Subgroups — #34078041

Order 19328.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752766	C₂	1	1	yes	maximal, characteristic
2	#23752342	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752340	C₂	1	2	yes	maximal, complemented
4	Q₈×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752336	C₂	1	2	yes	maximal, complemented
6	#23752336	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752237	C₂	1	2	yes	maximal
8	#23752234	C₂	1	1	yes	maximal, characteristic
9	#23752169	C₂	1	2	yes	maximal
10	C₄×C₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14558987	C₂²	1	2	yes	—
13	C₂×C₈×D₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₅₁	C₂²	1	1	yes	characteristic
15	#18012738	C₄	1	1	yes	characteristic
16	#18012737	C₄	1	1	yes	characteristic
17	#18012729	C₄	1	2	yes	—
18	#18012719	C₄	1	1	yes	characteristic
19	#18012717	C₄	1	2	yes	—
20	#18012715	C₄	1	1	yes	characteristic
21	C₄×Dic₃₀₂	C₂²	1	2	yes	—
22	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
23	#18012696	C₂²	1	2	yes	—
24	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₅₁⋊C₈	C₂²	1	2	yes	—
28	Dic₁₅₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₅₁⋊C₈	C₂²	1	2	yes	—
30	#18012649	C₂²	1	2	yes	—
31	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
32	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	—
33	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₅₁	C₂²	1	2	yes	—
35	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₅₁	C₂²	1	2	yes	—
39	#97×C₁₅₁	C₂²	1	2	yes	—
40	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₀₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₃₀₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₅₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₅₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
46	#14558987	—	2	4	no	—
47	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₈×D₁₅₁	—	2	2	no	—
51	C₈×D₁₅₁	Q₈	1	4	yes	complemented
52	C₄⋊Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₅₁	C₂×C₄	1	2	yes	—
54	Dic₁₅₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₅₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₅₁⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₅₁⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₁₅₁⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₅₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₅₁)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₅₁	C₂³	1	2	yes	—
65	C₂×Q₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
68	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₅₁	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
72	C₄²×C₁₅₁	C₂³	1	2	yes	—
73	C₂²×D₁₅₁	C₂²×C₄	1	1	yes	characteristic
74	Dic₃₀₂	—	2	2	no	—
75	Dic₃₀₂	—	2	4	no	—
76	C₂×Dic₁₅₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₅₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₅₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₅₁	—	2	2	no	—
82	C₄×D₁₅₁	—	2	2	no	—
83	C₄×D₁₅₁	C₂×Q₈	1	2	yes	—
84	C₄×D₁₅₁	#41	1	2	yes	—
85	C₁₅₁⋊C₈	—	2	1	no	—
86	C₁₅₁⋊C₈	—	2	2	no	—
87	C₁₅₁⋊C₈	C₂×Q₈	1	2	yes	—
88	Q₈×C₁₅₁	—	2	2	no	—
89	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₅₁	—	2	1	no	—
95	C₈×C₁₅₁	—	2	2	no	—
96	C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₅₁	—	2	1	no	—
99	C₂×D₁₅₁	—	2	1	no	—
100	C₂×D₁₅₁	C₄×Q₈	1	2	yes	—
101	C₄×C₁₅₁	—	2	2	no	—
102	C₄×C₁₅₁	—	2	1	no	—
103	C₄×C₁₅₁	—	2	2	no	—
104	C₄×C₁₅₁	#131	1	2	yes	—
105	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
107	Dic₁₅₁	—	2	1	no	—
108	Dic₁₅₁	—	2	2	no	—
109	Dic₁₅₁	—	2	2	no	—
110	Dic₁₅₁	C₄×Q₈	1	2	yes	—
111	Dic₁₅₁	#131	1	2	yes	—
112	C₂×C₁₅₁	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₅₁	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₅₁	C₂×#131	1	1	yes	characteristic
115	D₁₅₁	—	2	2	no	—
116	C₁₅₁	#2960	1	1	yes	characteristic, complemented, Sylow
117	#2960	—	151	1	no	maximal, Sylow
118	#533	—	151	1	no	—
119	#446	—	151	2	no	—
120	#446	—	151	1	no	—
121	#446	D₁₅₁	1	1	yes	characteristic, complemented
122	C₈×Q₈	—	151	1	no	—
123	C₈×Q₈	—	151	2	no	—
124	C₈×Q₈	—	151	1	no	—
125	#424	—	151	2	no	—
126	#423	—	151	1	no	—
127	#405	—	151	2	no	—
128	C₂×C₄×C₈	—	151	1	no	—
129	C₂×M₁₆	—	151	2	no	—
130	C₂²×C₈	—	151	1	no	—
131	C₂²×C₈	—	151	1	no	—
132	C₄⋊Q₈	—	151	1	no	—
133	C₄⋊Q₈	—	151	1	no	—
134	#125	—	151	2	no	—
135	C₂²⋊Q₈	—	151	1	no	—
136	C₂²⋊Q₈	—	151	2	no	—
137	C₂²⋊Q₈	—	151	1	no	—
138	C₄×Q₈	—	151	2	no	—
139	C₄×Q₈	—	151	1	no	—
140	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
141	#117	—	151	2	no	—
142	C₂×C₄²	—	151	1	no	—
143	C₄⋊C₈	—	151	2	no	—
144	C₄⋊C₈	—	151	2	no	—
145	C₄⋊C₈	—	151	2	no	—
146	C₄⋊C₈	—	151	1	no	—
147	C₄⋊C₈	—	151	2	no	—
148	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₁₅₁	1	2	yes	—
150	#97	—	151	2	no	—
151	#97	C₂×D₁₅₁	1	2	yes	—
152	C₄×C₈	—	151	1	no	—
153	C₄×C₈	—	151	1	no	—
154	C₄×C₈	—	151	2	no	—
155	C₄×C₈	—	151	1	no	—
156	C₄×C₈	—	151	2	no	—
157	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
158	C₂×Q₈	—	151	1	no	—
159	C₂×Q₈	—	151	2	no	—
160	C₂×Q₈	C₄×D₁₅₁	1	1	yes	characteristic
161	C₂²×C₄	—	151	1	no	—
162	C₂²×C₄	—	151	1	no	—
163	C₂²×C₄	—	151	1	no	—
164	M₁₆	—	302	4	no	—
165	C₂×C₈	—	302	2	no	—
166	C₂×C₈	—	151	1	no	—
167	C₂×C₈	—	151	2	no	—
168	C₂×C₈	—	151	1	no	—
169	C₂×C₈	—	151	4	no	—
170	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
172	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
173	C₄⋊C₄	—	151	2	no	—
174	C₄⋊C₄	—	151	2	no	—
175	C₄⋊C₄	—	151	2	no	—
176	C₄⋊C₄	—	151	2	no	—
177	C₄⋊C₄	—	151	2	no	—
178	C₄⋊C₄	—	151	2	no	—
179	C₄⋊C₄	—	151	1	no	—
180	C₄⋊C₄	C₄×D₁₅₁	1	2	yes	—
181	C₄⋊C₄	C₄×D₁₅₁	1	1	yes	characteristic
182	C₂²⋊C₄	—	151	2	no	—
183	C₂²⋊C₄	—	151	2	no	—
184	C₄²	—	151	1	no	—
185	C₄²	—	151	1	no	—
186	C₄²	—	151	1	no	—
187	C₄²	—	151	2	no	—
188	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
189	C₄²	C₂²×D₁₅₁	1	2	yes	—
190	C₂³	—	151	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	302	4	no	—
193	Q₈	—	302	2	no	—
194	C₂×C₄	—	302	2	no	—
195	C₂×C₄	—	302	2	no	—
196	C₂×C₄	—	151	1	no	—
197	C₂×C₄	—	151	2	no	—
198	C₂×C₄	—	151	2	no	—
199	C₂×C₄	—	151	1	no	—
200	C₂×C₄	—	151	1	no	—
201	C₂×C₄	—	151	2	no	—
202	C₂×C₄	—	151	2	no	—
203	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	302	1	no	—
211	C₈	—	302	2	no	—
212	C₈	—	151	2	no	—
213	C₈	Q₈×D₁₅₁	1	2	yes	—
214	C₂²	—	302	1	no	—
215	C₂²	—	302	1	no	—
216	C₂²	—	151	2	no	—
217	C₂²	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	302	2	no	—
222	C₄	—	302	1	no	—
223	C₄	—	302	2	no	—
224	C₄	—	151	2	no	—
225	C₄	—	151	2	no	—
226	C₄	#18012747	1	2	yes	—
227	C₄	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
228	C₄	#41×D₁₅₁	1	1	yes	characteristic
229	C₂	—	302	2	no	—
230	C₂	C₄×Q₈×D₁₅₁	1	1	yes	characteristic
231	C₂	#131×D₁₅₁	1	1	yes	characteristic
232	C₂	C₂×#18012747	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₁
commutator, socle	C₂²×C₁₅₁	C₂³×C₄
Fitting	#446×C₁₅₁	C₂
prime radical	C₂²×D₁₅₁	C₂²×C₄
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#446	D₁₅₁
hypocenter	C₁₅₁	#2960
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #23752342	C₂	semidirect
3. #23752340	C₂	semidirect
4. Q₈×C₁₅₁⋊C₈	C₂	semidirect
5. #23752336	C₂	semidirect
6. #23752336	C₂	semidirect
11. #446×C₁₅₁	C₂	semidirect
51. C₈×D₁₅₁	Q₈	semidirect
116. C₁₅₁	#2960	semidirect
121. #446	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752766	2
#23752342	2
#23752340	2
Q₈×C₁₅₁⋊C₈	2
#23752336	2
#23752237	2
#23752234	2
#23752169	2
C₄×C₈×D₁₅₁	2
#446×C₁₅₁	2
#2960	151