Subgroups — #46819833

Order 34432.

356 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	#31771159	C₂	1	1	yes	maximal, characteristic
2	#31770733	C₂	1	2	yes	maximal, complemented
3	#31770732	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770732	C₂	1	2	yes	maximal, complemented
5	Q₈×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770630	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
9	#31770562	C₂	1	2	yes	maximal
10	C₄×#16864385	C₂	1	1	yes	maximal, characteristic
11	#446×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864385	C₂²	1	1	yes	characteristic
13	C₂×#16864385	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₆₉	C₂²	1	2	yes	—
15	#22575452	C₄	1	1	yes	characteristic
16	#22575451	C₄	1	1	yes	characteristic
17	#22575443	C₄	1	2	yes	—
18	#22575433	C₄	1	1	yes	characteristic
19	#22575431	C₄	1	2	yes	—
20	#22575429	C₄	1	1	yes	characteristic
21	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic
22	C₄×Dic₅₃₈	C₂²	1	2	yes	—
23	#22575410	C₂²	1	2	yes	—
24	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
25	C₄⋊#14524130	C₂²	1	1	yes	characteristic
26	C₄⋊#14524130	C₂²	1	2	yes	—
27	#22575382	C₂²	1	2	yes	—
28	#22575382	C₂²	1	2	yes	—
29	#22575382	C₂²	1	2	yes	—
30	#22575363	C₂²	1	1	yes	characteristic
31	#22575363	C₂²	1	2	yes	—
32	#22575363	C₂²	1	1	yes	characteristic
33	C₄×#14524130	C₂²	1	1	yes	characteristic
34	C₄×#14524130	C₂²	1	2	yes	—
35	C₈×Dic₂₆₉	C₂²	1	2	yes	—
36	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₆₉	C₂²	1	2	yes	—
39	#97×C₂₆₉	C₂²	1	2	yes	—
40	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₅₃₈	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₅₃₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
46	#16864385	—	2	2	no	—
47	#16864385	Q₈	1	4	yes	complemented
48	C₂×#14524130	C₂³	1	2	yes	—
49	C₂×#14524130	C₂³	1	1	yes	characteristic
50	C₂×#14524130	C₂³	1	1	yes	characteristic
51	C₈×D₂₆₉	—	2	4	no	—
52	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₂₆₉	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	#16864381	C₂×C₄	1	2	yes	—
58	#16864381	C₂×C₄	1	2	yes	—
59	#16864379	C₂×C₄	1	2	yes	—
60	#16864379	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₂₆₉	C₂³	1	2	yes	—
64	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
67	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
72	C₄²×C₂₆₉	C₂³	1	2	yes	—
73	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
74	Dic₅₃₈	—	2	2	no	—
75	Dic₅₃₈	—	4	2	no	—
76	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₂₆₉	—	2	2	no	—
82	C₄×D₂₆₉	—	4	1	no	—
83	C₄×D₂₆₉	C₂×Q₈	1	2	yes	—
84	C₄×D₂₆₉	#41	1	2	yes	—
85	#14524130	—	2	2	no	—
86	#14524130	—	2	1	no	—
87	#14524130	C₂×Q₈	1	2	yes	—
88	Q₈×C₂₆₉	—	2	2	no	—
89	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₆₉	—	2	2	no	—
95	C₈×C₂₆₉	—	2	1	no	—
96	C₈×C₂₆₉	C₂×Q₈	1	2	yes	—
97	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₆₉	—	2	1	no	—
99	C₂×D₂₆₉	—	2	1	no	—
100	C₂×D₂₆₉	C₄×Q₈	1	2	yes	—
101	C₄×C₂₆₉	—	2	2	no	—
102	C₄×C₂₆₉	—	2	1	no	—
103	C₄×C₂₆₉	—	2	2	no	—
104	C₄×C₂₆₉	#131	1	2	yes	—
105	C₄×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
107	Dic₂₆₉	—	2	1	no	—
108	Dic₂₆₉	—	2	2	no	—
109	Dic₂₆₉	—	2	1	no	—
110	Dic₂₆₉	—	2	2	no	—
111	Dic₂₆₉	C₄×Q₈	1	2	yes	—
112	C₂×C₂₆₉	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
114	C₂×C₂₆₉	#568	1	1	yes	characteristic
115	D₂₆₉	—	2	2	no	—
116	C₂₆₉	#2962	1	1	yes	characteristic, complemented, Sylow
117	#2962	—	269	1	no	maximal, Sylow
118	#533	—	269	1	no	—
119	#446	—	269	1	no	—
120	#446	—	269	2	no	—
121	#446	—	269	2	no	—
122	#446	D₂₆₉	1	1	yes	characteristic, complemented
123	C₈×Q₈	—	269	1	no	—
124	C₈×Q₈	—	269	1	no	—
125	#424	—	269	2	no	—
126	C₂×C₄⋊C₈	—	269	1	no	—
127	#405	—	269	2	no	—
128	C₄×M₁₆	—	269	1	no	—
129	C₂×M₁₆	—	269	1	no	—
130	C₂×M₁₆	—	269	1	no	—
131	C₂²×C₈	—	269	2	no	—
132	C₄⋊Q₈	—	269	1	no	—
133	C₄⋊Q₈	—	269	1	no	—
134	#125	—	269	2	no	—
135	C₂²⋊Q₈	—	269	1	no	—
136	C₂²⋊Q₈	—	269	2	no	—
137	C₂²⋊Q₈	—	269	1	no	—
138	C₄×Q₈	—	269	1	no	—
139	C₄×Q₈	—	269	2	no	—
140	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
141	#117	—	269	2	no	—
142	C₂×C₄²	—	269	1	no	—
143	C₄⋊C₈	—	269	2	no	—
144	C₄⋊C₈	—	269	2	no	—
145	C₄⋊C₈	—	269	2	no	—
146	C₄⋊C₈	—	269	1	no	—
147	C₄⋊C₈	—	269	2	no	—
148	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₂₆₉	1	2	yes	—
150	#97	—	269	1	no	—
151	#97	—	269	1	no	—
152	#97	—	269	2	no	—
153	#97	C₂×D₂₆₉	1	2	yes	—
154	C₄×C₈	—	269	2	no	—
155	C₄×C₈	—	269	1	no	—
156	C₄×C₈	—	269	2	no	—
157	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
158	C₂×Q₈	—	269	1	no	—
159	C₂×Q₈	—	269	2	no	—
160	C₂×Q₈	C₄×D₂₆₉	1	1	yes	characteristic
161	C₂²×C₄	—	269	1	no	—
162	C₂²×C₄	—	269	1	no	—
163	C₂²×C₄	—	269	1	no	—
164	M₁₆	—	269	4	no	—
165	M₁₆	—	538	2	no	—
166	C₂×C₈	—	269	1	no	—
167	C₂×C₈	—	269	2	no	—
168	C₂×C₈	—	269	1	no	—
169	C₂×C₈	—	538	4	no	—
170	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
172	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
173	C₄⋊C₄	—	269	2	no	—
174	C₄⋊C₄	—	269	2	no	—
175	C₄⋊C₄	—	269	2	no	—
176	C₄⋊C₄	—	269	2	no	—
177	C₄⋊C₄	—	269	2	no	—
178	C₄⋊C₄	—	269	2	no	—
179	C₄⋊C₄	—	269	1	no	—
180	C₄⋊C₄	C₄×D₂₆₉	1	2	yes	—
181	C₄⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
182	C₂²⋊C₄	—	269	2	no	—
183	C₂²⋊C₄	—	269	2	no	—
184	C₄²	—	269	1	no	—
185	C₄²	—	269	1	no	—
186	C₄²	—	269	2	no	—
187	C₄²	—	269	1	no	—
188	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
189	C₄²	C₂²×D₂₆₉	1	2	yes	—
190	C₂³	—	269	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	538	2	no	—
193	Q₈	—	1076	2	no	—
194	C₂×C₄	—	269	1	no	—
195	C₂×C₄	—	269	2	no	—
196	C₂×C₄	—	269	2	no	—
197	C₂×C₄	—	269	1	no	—
198	C₂×C₄	—	269	1	no	—
199	C₂×C₄	—	269	2	no	—
200	C₂×C₄	—	269	2	no	—
201	C₂×C₄	—	538	2	no	—
202	C₂×C₄	—	1076	1	no	—
203	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	269	2	no	—
211	C₈	—	538	2	no	—
212	C₈	—	538	1	no	—
213	C₈	Q₈×D₂₆₉	1	2	yes	—
214	C₂²	—	269	2	no	—
215	C₂²	—	538	1	no	—
216	C₂²	—	538	1	no	—
217	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	269	2	no	—
222	C₄	—	538	1	no	—
223	C₄	—	538	2	no	—
224	C₄	—	538	1	no	—
225	C₄	—	538	2	no	—
226	C₄	#22575461	1	2	yes	—
227	C₄	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
228	C₄	#41×D₂₆₉	1	1	yes	characteristic
229	C₂	—	538	2	no	—
230	C₂	C₄×Q₈×D₂₆₉	1	1	yes	characteristic
231	C₂	C₂×#22575461	1	1	yes	characteristic
232	C₂	#31771370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	#446×C₂₆₉	C₂
prime radical	C₂²×D₂₆₉	C₂²×C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#446	D₂₆₉
hypocenter	C₂₆₉	#2962
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #31770733	C₂	semidirect
3. #31770732	C₂	semidirect
4. #31770732	C₂	semidirect
5. Q₈×#14524130	C₂	semidirect
6. C₈×Dic₅₃₈	C₂	semidirect
11. #446×C₂₆₉	C₂	semidirect
47. #16864385	Q₈	semidirect
116. C₂₆₉	#2962	semidirect
122. #446	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771159	2
#31770733	2
#31770732	2
Q₈×#14524130	2
C₈×Dic₅₃₈	2
#31770630	2
C₄⋊C₈×D₂₆₉	2
#31770562	2
C₄×#16864385	2
#446×C₂₆₉	2
#2962	269