Subgroups — #34077632

Order 19328.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18012785	C₂	1	1	yes	maximal, characteristic
2	C₂×#18012745	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752692	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752216	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752215	C₂	1	2	yes	maximal, complemented
6	#23752213	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752212	C₂	1	2	yes	maximal, complemented
8	#23752207	C₂	1	2	yes	maximal, complemented
9	C₂×#18012661	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₁₅₁	C₂	1	2	yes	maximal, complemented
11	#416×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#18012785	—	2	4	no	—
13	#18012785	C₄	1	8	yes	complemented
14	C₂×#14559005	C₂²	1	2	yes	—
15	C₂×#14559004	—	2	1	no	—
16	C₂×#14559004	C₂²	1	1	yes	characteristic
17	C₂²×D₆₀₄	—	2	1	no	—
18	C₂²×D₆₀₄	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₁₅₁	—	2	2	no	—
20	C₂×D₄×D₁₅₁	C₂²	1	2	yes	—
21	#18012745	—	2	4	no	—
22	C₂×#14558988	C₂²	1	1	yes	characteristic, complemented
23	C₂×#14558987	C₂²	1	2	yes	complemented
24	C₂×C₈×D₁₅₁	C₂²	1	2	yes	complemented
25	C₄×Dic₃₀₂	—	2	1	no	—
26	C₄×D₆₀₄	—	2	1	no	—
27	(C₂²⋊D₁₅₁)⋊C₄	—	2	2	no	—
28	#18012697	C₂²	1	2	yes	complemented
29	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₄×D₁₅₁	C₂²	1	2	yes	complemented
31	#18012663	C₂²	1	2	yes	complemented
32	#18012661	C₂²	1	2	yes	complemented
33	#18012661	C₂²	1	2	yes	complemented
34	#18012660	C₂²	1	2	yes	complemented
35	#18012659	C₂²	1	2	yes	complemented
36	#18012659	C₂²	1	2	yes	complemented
37	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
39	#117×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
40	#103×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
41	#103×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
42	#102×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
43	#102×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
44	C₂³×D₁₅₁	—	2	2	no	—
45	C₂³×D₁₅₁	—	2	1	no	—
46	#14559005	—	2	4	no	—
47	#14559005	C₂×C₄	1	8	yes	complemented
48	#14559004	—	2	8	no	—
49	#14559004	C₂×C₄	1	4	yes	—
50	#14559004	D₄	1	4	yes	—
51	C₂×Dic₃₀₂	D₄	1	1	yes	characteristic
52	C₂×D₆₀₄	—	2	2	no	—
53	C₂×D₆₀₄	—	2	8	no	—
54	C₂×D₆₀₄	—	2	1	no	—
55	C₂×D₆₀₄	—	2	2	no	—
56	C₂×D₆₀₄	C₂×C₄	1	4	yes	complemented
57	C₂×D₆₀₄	D₄	1	1	yes	characteristic
58	C₂×D₆₀₄	C₂³	1	2	yes	—
59	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
60	C₂×C₂²⋊D₁₅₁	D₄	1	2	yes	—
61	D₄×D₁₅₁	—	2	16	no	—
62	D₄×D₁₅₁	—	2	4	no	—
63	D₄×D₁₅₁	C₂×C₄	1	8	yes	complemented
64	C₂²×Dic₁₅₁	—	2	1	no	—
65	C₂×C₄×D₁₅₁	—	2	2	no	—
66	C₂×C₄×D₁₅₁	—	2	2	no	—
67	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
68	#14558988	—	2	2	no	—
69	#14558987	—	2	4	no	—
70	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
71	C₈×D₁₅₁	—	2	4	no	—
72	C₄⋊Dic₁₅₁	—	2	1	no	—
73	C₄⋊Dic₁₅₁	C₂³	1	2	yes	—
74	Dic₁₅₁⋊C₄	—	2	2	no	—
75	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
76	C₄×Dic₁₅₁	—	2	2	no	—
77	#41×C₁₅₁	—	2	2	no	—
78	#41×C₁₅₁	C₂×C₄	1	4	yes	complemented
79	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₅₁	—	2	1	no	—
81	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₅₁	—	2	1	no	—
83	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
90	C₄²×C₁₅₁	—	2	1	no	—
91	C₂²×D₁₅₁	—	2	2	no	—
92	C₂²×D₁₅₁	—	2	4	no	—
93	C₂²×D₁₅₁	—	2	8	no	—
94	C₂²×D₁₅₁	—	8	2	no	—
95	C₂²×D₁₅₁	—	8	2	no	—
96	C₂²×D₁₅₁	—	8	1	no	—
97	C₂²×D₁₅₁	—	8	1	no	—
98	C₂²×D₁₅₁	C₂×D₄	1	2	yes	—
99	Dic₃₀₂	C₂²⋊C₄	1	4	yes	—
100	D₆₀₄	—	2	8	no	—
101	D₆₀₄	—	2	4	no	—
102	D₆₀₄	—	2	2	no	—
103	D₆₀₄	C₂²⋊C₄	1	4	yes	—
104	D₆₀₄	C₂²×C₄	1	4	yes	—
105	C₂²⋊D₁₅₁	—	2	8	no	—
106	C₂²⋊D₁₅₁	C₂²⋊C₄	1	8	yes	—
107	C₂×Dic₁₅₁	—	2	2	no	—
108	C₂×Dic₁₅₁	—	8	1	no	—
109	C₂×Dic₁₅₁	C₂×D₄	1	2	yes	—
110	C₄×D₁₅₁	—	2	8	no	—
111	C₄×D₁₅₁	—	8	2	no	—
112	C₄×D₁₅₁	C₂²×C₄	1	4	yes	—
113	C₄×D₁₅₁	C₂×D₄	1	4	yes	—
114	C₁₅₁⋊C₈	—	2	2	no	—
115	C₂³×C₁₅₁	—	2	1	no	—
116	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₁₅₁	—	2	1	no	—
118	Q₈×C₁₅₁	C₂²×C₄	1	2	yes	—
119	D₄×C₁₅₁	—	2	4	no	—
120	D₄×C₁₅₁	—	2	1	no	—
121	D₄×C₁₅₁	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₁₅₁	—	2	1	no	—
123	C₂×C₄×C₁₅₁	—	2	4	no	—
124	C₂×C₄×C₁₅₁	—	2	1	no	—
125	C₂×C₄×C₁₅₁	—	2	1	no	—
126	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
127	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₁₅₁	—	2	1	no	—
131	C₈×C₁₅₁	—	2	1	no	—
132	C₂²×C₁₅₁	—	2	1	no	—
133	C₂²×C₁₅₁	—	2	1	no	—
134	C₂²×C₁₅₁	—	2	2	no	—
135	C₂²×C₁₅₁	—	4	1	no	—
136	C₂²×C₁₅₁	—	4	1	no	—
137	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₅₁	—	2	2	no	—
140	C₂×D₁₅₁	—	2	2	no	—
141	C₂×D₁₅₁	—	2	4	no	—
142	C₂×D₁₅₁	—	4	2	no	—
143	C₂×D₁₅₁	—	4	2	no	—
144	C₂×D₁₅₁	—	8	8	no	—
145	C₂×D₁₅₁	—	8	4	no	—
146	C₂×D₁₅₁	C₂×C₂²⋊C₄	1	4	yes	—
147	C₄×C₁₅₁	—	2	2	no	—
148	C₄×C₁₅₁	—	4	1	no	—
149	C₄×C₁₅₁	—	4	1	no	—
150	C₄×C₁₅₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
152	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
153	Dic₁₅₁	—	4	2	no	—
154	Dic₁₅₁	C₂×C₂²⋊C₄	1	4	yes	—
155	C₂×C₁₅₁	—	2	2	no	—
156	C₂×C₁₅₁	—	4	2	no	—
157	C₂×C₁₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₁₅₁	#576	1	1	yes	characteristic
159	C₂×C₁₅₁	#577	1	1	yes	characteristic
160	D₁₅₁	—	2	4	no	—
161	D₁₅₁	—	4	4	no	—
162	C₁₅₁	#2897	1	1	yes	characteristic, complemented, Sylow
163	#2897	—	151	1	no	maximal, Sylow
164	C₂×H(5,𝔽₂)	—	151	1	no	—
165	C₂×#131	—	151	1	no	—
166	#520	—	151	1	no	—
167	#418	—	151	1	no	—
168	#418	—	151	2	no	—
169	#417	—	151	1	no	—
170	#417	—	151	2	no	—
171	#416	—	151	2	no	—
172	#416	D₁₅₁	1	1	yes	characteristic, complemented
173	C₂×#102	—	151	1	no	—
174	C₂×#102	—	151	2	no	—
175	H(5,𝔽₂)	—	302	4	no	—
176	H(5,𝔽₂)	—	151	8	no	—
177	C₂×#41	—	302	1	no	—
178	C₂×#41	—	151	2	no	—
179	C₂×#41	—	151	1	no	—
180	C₂×#41	C₂×D₁₅₁	1	1	yes	characteristic
181	C₂²×D₄	—	302	1	no	—
182	C₂²×D₄	—	302	2	no	—
183	C₂²×D₄	—	151	2	no	—
184	C₂²×D₄	—	151	1	no	—
185	#131	—	302	4	no	—
186	C₂×M₁₆	—	151	1	no	—
187	C₂×M₁₆	—	151	2	no	—
188	C₂²×C₈	—	151	2	no	—
189	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic, complemented
190	C₄×Q₈	—	302	1	no	—
191	C₄×D₄	—	302	1	no	—
192	C₄×D₄	—	302	2	no	—
193	#117	—	151	2	no	—
194	#117	C₂×D₁₅₁	1	1	yes	characteristic, complemented
195	C₂×C₄⋊C₄	—	151	1	no	—
196	C₂×C₄⋊C₄	—	151	2	no	—
197	#103	—	151	2	no	—
198	#103	C₂×D₁₅₁	1	1	yes	characteristic, complemented
199	#103	C₂×D₁₅₁	1	1	yes	characteristic, complemented
200	#102	—	151	2	no	—
201	#102	—	151	2	no	—
202	#102	—	151	2	no	—
203	#102	—	151	2	no	—
204	#102	—	151	2	no	—
205	#102	C₂×D₁₅₁	1	1	yes	characteristic, complemented
206	#102	C₂×D₁₅₁	1	1	yes	characteristic, complemented
207	C₂⁴	—	302	2	no	—
208	C₂⁴	—	302	1	no	—
209	#41	—	2	2	no	—
210	#41	—	302	8	no	—
211	#41	—	302	4	no	—
212	#41	—	151	8	no	—
213	#41	—	151	4	no	—
214	#41	—	151	4	no	—
215	#41	C₄×D₁₅₁	1	4	yes	complemented
216	C₂×Q₈	—	151	1	no	—
217	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	302	2	no	—
220	C₂×D₄	—	302	8	no	—
221	C₂×D₄	—	302	16	no	—
222	C₂×D₄	—	302	4	no	—
223	C₂×D₄	—	302	2	no	—
224	C₂×D₄	—	302	1	no	—
225	C₂×D₄	—	302	2	no	—
226	C₂×D₄	—	151	8	no	—
227	C₂×D₄	—	151	2	no	—
228	C₂×D₄	—	151	4	no	—
229	C₂×D₄	—	151	1	no	—
230	C₂×D₄	—	151	2	no	—
231	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	302	2	no	—
234	C₂²×C₄	—	302	2	no	—
235	C₂²×C₄	—	302	1	no	—
236	C₂²×C₄	—	151	2	no	—
237	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
238	M₁₆	—	302	4	no	—
239	M₁₆	—	302	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	302	4	no	—
242	C₂×C₈	—	151	2	no	—
243	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
245	C₄⋊C₄	—	302	2	no	—
246	C₄⋊C₄	—	302	1	no	—
247	C₄⋊C₄	—	151	2	no	—
248	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	302	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	302	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	302	8	no	—
256	C₂³	—	302	2	no	—
257	C₂³	—	302	4	no	—
258	C₂³	—	151	2	no	—
259	C₂³	—	1208	1	no	—
260	C₂³	—	1208	1	no	—
261	C₂³	—	1208	2	no	—
262	C₂³	—	1208	2	no	—
263	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	151	4	no	—
266	Q₈	C₂×C₄×D₁₅₁	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	1	no	—
269	D₄	—	302	8	no	—
270	D₄	—	302	8	no	—
271	D₄	—	302	4	no	—
272	D₄	—	302	2	no	—
273	D₄	—	151	8	no	—
274	D₄	—	151	4	no	—
275	D₄	—	151	4	no	—
276	D₄	C₂×C₄×D₁₅₁	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	302	8	no	—
282	C₂×C₄	—	302	2	no	—
283	C₂×C₄	—	151	2	no	—
284	C₂×C₄	—	151	4	no	—
285	C₂×C₄	—	151	4	no	—
286	C₂×C₄	—	1208	2	no	—
287	C₂×C₄	—	1208	1	no	—
288	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
289	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
291	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	302	2	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	302	2	no	—
301	C₂²	—	302	2	no	—
302	C₂²	—	302	4	no	—
303	C₂²	—	604	2	no	—
304	C₂²	—	604	2	no	—
305	C₂²	—	151	4	no	—
306	C₂²	—	1208	4	no	—
307	C₂²	—	1208	8	no	—
308	C₂²	C₂²⋊C₄×D₁₅₁	1	2	yes	—
309	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	604	2	no	—
314	C₄	—	151	4	no	—
315	C₄	C₂²⋊C₄×D₁₅₁	1	2	yes	—
316	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
317	C₄	C₂×D₄×D₁₅₁	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	302	4	no	—
321	C₂	—	604	4	no	—
322	C₂	C₂×C₂²⋊C₄×D₁₅₁	1	1	yes	characteristic
323	C₂	#23753014	1	1	yes	characteristic
324	C₂	#23753019	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator	C₄×C₁₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#416×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#18012785	C₂
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#416	D₁₅₁
hypocenter	C₁₅₁	#2897
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. C₂×#18012745	C₂	semidirect
3. #23752692	C₂	semidirect
4. #23752216	C₂	semidirect
5. #23752215	C₂	semidirect
6. #23752213	C₂	semidirect
7. #23752212	C₂	semidirect
8. #23752207	C₂	semidirect
9. C₂×#18012661	C₂	semidirect
10. #102×D₁₅₁	C₂	semidirect
11. #416×C₁₅₁	C₂	semidirect
13. #18012785	C₄	semidirect
22. C₂×#14558988	C₂²	semidirect
23. C₂×#14558987	C₂²	semidirect
24. C₂×C₈×D₁₅₁	C₂²	semidirect
28. #18012697	C₂²	semidirect
29. C₂×C₄⋊Dic₁₅₁	C₂²	semidirect
30. C₄⋊C₄×D₁₅₁	C₂²	semidirect
31. #18012663	C₂²	semidirect
32. #18012661	C₂²	semidirect
33. #18012661	C₂²	semidirect
34. #18012660	C₂²	semidirect
35. #18012659	C₂²	semidirect
36. #18012659	C₂²	semidirect
38. C₂²×C₈×C₁₅₁	C₂²	semidirect
39. #117×C₁₅₁	C₂²	semidirect
40. #103×C₁₅₁	C₂²	semidirect
41. #103×C₁₅₁	C₂²	semidirect
42. #102×C₁₅₁	C₂²	semidirect
43. #102×C₁₅₁	C₂²	semidirect
47. #14559005	C₂×C₄	semidirect
56. C₂×D₆₀₄	C₂×C₄	semidirect
63. D₄×D₁₅₁	C₂×C₄	semidirect
78. #41×C₁₅₁	C₂×C₄	semidirect
162. C₁₅₁	#2897	semidirect
172. #416	D₁₅₁	semidirect
189. C₂²×C₈	C₂×D₁₅₁	semidirect
194. #117	C₂×D₁₅₁	semidirect
198. #103	C₂×D₁₅₁	semidirect
199. #103	C₂×D₁₅₁	semidirect
205. #102	C₂×D₁₅₁	semidirect
206. #102	C₂×D₁₅₁	semidirect
215. #41	C₄×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18012785	2
C₂×#18012745	2
#23752692	2
#23752216	2
#23752215	2
#23752213	2
#23752212	2
#23752207	2
C₂×#18012661	2
#102×D₁₅₁	2
#416×C₁₅₁	2
#2897	151