₂²	C₂²×C₂²⋊D₁₄₉	1	8	yes	—
253	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	298	4	no	—
257	C₄	—	298	8	no	—
258	C₄	C₂²×C₂²⋊D₁₄₉	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	596	4	no	—
263	C₂	C₂³×C₂²⋊D₁₄₉	1	1	yes	characteristic
264	C₂	#23750461	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	C₂²×#41×C₁₄₉	C₂
hypercenter	C₂²×#41	D₁₄₉
hypocenter	C₁₄₉	#3485
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#18011956	C₂	semidirect
2. #23750274	C₂	semidirect
3. #23750263	C₂	semidirect
4. #23750199	C₂	semidirect
5. #23750184	C₂	semidirect
6. #23750136	C₂	semidirect
7. C₂×#18011894	C₂	semidirect
8. #23750100	C₂	semidirect
9. C₂²×#41×C₁₄₉	C₂	semidirect
12. C₂×#14558735	C₂²	semidirect
14. C₂×D₄×D₁₄₉	C₂²	semidirect
15. #18011909	C₂²	semidirect
16. #18011900	C₂²	semidirect
17. #18011894	C₂²	semidirect
18. #18011894	C₂²	semidirect
19. #18011894	C₂²	semidirect
20. #18011892	C₂²	semidirect
21. #18011890	C₂²	semidirect
22. #18011887	C₂²	semidirect
23. #18011885	C₂²	semidirect
24. #18011883	C₂²	semidirect
25. #18011880	C₂²	semidirect
26. #18011879	C₂²	semidirect
27. D₄×Dic₁₄₉	C₂²	semidirect
28. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
29. #18011870	C₂²	semidirect
30. C₂×#14558715	C₂²	semidirect
31. C₂×C₂²⋊Dic₁₄₉	C₂²	semidirect
32. C₂×#14558713	C₂²	semidirect
34. C₂×#41×C₁₄₉	C₂²	semidirect
35. C₂²×Q₈×C₁₄₉	C₂²	semidirect
37. C₂²×D₄×C₁₄₉	C₂²	semidirect
38. C₂³×C₄×C₁₄₉	C₂²	semidirect
39. C₂³×C₄×C₁₄₉	C₂²	semidirect
68. C₂×D₄×C₁₄₉	D₄	semidirect
132. C₁₄₉	#3485	semidirect
135. C₂²×#41	D₁₄₉	semidirect
147. C₂×#41	C₂×D₁₄₉	semidirect
148. C₂²×Q₈	C₂×D₁₄₉	semidirect
152. C₂²×D₄	C₂×D₁₄₉	semidirect
153. C₂³×C₄	C₂×D₁₄₉	semidirect
154. C₂³×C₄	C₂×D₁₄₉	semidirect
189. C₂×D₄	C₂²⋊D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011956	2
#23750274	2
#23750263	2
#23750199	2
#23750184	2
#23750136	2
C₂×#18011894	2
#23750100	2
C₂²×#41×C₁₄₉	2
#3485	149