Subgroups — #34083207

Order 19328.

992 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752941	C₂	1	3	yes	maximal, complemented
2	#23752936	C₂	1	3	yes	maximal, complemented
3	#23752900	C₂	1	6	yes	maximal, complemented
4	Q₈×D₆₀₄	C₂	1	2	yes	maximal, complemented
5	Q₈×D₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752825	C₂	1	6	yes	maximal, complemented
7	#23752767	C₂	1	3	yes	maximal, complemented
8	#23752749	C₂	1	3	yes	maximal
9	#23752688	C₂	1	2	yes	maximal, complemented
10	C₄×Q₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#557×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₅₁	C₂²	1	2	yes	—
14	#18012739	C₂²	1	3	yes	—
15	#18012739	C₂²	1	6	yes	—
16	#18012734	C₂²	1	6	yes	—
17	#18012734	C₂²	1	12	yes	—
18	#18012730	C₂²	1	3	yes	—
19	#18012730	C₂²	1	6	yes	—
20	#18012730	C₂²	1	12	yes	—
21	#18012728	C₂²	1	3	yes	—
22	#18012724	C₂²	1	6	yes	—
23	#18012719	C₂²	1	6	yes	—
24	#18012719	C₂²	1	6	yes	—
25	#18012719	C₂²	1	6	yes	—
26	#18012715	C₂²	1	3	yes	—
27	#18012714	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₅₁	C₂²	1	6	yes	—
29	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₀₂	C₂²	1	6	yes	—
31	C₄×Dic₃₀₂	C₂²	1	3	yes	—
32	#18012704	C₂²	1	6	yes	—
33	#18012704	C₂²	1	3	yes	—
34	C₄×D₆₀₄	C₂²	1	6	yes	—
35	C₄×D₆₀₄	C₂²	1	3	yes	—
36	#18012697	C₂²	1	6	yes	—
37	#18012697	C₂²	1	6	yes	—
38	#18012696	C₂²	1	6	yes	—
39	C₄⋊C₄×D₁₅₁	C₂²	1	3	yes	—
40	C₄²×D₁₅₁	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₁₅₁	C₂²	1	3	yes	—
42	#125×C₁₅₁	C₂²	1	6	yes	—
43	C₄×Q₈×C₁₅₁	C₂²	1	3	yes	—
44	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₁₅₁	C₂²	1	2	yes	—
46	C₂×Dic₃₀₂	C₂³	1	6	yes	—
47	C₂×Dic₃₀₂	C₂³	1	3	yes	—
48	Q₈×D₁₅₁	—	2	4	no	—
49	Q₈×D₁₅₁	—	2	8	no	—
50	C₂×D₆₀₄	C₂³	1	2	yes	—
51	C₂×D₆₀₄	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₅₁	C₂³	1	3	yes	—
54	C₂×C₄×D₁₅₁	C₂³	1	3	yes	—
55	C₂×C₄×D₁₅₁	C₂³	1	6	yes	—
56	C₄⋊Dic₁₅₁	C₂³	1	3	yes	—
57	C₄⋊Dic₁₅₁	C₂³	1	6	yes	—
58	C₄⋊Dic₁₅₁	C₂³	1	3	yes	—
59	C₄⋊Dic₁₅₁	C₂³	1	6	yes	—
60	C₄⋊Dic₁₅₁	C₂³	1	12	yes	—
61	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
62	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
63	Dic₁₅₁⋊C₄	C₂³	1	6	yes	—
64	(C₂×D₁₅₁)⋊C₄	C₂³	1	2	yes	—
65	(C₂×D₁₅₁)⋊C₄	C₂³	1	6	yes	—
66	(C₂×D₁₅₁)⋊C₄	C₂³	1	6	yes	—
67	(C₂×D₁₅₁)⋊C₄	C₂³	1	6	yes	—
68	(C₂×D₁₅₁)⋊C₄	C₂³	1	6	yes	—
69	C₄×Dic₁₅₁	C₂³	1	3	yes	—
70	C₄×Dic₁₅₁	C₂³	1	3	yes	—
71	C₄×Dic₁₅₁	C₂³	1	3	yes	—
72	C₄×Dic₁₅₁	C₂³	1	12	yes	—
73	C₂×Q₈×C₁₅₁	C₂³	1	3	yes	—
74	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₅₁	C₂³	1	3	yes	—
76	C₄⋊C₄×C₁₅₁	C₂³	1	6	yes	—
77	C₄⋊C₄×C₁₅₁	C₂³	1	6	yes	—
78	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₅₁	C₂³	1	6	yes	—
80	C₄²×C₁₅₁	C₂³	1	3	yes	—
81	C₄²×C₁₅₁	C₂³	1	6	yes	—
82	C₂²×D₁₅₁	C₂⁴	1	2	yes	—
83	C₂²×D₁₅₁	C₂⁴	1	1	yes	characteristic
84	Dic₃₀₂	—	2	6	no	—
85	Dic₃₀₂	—	2	12	no	—
86	D₆₀₄	—	2	4	no	—
87	D₆₀₄	—	2	2	no	—
88	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₁₅₁	C₂⁴	1	3	yes	—
90	C₂×Dic₁₅₁	C₂⁴	1	3	yes	—
91	C₂×Dic₁₅₁	C₂⁴	1	6	yes	—
92	C₄×D₁₅₁	—	2	6	no	—
93	C₄×D₁₅₁	—	2	6	no	—
94	C₄×D₁₅₁	—	2	12	no	—
95	C₄×D₁₅₁	—	2	2	no	—
96	Q₈×C₁₅₁	—	2	6	no	—
97	Q₈×C₁₅₁	#41	1	4	yes	—
98	C₂×C₄×C₁₅₁	C₂⁴	1	6	yes	—
99	C₂×C₄×C₁₅₁	C₂⁴	1	3	yes	—
100	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₅₁	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
103	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
104	C₂×D₁₅₁	—	2	1	no	—
105	C₂×D₁₅₁	—	2	1	no	—
106	C₂×D₁₅₁	—	2	2	no	—
107	C₂×D₁₅₁	—	2	2	no	—
108	C₂×D₁₅₁	—	2	2	no	—
109	C₂×D₁₅₁	2₋¹⁺⁴	1	2	yes	—
110	C₄×C₁₅₁	—	2	6	no	—
111	C₄×C₁₅₁	—	2	3	no	—
112	C₄×C₁₅₁	—	2	2	no	—
113	C₄×C₁₅₁	C₂×#41	1	6	yes	—
114	C₄×C₁₅₁	2₋¹⁺⁴	1	2	yes	—
115	Dic₁₅₁	—	2	3	no	—
116	Dic₁₅₁	—	2	6	no	—
117	Dic₁₅₁	—	2	3	no	—
118	Dic₁₅₁	—	2	1	no	—
119	C₂×C₁₅₁	C₂²×#41	1	1	yes	characteristic
120	C₂×C₁₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×C₁₅₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
122	D₁₅₁	—	2	2	no	—
123	D₁₅₁	—	4	2	no	—
124	C₁₅₁	#3523	1	1	yes	characteristic, complemented, Sylow
125	#3523	—	151	1	no	maximal, Sylow
126	#557	—	151	3	no	—
127	#557	D₁₅₁	1	1	yes	characteristic, complemented
128	#556	—	151	3	no	—
129	#552	—	151	6	no	—
130	D₄×Q₈	—	151	2	no	—
131	D₄×Q₈	—	151	1	no	—
132	#541	—	151	6	no	—
133	#533	—	151	3	no	—
134	#529	—	151	3	no	—
135	#519	—	151	2	no	—
136	C₂×C₄×Q₈	—	151	1	no	—
137	C₂²×Q₈	—	151	1	no	—
138	C₂²×Q₈	—	151	2	no	—
139	C₄⋊Q₈	—	151	3	no	—
140	C₄⋊Q₈	—	151	6	no	—
141	C₄⋊Q₈	C₂×D₁₅₁	1	3	yes	—
142	#126	—	151	6	no	—
143	#126	—	151	12	no	—
144	#125	—	151	3	no	—
145	#125	—	151	6	no	—
146	#125	—	151	12	no	—
147	#125	C₂×D₁₅₁	1	6	yes	—
148	#124	—	151	3	no	—
149	#123	—	151	6	no	—
150	C₂²⋊Q₈	—	151	3	no	—
151	C₂²⋊Q₈	—	151	6	no	—
152	C₂²⋊Q₈	—	151	6	no	—
153	C₂²⋊Q₈	—	151	6	no	—
154	#121	—	151	1	no	—
155	C₄×Q₈	—	151	1	no	—
156	C₄×Q₈	—	151	6	no	—
157	C₄×Q₈	—	151	3	no	—
158	C₄×Q₈	—	151	6	no	—
159	C₄×Q₈	—	151	3	no	—
160	C₄×Q₈	—	151	6	no	—
161	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
162	C₄×Q₈	C₂×D₁₅₁	1	2	yes	—
163	C₄×Q₈	C₂×D₁₅₁	1	3	yes	—
164	C₄×D₄	—	151	6	no	—
165	C₄×D₄	—	151	3	no	—
166	#117	—	151	6	no	—
167	#117	—	151	6	no	—
168	#117	—	151	6	no	—
169	C₂×C₄⋊C₄	—	151	3	no	—
170	C₂×C₄²	—	151	3	no	—
171	C₂×Q₈	—	302	4	no	—
172	C₂×Q₈	—	302	8	no	—
173	C₂×Q₈	—	151	3	no	—
174	C₂×Q₈	—	151	6	no	—
175	C₂×Q₈	C₂²×D₁₅₁	1	3	yes	—
176	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
177	C₂×D₄	—	151	2	no	—
178	C₂×D₄	—	151	1	no	—
179	C₂²×C₄	—	151	1	no	—
180	C₂²×C₄	—	151	3	no	—
181	C₂²×C₄	—	151	3	no	—
182	C₂²×C₄	—	151	6	no	—
183	C₄⋊C₄	—	151	6	no	—
184	C₄⋊C₄	—	151	6	no	—
185	C₄⋊C₄	—	151	6	no	—
186	C₄⋊C₄	—	151	3	no	—
187	C₄⋊C₄	—	151	6	no	—
188	C₄⋊C₄	—	151	3	no	—
189	C₄⋊C₄	—	151	6	no	—
190	C₄⋊C₄	—	151	12	no	—
191	C₄⋊C₄	C₂²×D₁₅₁	1	3	yes	—
192	C₄⋊C₄	C₂²×D₁₅₁	1	6	yes	—
193	C₄⋊C₄	C₂²×D₁₅₁	1	6	yes	—
194	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₁₅₁	1	6	yes	—
196	C₂²⋊C₄	—	151	2	no	—
197	C₂²⋊C₄	—	151	6	no	—
198	C₂²⋊C₄	—	151	6	no	—
199	C₂²⋊C₄	—	151	6	no	—
200	C₂²⋊C₄	—	151	6	no	—
201	C₄²	—	151	3	no	—
202	C₄²	—	151	3	no	—
203	C₄²	—	151	3	no	—
204	C₄²	—	151	12	no	—
205	C₄²	C₂²×D₁₅₁	1	3	yes	—
206	C₄²	C₂²×D₁₅₁	1	6	yes	—
207	C₂³	—	151	2	no	—
208	C₂³	—	151	1	no	—
209	Q₈	—	2	6	no	—
210	Q₈	—	302	6	no	—
211	Q₈	—	302	12	no	—
212	Q₈	#14559005	1	4	yes	—
213	D₄	—	302	4	no	—
214	D₄	—	302	2	no	—
215	C₂×C₄	—	302	6	no	—
216	C₂×C₄	—	302	6	no	—
217	C₂×C₄	—	302	12	no	—
218	C₂×C₄	—	302	2	no	—
219	C₂×C₄	—	151	3	no	—
220	C₂×C₄	—	151	1	no	—
221	C₂×C₄	—	151	3	no	—
222	C₂×C₄	—	151	6	no	—
223	C₂×C₄	C₂³×D₁₅₁	1	6	yes	—
224	C₂×C₄	C₂³×D₁₅₁	1	3	yes	—
225	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
226	C₂×C₄	C₂³×D₁₅₁	1	3	yes	—
227	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
228	C₂²	—	302	1	no	—
229	C₂²	—	302	1	no	—
230	C₂²	—	302	2	no	—
231	C₂²	—	302	2	no	—
232	C₂²	—	302	2	no	—
233	C₂²	—	151	2	no	—
234	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
235	C₄	—	2	3	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	6	no	—
238	C₄	—	302	3	no	—
239	C₄	—	302	6	no	—
240	C₄	—	302	3	no	—
241	C₄	—	302	1	no	—
242	C₄	C₂×#14559005	1	6	yes	—
243	C₄	#18012786	1	2	yes	—
244	C₂	—	302	2	no	—
245	C₂	—	604	2	no	—
246	C₂	C₂²×#14559005	1	1	yes	characteristic
247	C₂	2₋¹⁺⁴×D₁₅₁	1	1	yes	characteristic
248	C₂	C₂×#18012786	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Fitting	#557×C₁₅₁	C₂
prime radical	#18012714	C₂²
hypercenter	#557	D₁₅₁
hypocenter	C₁₅₁	#3523
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. #23752941	C₂	semidirect
2. #23752936	C₂	semidirect
3. #23752900	C₂	semidirect
4. Q₈×D₆₀₄	C₂	semidirect
5. Q₈×D₆₀₄	C₂	semidirect
6. #23752825	C₂	semidirect
7. #23752767	C₂	semidirect
9. #23752688	C₂	semidirect
10. C₄×Q₈×D₁₅₁	C₂	semidirect
11. #557×C₁₅₁	C₂	semidirect
124. C₁₅₁	#3523	semidirect
127. #557	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752941	2
#23752936	2
#23752900	2
Q₈×D₆₀₄	2
#23752825	2
#23752767	2
#23752749	2
#23752688	2
C₄×Q₈×D₁₅₁	2
#557×C₁₅₁	2
#3523	151