Subgroups — C₂×#23750242 #34058065

Order 19072.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750242	C₂	1	16	yes	maximal, direct product factor
2	C₂×#18011910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#18011904	C₂	1	2	yes	maximal, complemented
4	C₂×#18011898	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#18011898	C₂	1	2	yes	maximal, complemented
6	C₂×#18011892	C₂	1	2	yes	maximal, complemented
7	C₂×#18011890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#18011888	C₂	1	2	yes	maximal, complemented
9	C₂×#18011876	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#18011870	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#121×C₁₄₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
15	C₂³×Dic₁₄₉	C₂²	1	2	yes	—
16	#18011910	C₂²	1	8	yes	complemented
17	#18011904	C₂²	1	16	yes	complemented
18	#18011898	C₂²	1	8	yes	complemented
19	#18011898	C₂²	1	16	yes	complemented
20	#18011892	C₂²	1	16	yes	complemented
21	#18011890	C₂²	1	8	yes	complemented
22	#18011888	C₂²	1	16	yes	complemented
23	#18011876	C₂²	1	8	yes	complemented
24	D₄×Dic₁₄₉	C₂²	1	8	yes	complemented
25	#18011870	C₂²	1	8	yes	complemented
26	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
29	C₂×#14558715	C₂²	1	2	yes	—
30	C₂×#14558715	C₂²	1	2	yes	—
31	C₂×#14558715	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
37	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₁₄₉	C₂²	1	2	yes	—
43	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
44	#121×C₁₄₉	C₂²	1	8	yes	—
45	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	—
47	C₂×Dic₂₉₈	—	2	4	no	—
48	C₂×Dic₂₉₈	—	4	4	no	—
49	C₂×Dic₂₉₈	C₂³	1	4	yes	—
50	C₂×Dic₂₉₈	C₂³	1	8	yes	—
51	C₂²×Dic₁₄₉	—	4	4	no	—
52	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
54	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
59	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
60	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
61	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
62	#14558715	C₂³	1	8	yes	—
63	#14558715	C₂³	1	8	yes	—
64	#14558715	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
66	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
67	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
68	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
69	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
71	C₄×Dic₁₄₉	C₂³	1	8	yes	—
72	C₄×Dic₁₄₉	C₂³	1	4	yes	—
73	C₄×Dic₁₄₉	C₂³	1	4	yes	—
74	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
75	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₄₉	—	2	4	no	—
77	C₂×D₄×C₁₄₉	—	4	4	no	—
78	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
79	C₂×D₄×C₁₄₉	C₂³	1	8	yes	—
80	C₂²×C₄×C₁₄₉	—	2	4	no	—
81	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
84	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
87	Dic₂₉₈	—	2	8	no	—
88	Dic₂₉₈	—	4	8	no	—
89	C₂×Dic₁₄₉	—	2	4	no	—
90	C₂×Dic₁₄₉	—	2	2	no	—
91	C₂×Dic₁₄₉	—	2	4	no	—
92	C₂×Dic₁₄₉	—	2	2	no	—
93	C₂×Dic₁₄₉	—	2	2	no	—
94	C₂×Dic₁₄₉	—	2	2	no	—
95	C₂×Dic₁₄₉	—	4	8	no	—
96	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
97	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
98	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
99	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
100	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
101	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
102	C₂³×C₁₄₉	—	2	2	no	—
103	C₂³×C₁₄₉	—	2	4	no	—
104	C₂³×C₁₄₉	—	2	4	no	—
105	C₂³×C₁₄₉	—	2	2	no	—
106	C₂³×C₁₄₉	—	2	4	no	—
107	C₂³×C₁₄₉	—	2	2	no	—
108	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
109	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
110	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
111	D₄×C₁₄₉	—	2	8	no	—
112	D₄×C₁₄₉	—	4	8	no	—
113	C₂×C₄×C₁₄₉	—	2	2	no	—
114	C₂×C₄×C₁₄₉	—	2	4	no	—
115	C₂×C₄×C₁₄₉	—	2	8	no	—
116	C₂×C₄×C₁₄₉	#41	1	4	yes	—
117	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
119	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
120	C₂²×C₁₄₉	—	2	2	no	—
121	C₂²×C₁₄₉	—	2	4	no	—
122	C₂²×C₁₄₉	—	2	2	no	—
123	C₂²×C₁₄₉	—	2	4	no	—
124	C₂²×C₁₄₉	—	2	4	no	—
125	C₂²×C₁₄₉	—	2	2	no	—
126	C₂²×C₁₄₉	—	4	4	no	—
127	C₂²×C₁₄₉	—	4	8	no	—
128	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
129	C₂²×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
131	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
132	C₄×C₁₄₉	—	2	2	no	—
133	C₄×C₁₄₉	—	2	4	no	—
134	C₄×C₁₄₉	C₂×#41	1	4	yes	—
135	Dic₁₄₉	—	2	4	no	—
136	Dic₁₄₉	—	2	2	no	—
137	Dic₁₄₉	—	2	4	no	—
138	Dic₁₄₉	—	2	2	no	—
139	Dic₁₄₉	—	2	2	no	—
140	Dic₁₄₉	—	2	2	no	—
141	C₂×C₁₄₉	—	4	2	no	—
142	C₂×C₁₄₉	—	4	4	no	—
143	C₂×C₁₄₉	#542	1	4	yes	complemented
144	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
147	C₁₄₉	C₂×#542	1	1	yes	characteristic, complemented, Sylow
148	C₂×#542	—	149	1	no	maximal, Sylow
149	#542	—	149	16	no	—
150	C₂×C₄⋊Q₈	—	149	1	no	—
151	C₂×#126	—	149	2	no	—
152	C₂×#124	—	149	1	no	—
153	C₂×#124	—	149	2	no	—
154	C₂×#123	—	149	2	no	—
155	C₂×C₂²⋊Q₈	—	149	1	no	—
156	C₂×C₂²⋊Q₈	—	149	2	no	—
157	C₂×#121	D₁₄₉	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	149	1	no	—
159	C₂×C₄×D₄	—	149	1	no	—
160	C₂×#117	—	149	1	no	—
161	C₂²×Q₈	—	149	1	no	—
162	C₂²×Q₈	—	149	2	no	—
163	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
164	C₂²×D₄	C₂×D₁₄₉	1	2	yes	—
165	C₂³×C₄	—	149	2	no	—
166	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
167	C₄⋊Q₈	—	149	8	no	—
168	#126	—	149	16	no	—
169	#124	—	149	8	no	—
170	#124	—	149	16	no	—
171	#123	—	149	16	no	—
172	C₂²⋊Q₈	—	149	8	no	—
173	C₂²⋊Q₈	—	149	16	no	—
174	#121	C₂×D₁₄₉	1	8	yes	—
175	C₄×Q₈	—	149	8	no	—
176	C₄×D₄	—	149	8	no	—
177	#117	—	149	8	no	—
178	C₂×C₄⋊C₄	—	149	1	no	—
179	C₂×C₄⋊C₄	—	149	1	no	—
180	C₂×C₄⋊C₄	—	149	1	no	—
181	C₂×C₄⋊C₄	—	149	2	no	—
182	C₂×C₄⋊C₄	—	149	2	no	—
183	C₂×C₄⋊C₄	—	149	2	no	—
184	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	149	1	no	—
186	C₂×C₂²⋊C₄	—	149	1	no	—
187	C₂×C₂²⋊C₄	—	149	2	no	—
188	C₂×C₂²⋊C₄	—	149	2	no	—
189	C₂×C₂²⋊C₄	—	149	2	no	—
190	C₂×C₂²⋊C₄	—	149	2	no	—
191	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	—
192	C₂×C₄²	—	149	1	no	—
193	C₂×C₄²	—	149	1	no	—
194	C₂×C₄²	—	149	2	no	—
195	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
196	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
197	C₂×Q₈	—	298	4	no	—
198	C₂×Q₈	—	596	4	no	—
199	C₂×Q₈	—	149	4	no	—
200	C₂×Q₈	—	149	8	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
204	C₂×D₄	C₂²×D₁₄₉	1	8	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	596	4	no	—
207	C₂²×C₄	—	149	1	no	—
208	C₂²×C₄	—	149	1	no	—
209	C₂²×C₄	—	149	2	no	—
210	C₂²×C₄	—	149	1	no	—
211	C₂²×C₄	—	149	1	no	—
212	C₂²×C₄	—	149	2	no	—
213	C₂²×C₄	—	149	8	no	—
214	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
215	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
216	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
218	C₄⋊C₄	—	149	4	no	—
219	C₄⋊C₄	—	149	4	no	—
220	C₄⋊C₄	—	149	4	no	—
221	C₄⋊C₄	—	149	8	no	—
222	C₄⋊C₄	—	149	8	no	—
223	C₄⋊C₄	—	149	8	no	—
224	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
225	C₂²⋊C₄	—	149	4	no	—
226	C₂²⋊C₄	—	149	4	no	—
227	C₂²⋊C₄	—	149	8	no	—
228	C₂²⋊C₄	—	149	8	no	—
229	C₂²⋊C₄	—	149	8	no	—
230	C₂²⋊C₄	—	149	8	no	—
231	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
232	C₄²	—	149	4	no	—
233	C₄²	—	149	4	no	—
234	C₄²	—	149	8	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	C₂³×D₁₄₉	1	2	yes	—
242	C₂³	C₂³×D₁₄₉	1	4	yes	—
243	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
244	Q₈	—	298	8	no	—
245	Q₈	—	596	8	no	—
246	D₄	—	2	8	no	—
247	D₄	—	4	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	298	4	no	—
252	C₂×C₄	—	298	2	no	—
253	C₂×C₄	—	298	4	no	—
254	C₂×C₄	—	298	2	no	—
255	C₂×C₄	—	298	2	no	—
256	C₂×C₄	—	298	2	no	—
257	C₂×C₄	—	596	8	no	—
258	C₂×C₄	—	149	2	no	—
259	C₂×C₄	—	149	2	no	—
260	C₂×C₄	—	149	4	no	—
261	C₂×C₄	—	149	2	no	—
262	C₂×C₄	—	149	2	no	—
263	C₂×C₄	—	149	4	no	—
264	C₂×C₄	#14558735	1	4	yes	—
265	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
266	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
267	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	C₂×#14558735	1	2	yes	—
277	C₂²	#18011956	1	2	yes	—
278	C₂²	#18011958	1	2	yes	—
279	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	298	4	no	—
283	C₄	—	298	2	no	—
284	C₄	—	298	4	no	—
285	C₄	—	298	2	no	—
286	C₄	—	298	2	no	—
287	C₄	—	298	2	no	—
288	C₄	C₂×#14558735	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	#23750242	1	4	yes	direct product factor
292	C₂	C₂²×#14558735	1	1	yes	characteristic
293	C₂	C₂×#18011956	1	1	yes	characteristic
294	C₂	C₂×#18011958	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting, prime radical	C₂×#121×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
hypercenter	C₂×#121	D₁₄₉
hypocenter	C₁₄₉	C₂×#542
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. #23750242	C₂	direct (N × H)
2. C₂×#18011910	C₂	semidirect
3. C₂×#18011904	C₂	semidirect
4. C₂×#18011898	C₂	semidirect
5. C₂×#18011898	C₂	semidirect
6. C₂×#18011892	C₂	semidirect
7. C₂×#18011890	C₂	semidirect
8. C₂×#18011888	C₂	semidirect
9. C₂×#18011876	C₂	semidirect
10. C₂×D₄×Dic₁₄₉	C₂	semidirect
11. C₂×#18011870	C₂	semidirect
16. #18011910	C₂²	semidirect
17. #18011904	C₂²	semidirect
18. #18011898	C₂²	semidirect
19. #18011898	C₂²	semidirect
20. #18011892	C₂²	semidirect
21. #18011890	C₂²	semidirect
22. #18011888	C₂²	semidirect
23. #18011876	C₂²	semidirect
24. D₄×Dic₁₄₉	C₂²	semidirect
25. #18011870	C₂²	semidirect
143. C₂×C₁₄₉	#542	semidirect
147. C₁₄₉	C₂×#542	semidirect
291. C₂	#23750242	direct (N × H)

Maximal subgroups

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

subgroup	index
#23750242	2
C₂×#18011910	2
C₂×#18011904	2
C₂×#18011898	2
C₂×#18011892	2
C₂×#18011890	2
C₂×#18011888	2
C₂×#18011876	2
C₂×D₄×Dic₁₄₉	2
C₂×#18011870	2
C₂×#121×C₁₄₉	2
C₂×#542	149

Subgroups — C2×#23750242 #34058065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750242 #34058065