Subgroups — #46820041

Order 34432.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771139	C₂	1	1	yes	maximal, characteristic
2	#31770732	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770729	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770682	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770676	C₂	1	2	yes	maximal, complemented
6	#31770670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₂₆₉	C₂	1	2	yes	maximal, complemented
8	#31770665	C₂	1	2	yes	maximal, complemented
9	#31770658	C₂	1	1	yes	maximal, characteristic, complemented
10	#31770630	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864385	C₂²	1	2	yes	—
13	C₂×C₈×D₂₆₉	C₂²	1	2	yes	—
14	#22575447	C₄	1	2	yes	—
15	#22575445	C₄	1	1	yes	characteristic
16	#22575441	C₄	1	1	yes	characteristic
17	#22575436	C₄	1	2	yes	—
18	#22575432	C₄	1	1	yes	characteristic
19	#22575427	C₄	1	1	yes	characteristic
20	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic
21	C₄×D₁₀₇₆	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
23	#22575410	C₂²	1	2	yes	—
24	C₄⋊#14524130	C₂²	1	1	yes	characteristic
25	C₄⋊#14524130	C₂²	1	1	yes	characteristic
26	#22575382	C₂²	1	2	yes	—
27	#22575382	C₂²	1	2	yes	—
28	C₂²⋊#14524130	C₂²	1	1	yes	characteristic
29	C₂²⋊#14524130	C₂²	1	1	yes	characteristic
30	#22575365	C₂²	1	2	yes	—
31	#22575365	C₂²	1	2	yes	—
32	#22575363	C₂²	1	2	yes	—
33	C₄×#14524130	C₂²	1	1	yes	characteristic
34	C₄×#14524130	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₆₉	C₂²	1	2	yes	—
36	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	—
37	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₆₉	C₂²	1	2	yes	—
39	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₅₃₈	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₀₇₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₆₉	C₂³	1	2	yes	—
45	#16864385	—	4	2	no	—
46	C₂×#14524130	C₂³	1	1	yes	characteristic
47	C₂×#14524130	C₂³	1	1	yes	characteristic
48	C₂×#14524130	C₂³	1	1	yes	characteristic
49	C₂×#14524130	C₂³	1	1	yes	characteristic
50	C₈×D₂₆₉	—	4	2	no	—
51	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
53	#16864381	C₂×C₄	1	2	yes	—
54	#16864381	C₂×C₄	1	2	yes	—
55	#16864381	C₂×C₄	1	2	yes	—
56	#16864381	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₂₆₉	C₂×C₄	1	2	yes	—
58	#16864379	C₂×C₄	1	2	yes	—
59	#16864379	C₂×C₄	1	2	yes	—
60	#16864379	C₂×C₄	1	2	yes	—
61	#16864379	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₆₉	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₆₉	—	4	2	no	—
67	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
68	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
69	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
72	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
73	C₂²×D₂₆₉	C₂²×C₄	1	2	yes	—
74	Dic₅₃₈	—	2	2	no	—
75	D₁₀₇₆	—	2	2	no	—
76	C₂²⋊D₂₆₉	—	2	4	no	—
77	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
80	C₄×D₂₆₉	—	2	2	no	—
81	C₄×D₂₆₉	—	2	2	no	—
82	#14524130	—	2	1	no	—
83	#14524130	—	2	1	no	—
84	#14524130	—	2	1	no	—
85	#14524130	—	2	1	no	—
86	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₆₉	—	2	2	no	—
88	C₂×C₄×C₂₆₉	—	2	2	no	—
89	C₂×C₄×C₂₆₉	—	2	1	no	—
90	C₂×C₄×C₂₆₉	—	2	1	no	—
91	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₆₉	—	2	2	no	—
98	C₈×C₂₆₉	—	2	2	no	—
99	C₂²×C₂₆₉	—	2	1	no	—
100	C₂²×C₂₆₉	—	2	1	no	—
101	C₂²×C₂₆₉	#131	1	2	yes	—
102	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₂₆₉	—	2	2	no	—
104	C₂×D₂₆₉	—	2	2	no	—
105	C₂×D₂₆₉	—	2	2	no	—
106	C₄×C₂₆₉	—	2	1	no	—
107	C₄×C₂₆₉	—	2	1	no	—
108	C₄×C₂₆₉	—	2	1	no	—
109	C₄×C₂₆₉	—	2	1	no	—
110	C₄×C₂₆₉	#131	1	2	yes	—
111	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₂₆₉	—	2	2	no	—
114	Dic₂₆₉	—	2	2	no	—
115	Dic₂₆₉	—	2	2	no	—
116	C₂×C₂₆₉	—	2	2	no	—
117	C₂×C₂₆₉	#520	1	1	yes	characteristic
118	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
119	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
120	D₂₆₉	—	4	2	no	—
121	C₂₆₉	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	269	1	no	maximal, Sylow
123	#529	—	269	1	no	—
124	#446	—	269	1	no	—
125	C₈×Q₈	—	269	1	no	—
126	#436	—	269	1	no	—
127	#435	—	269	2	no	—
128	C₈×D₄	—	269	2	no	—
129	C₈×D₄	—	269	1	no	—
130	#433	—	269	2	no	—
131	#431	—	269	1	no	—
132	#424	—	269	2	no	—
133	C₂×C₄⋊C₈	D₂₆₉	1	1	yes	characteristic, complemented
134	C₂×M₁₆	—	269	2	no	—
135	C₂²×C₈	—	269	2	no	—
136	C₂²×C₈	C₂×D₂₆₉	1	2	yes	—
137	#126	—	269	2	no	—
138	#125	—	269	1	no	—
139	#124	—	269	1	no	—
140	#123	—	269	2	no	—
141	C₂²⋊Q₈	—	269	1	no	—
142	#121	—	269	1	no	—
143	C₄×Q₈	—	269	1	no	—
144	C₄×D₄	—	269	2	no	—
145	C₄×D₄	—	269	1	no	—
146	#117	—	269	2	no	—
147	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
148	C₄⋊C₈	—	269	2	no	—
149	C₄⋊C₈	—	269	2	no	—
150	C₄⋊C₈	—	269	1	no	—
151	C₄⋊C₈	—	269	1	no	—
152	C₄⋊C₈	C₂×D₂₆₉	1	2	yes	—
153	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
155	C₂²⋊C₈	—	269	1	no	—
156	C₂²⋊C₈	—	269	1	no	—
157	C₂²⋊C₈	—	269	2	no	—
158	C₂²⋊C₈	—	269	2	no	—
159	#97	—	269	2	no	—
160	C₄×C₈	—	269	2	no	—
161	C₄×C₈	—	269	1	no	—
162	C₄×C₈	—	269	1	no	—
163	C₂×Q₈	—	269	1	no	—
164	C₂×D₄	—	269	2	no	—
165	C₂×D₄	—	269	1	no	—
166	C₂²×C₄	—	269	2	no	—
167	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
170	M₁₆	—	1076	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	269	1	no	—
173	C₂×C₈	—	269	1	no	—
174	C₂×C₈	—	269	1	no	—
175	C₂×C₈	—	269	1	no	—
176	C₂×C₈	—	1076	2	no	—
177	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
178	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
179	C₄⋊C₄	—	269	2	no	—
180	C₄⋊C₄	—	269	2	no	—
181	C₄⋊C₄	—	269	1	no	—
182	C₄⋊C₄	—	269	2	no	—
183	C₄⋊C₄	—	269	1	no	—
184	C₄⋊C₄	—	269	2	no	—
185	C₂²⋊C₄	—	269	2	no	—
186	C₂²⋊C₄	—	269	2	no	—
187	C₂²⋊C₄	—	269	2	no	—
188	C₂²⋊C₄	—	269	2	no	—
189	C₂²⋊C₄	—	269	2	no	—
190	C₄²	—	269	2	no	—
191	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
192	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
193	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
194	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
195	C₂³	—	269	2	no	—
196	C₂³	C₂×C₄×D₂₆₉	1	1	yes	characteristic
197	Q₈	—	538	2	no	—
198	D₄	—	538	4	no	—
199	D₄	—	538	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	269	2	no	—
205	C₂×C₄	—	269	2	no	—
206	C₂×C₄	—	269	2	no	—
207	C₂×C₄	—	538	2	no	—
208	C₂×C₄	—	538	2	no	—
209	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	538	1	no	—
218	C₈	—	538	1	no	—
219	C₈	—	538	1	no	—
220	C₈	—	538	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	538	2	no	—
224	C₂²	—	538	2	no	—
225	C₂²	—	538	2	no	—
226	C₂²	#22575461	1	2	yes	—
227	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	538	2	no	—
233	C₄	—	538	2	no	—
234	C₄	—	538	2	no	—
235	C₄	#22575459	1	2	yes	—
236	C₄	#22575498	1	1	yes	characteristic
237	C₄	#22575501	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	1076	2	no	—
240	C₂	#31771086	1	1	yes	characteristic
241	C₂	C₂×#22575459	1	1	yes	characteristic
242	C₂	C₂×#22575461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₂₆₉	C₂
prime radical	#22575427	C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₆₉
hypocenter	C₂₆₉	#2994
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #31770732	C₂	semidirect
3. #31770729	C₂	semidirect
4. #31770682	C₂	semidirect
5. #31770676	C₂	semidirect
6. #31770670	C₂	semidirect
7. C₈×C₂²⋊D₂₆₉	C₂	semidirect
8. #31770665	C₂	semidirect
9. #31770658	C₂	semidirect
10. #31770630	C₂	semidirect
11. C₂×C₄⋊C₈×C₂₆₉	C₂	semidirect
121. C₂₆₉	#2994	semidirect
133. C₂×C₄⋊C₈	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771139	2
#31770732	2
#31770729	2
#31770682	2
#31770676	2
#31770670	2
C₈×C₂²⋊D₂₆₉	2
#31770665	2
#31770658	2
#31770630	2
C₂×C₄⋊C₈×C₂₆₉	2
#2994	269