Subgroups — #34058553

Order 19072.

1012 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750361	C₂	1	2	yes	maximal, complemented
2	#23750350	C₂	1	2	yes	maximal, complemented
3	#23750345	C₂	1	2	yes	maximal, complemented
4	#23750344	C₂	1	4	yes	maximal, complemented
5	#23750332	C₂	1	2	yes	maximal, complemented
6	#23750315	C₂	1	4	yes	maximal, complemented
7	#23750220	C₂	1	4	yes	maximal, complemented
8	#23750196	C₂	1	2	yes	maximal, complemented
9	#23750182	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750156	C₂	1	4	yes	maximal, complemented
11	#23750108	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#532×C₁₄₉	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
15	#18011911	C₂²	1	1	yes	characteristic
16	#18011911	C₂²	1	1	yes	characteristic
17	#18011909	C₂²	1	4	yes	—
18	#18011904	C₂²	1	8	yes	—
19	#18011904	C₂²	1	8	yes	—
20	#18011903	C₂²	1	2	yes	—
21	#18011902	C₂²	1	4	yes	—
22	#18011901	C₂²	1	8	yes	—
23	#18011898	C₂²	1	4	yes	—
24	#18011898	C₂²	1	4	yes	—
25	#18011892	C₂²	1	8	yes	—
26	#18011892	C₂²	1	8	yes	—
27	#18011890	C₂²	1	2	yes	—
28	#18011890	C₂²	1	2	yes	—
29	#18011890	C₂²	1	4	yes	—
30	#18011890	C₂²	1	4	yes	—
31	#18011888	C₂²	1	8	yes	—
32	Q₈×Dic₁₄₉	C₂²	1	4	yes	—
33	C₄×Dic₂₉₈	C₂²	1	2	yes	—
34	C₄×Dic₂₉₈	C₂²	1	2	yes	—
35	C₄×Dic₂₉₈	C₂²	1	2	yes	—
36	C₄×Dic₂₉₈	C₂²	1	2	yes	—
37	C₄×Dic₂₉₈	C₂²	1	2	yes	—
38	C₄×Dic₂₉₈	C₂²	1	2	yes	—
39	#18011876	C₂²	1	4	yes	—
40	D₄×Dic₁₄₉	C₂²	1	8	yes	—
41	D₄×Dic₁₄₉	C₂²	1	4	yes	—
42	D₄×Dic₁₄₉	C₂²	1	4	yes	—
43	#18011870	C₂²	1	2	yes	—
44	#18011870	C₂²	1	4	yes	—
45	#18011867	C₂²	1	8	yes	—
46	C₂×C₄⋊Dic₁₄₉	C₂²	1	4	yes	—
47	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
48	C₂×#14558715	C₂²	1	2	yes	—
49	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
50	C₂×#41×C₁₄₉	C₂²	1	2	yes	—
51	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
52	#127×C₁₄₉	C₂²	1	1	yes	characteristic
53	#124×C₁₄₉	C₂²	1	2	yes	—
54	#121×C₁₄₉	C₂²	1	2	yes	—
55	#121×C₁₄₉	C₂²	1	2	yes	—
56	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
57	C₄×D₄×C₁₄₉	C₂²	1	2	yes	—
58	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₉₈	—	2	4	no	—
60	C₂×Dic₂₉₈	C₂³	1	2	yes	—
61	C₂×Dic₂₉₈	C₂³	1	2	yes	—
62	C₂×Dic₂₉₈	C₂³	1	2	yes	—
63	C₂×Dic₂₉₈	C₂³	1	2	yes	—
64	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
65	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
66	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
67	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
68	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
69	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
70	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
71	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
72	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
73	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
74	#14558715	C₂³	1	8	yes	—
75	#14558715	C₂³	1	4	yes	—
76	#14558715	C₂³	1	4	yes	—
77	#14558715	C₂³	1	8	yes	—
78	#14558715	C₂³	1	4	yes	—
79	#14558715	C₂³	1	4	yes	—
80	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
82	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
83	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
84	C₄×Dic₁₄₉	C₂³	1	4	yes	—
85	C₄×Dic₁₄₉	C₂³	1	8	yes	—
86	C₄×Dic₁₄₉	C₂³	1	4	yes	—
87	C₄×Dic₁₄₉	C₂³	1	4	yes	—
88	#41×C₁₄₉	—	2	8	no	—
89	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
90	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
91	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
92	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
93	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
96	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
99	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
100	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
101	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
102	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
103	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
105	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
106	Dic₂₉₈	—	2	4	no	—
107	Dic₂₉₈	—	2	4	no	—
108	Dic₂₉₈	—	2	4	no	—
109	Dic₂₉₈	—	4	2	no	—
110	C₂×Dic₁₄₉	—	2	4	no	—
111	C₂×Dic₁₄₉	—	4	2	no	—
112	C₂×Dic₁₄₉	—	4	8	no	—
113	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
114	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
115	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
116	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
117	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
118	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
119	Q₈×C₁₄₉	—	2	4	no	—
120	D₄×C₁₄₉	—	2	8	no	—
121	D₄×C₁₄₉	—	2	4	no	—
122	D₄×C₁₄₉	—	2	4	no	—
123	D₄×C₁₄₉	—	4	2	no	—
124	C₂×C₄×C₁₄₉	—	2	8	no	—
125	C₂×C₄×C₁₄₉	—	2	2	no	—
126	C₂×C₄×C₁₄₉	—	4	1	no	—
127	C₂×C₄×C₁₄₉	#41	1	4	yes	—
128	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
131	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
132	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
133	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
135	C₂²×C₁₄₉	—	2	1	no	—
136	C₂²×C₁₄₉	—	2	1	no	—
137	C₂²×C₁₄₉	—	2	4	no	—
138	C₂²×C₁₄₉	—	2	4	no	—
139	C₂²×C₁₄₉	—	2	4	no	—
140	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
141	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
142	C₄×C₁₄₉	—	2	4	no	—
143	C₄×C₁₄₉	—	2	1	no	—
144	C₄×C₁₄₉	—	2	1	no	—
145	C₄×C₁₄₉	—	2	1	no	—
146	C₄×C₁₄₉	C₂×#41	1	2	yes	—
147	C₄×C₁₄₉	C₂×#41	1	2	yes	—
148	C₄×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
149	Dic₁₄₉	—	2	4	no	—
150	Dic₁₄₉	—	2	4	no	—
151	Dic₁₄₉	—	2	4	no	—
152	Dic₁₄₉	—	2	4	no	—
153	C₂×C₁₄₉	—	2	2	no	—
154	C₂×C₁₄₉	—	4	4	no	—
155	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
156	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×C₁₄₉	#585	1	1	yes	characteristic
158	C₁₄₉	#3507	1	1	yes	characteristic, complemented, Sylow
159	#3507	—	149	1	no	maximal, Sylow
160	#559	—	149	2	no	—
161	#557	—	149	2	no	—
162	#556	—	149	4	no	—
163	#556	—	149	2	no	—
164	#554	—	149	2	no	—
165	#552	—	149	4	no	—
166	#539	—	149	4	no	—
167	#536	—	149	2	no	—
168	#533	—	149	1	no	—
169	#532	D₁₄₉	1	1	yes	characteristic
170	#529	—	149	4	no	—
171	#520	—	149	2	no	—
172	C₂×C₄×Q₈	—	149	1	no	—
173	C₂×#41	C₂×D₁₄₉	1	2	yes	—
174	C₂²×Q₈	—	149	1	no	—
175	C₄⋊Q₈	—	149	4	no	—
176	C₄⋊Q₈	—	149	1	no	—
177	C₄⋊Q₈	—	149	1	no	—
178	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
179	#127	C₂×D₁₄₉	1	1	yes	characteristic
180	#126	—	149	8	no	—
181	#126	—	149	8	no	—
182	#125	—	149	4	no	—
183	#125	—	149	8	no	—
184	#125	—	149	2	no	—
185	#124	—	149	4	no	—
186	#124	—	149	4	no	—
187	#124	C₂×D₁₄₉	1	2	yes	—
188	#123	—	149	8	no	—
189	#123	—	149	8	no	—
190	C₂²⋊Q₈	—	149	2	no	—
191	C₂²⋊Q₈	—	149	2	no	—
192	C₂²⋊Q₈	—	149	4	no	—
193	C₂²⋊Q₈	—	149	4	no	—
194	C₂²⋊Q₈	—	149	8	no	—
195	#121	C₂×D₁₄₉	1	2	yes	—
196	#121	C₂×D₁₄₉	1	2	yes	—
197	C₄×Q₈	—	149	4	no	—
198	C₄×Q₈	—	149	4	no	—
199	C₄×Q₈	—	149	2	no	—
200	C₄×Q₈	—	149	2	no	—
201	C₄×Q₈	—	149	2	no	—
202	C₄×Q₈	—	149	2	no	—
203	C₄×Q₈	—	149	2	no	—
204	C₄×Q₈	—	149	2	no	—
205	C₄×D₄	—	149	8	no	—
206	C₄×D₄	—	149	4	no	—
207	C₄×D₄	—	149	4	no	—
208	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
209	C₄×D₄	C₂×D₁₄₉	1	2	yes	—
210	#117	—	149	2	no	—
211	#117	—	149	4	no	—
212	#117	—	149	8	no	—
213	C₂×C₄⋊C₄	—	149	1	no	—
214	C₂×C₄⋊C₄	—	149	2	no	—
215	C₂×C₄⋊C₄	—	149	4	no	—
216	C₂×C₄²	—	149	2	no	—
217	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
218	#41	—	2	8	no	—
219	C₂×Q₈	—	298	4	no	—
220	C₂×Q₈	—	149	2	no	—
221	C₂×Q₈	—	149	2	no	—
222	C₂×Q₈	—	149	2	no	—
223	C₂×Q₈	—	149	2	no	—
224	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
225	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
226	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
227	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
228	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
229	C₂²×C₄	—	149	2	no	—
230	C₂²×C₄	—	149	2	no	—
231	C₂²×C₄	—	149	8	no	—
232	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
234	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
236	C₄⋊C₄	—	149	2	no	—
237	C₄⋊C₄	—	149	2	no	—
238	C₄⋊C₄	—	149	2	no	—
239	C₄⋊C₄	—	149	8	no	—
240	C₄⋊C₄	—	149	4	no	—
241	C₄⋊C₄	—	149	4	no	—
242	C₄⋊C₄	—	149	2	no	—
243	C₄⋊C₄	—	149	4	no	—
244	C₄⋊C₄	—	149	8	no	—
245	C₄⋊C₄	—	149	2	no	—
246	C₄⋊C₄	—	149	4	no	—
247	C₄⋊C₄	—	149	8	no	—
248	C₄⋊C₄	—	149	2	no	—
249	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
250	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
251	C₂²⋊C₄	—	149	4	no	—
252	C₂²⋊C₄	—	149	8	no	—
253	C₂²⋊C₄	—	149	8	no	—
254	C₂²⋊C₄	—	149	8	no	—
255	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
256	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
257	C₄²	—	149	4	no	—
258	C₄²	—	149	4	no	—
259	C₄²	—	149	8	no	—
260	C₄²	—	149	4	no	—
261	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
262	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
263	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
264	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
265	C₂³	C₂³×D₁₄₉	1	4	yes	—
266	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	298	4	no	—
269	Q₈	—	298	4	no	—
270	Q₈	—	298	4	no	—
271	Q₈	—	596	2	no	—
272	D₄	—	2	8	no	—
273	D₄	—	2	4	no	—
274	D₄	—	2	4	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	298	4	no	—
280	C₂×C₄	—	596	2	no	—
281	C₂×C₄	—	596	8	no	—
282	C₂×C₄	—	149	4	no	—
283	C₂×C₄	—	149	4	no	—
284	C₂×C₄	—	149	4	no	—
285	C₂×C₄	—	149	4	no	—
286	C₂×C₄	#14558735	1	4	yes	—
287	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
289	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂×#14558735	1	2	yes	—
300	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	298	4	no	—
306	C₄	—	298	4	no	—
307	C₄	—	298	4	no	—
308	C₄	—	298	4	no	—
309	C₄	C₂×#14558735	1	2	yes	—
310	C₄	C₂×#14558735	1	2	yes	—
311	C₄	#18011958	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	4	no	—
314	C₂	C₂²×#14558735	1	1	yes	characteristic
315	C₂	C₂×#18011958	1	1	yes	characteristic
316	C₂	#23750461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting, prime radical	#532×C₁₄₉	C₂
hypercenter	#532	D₁₄₉
hypocenter	C₁₄₉	#3507
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #23750361	C₂	semidirect
2. #23750350	C₂	semidirect
3. #23750345	C₂	semidirect
4. #23750344	C₂	semidirect
5. #23750332	C₂	semidirect
6. #23750315	C₂	semidirect
7. #23750220	C₂	semidirect
8. #23750196	C₂	semidirect
9. #23750182	C₂	semidirect
10. #23750156	C₂	semidirect
11. #23750108	C₂	semidirect
12. C₂×C₄×Dic₂₉₈	C₂	semidirect
158. C₁₄₉	#3507	semidirect

Maximal subgroups

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

subgroup	index
#23750361	2
#23750350	2
#23750345	2
#23750344	2
#23750332	2
#23750315	2
#23750220	2
#23750196	2
#23750182	2
#23750156	2
#23750108	2
C₂×C₄×Dic₂₉₈	2
#532×C₁₄₉	2
#3507	149