Subgroups — #2378×C₂₆₉ #46807485

Order 34432.

464 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
2	#391×C₂₆₉	C₂	1	2	yes	maximal, complemented
3	#389×C₂₆₉	C₂	1	2	yes	maximal, complemented
4	#388×C₂₆₉	C₂	1	2	yes	maximal
5	#388×C₂₆₉	C₂	1	2	yes	maximal
6	#387×C₂₆₉	C₂	1	2	yes	maximal, complemented
7	#387×C₂₆₉	C₂	1	2	yes	maximal, complemented
8	#381×C₂₆₉	C₂	1	1	yes	maximal, characteristic
9	#380×C₂₆₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈×C₂₆₉	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
24	#95×C₂₆₉	C₂²	1	2	yes	—
25	#95×C₂₆₉	C₂²	1	1	yes	characteristic
26	#95×C₂₆₉	C₂²	1	2	yes	—
27	#95×C₂₆₉	C₂²	1	1	yes	characteristic
28	#95×C₂₆₉	C₂²	1	2	yes	—
29	#95×C₂₆₉	C₂²	1	2	yes	—
30	#95×C₂₆₉	C₂²	1	2	yes	—
31	#95×C₂₆₉	C₂²	1	2	yes	—
32	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₂₆₉	—	4	2	no	—
34	C₂×Q₈×C₂₆₉	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₂₆₉	—	4	2	no	—
36	C₂²×C₄×C₂₆₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₆₉	—	4	2	no	—
46	C₄⋊C₄×C₂₆₉	—	4	2	no	—
47	C₄⋊C₄×C₂₆₉	—	4	2	no	—
48	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	—
49	C₄⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
50	C₂²⋊C₄×C₂₆₉	—	4	1	no	—
51	C₂²⋊C₄×C₂₆₉	—	4	2	no	—
52	C₂²⋊C₄×C₂₆₉	—	4	1	no	—
53	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	—
54	C₄²×C₂₆₉	—	4	2	no	—
55	C₄²×C₂₆₉	—	4	2	no	—
56	C₂³×C₂₆₉	—	2	1	no	—
57	C₂³×C₂₆₉	—	2	2	no	—
58	C₂³×C₂₆₉	—	2	2	no	—
59	C₂³×C₂₆₉	—	2	1	no	—
60	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
61	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
62	Q₈×C₂₆₉	—	4	4	no	—
63	C₂×C₄×C₂₆₉	—	2	2	no	—
64	C₂×C₄×C₂₆₉	—	2	2	no	—
65	C₂×C₄×C₂₆₉	—	2	2	no	—
66	C₂×C₄×C₂₆₉	—	2	2	no	—
67	C₂×C₄×C₂₆₉	—	2	2	no	—
68	C₂×C₄×C₂₆₉	—	2	2	no	—
69	C₂×C₄×C₂₆₉	—	2	2	no	—
70	C₂×C₄×C₂₆₉	—	2	2	no	—
71	C₂×C₄×C₂₆₉	—	2	2	no	—
72	C₂×C₄×C₂₆₉	—	2	2	no	—
73	C₂×C₄×C₂₆₉	—	2	2	no	—
74	C₂×C₄×C₂₆₉	—	2	2	no	—
75	C₂×C₄×C₂₆₉	—	2	2	no	—
76	C₂×C₄×C₂₆₉	—	2	2	no	—
77	C₂×C₄×C₂₆₉	—	2	2	no	—
78	C₂×C₄×C₂₆₉	—	2	2	no	—
79	C₂×C₄×C₂₆₉	—	2	2	no	—
80	C₂×C₄×C₂₆₉	—	2	2	no	—
81	C₂×C₄×C₂₆₉	—	4	4	no	—
82	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
83	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₂₆₉	—	2	2	no	—
87	C₂²×C₂₆₉	—	2	2	no	—
88	C₂²×C₂₆₉	—	2	2	no	—
89	C₂²×C₂₆₉	—	4	2	no	—
90	C₂²×C₂₆₉	—	4	2	no	—
91	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
92	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
93	C₂²×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
96	C₄×C₂₆₉	—	2	2	no	—
97	C₄×C₂₆₉	—	2	2	no	—
98	C₄×C₂₆₉	—	2	4	no	—
99	C₄×C₂₆₉	—	2	4	no	—
100	C₄×C₂₆₉	—	4	2	no	—
101	C₄×C₂₆₉	—	4	2	no	—
102	C₄×C₂₆₉	—	4	2	no	—
103	C₄×C₂₆₉	—	4	2	no	—
104	C₂×C₂₆₉	—	4	2	no	—
105	C₂×C₂₆₉	#518	1	1	yes	characteristic
106	C₂×C₂₆₉	#519	1	1	yes	characteristic
107	C₂×C₂₆₉	#520	1	1	yes	characteristic
108	C₂×C₂₆₉	#562	1	2	yes	—
109	C₂×C₂₆₉	#563	1	2	yes	—
110	C₂₆₉	#2378	1	1	yes	characteristic, direct product factor, Sylow
111	#2378	C₂₆₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₂²⋊Q₈	C₂×C₂₆₉	1	1	yes	characteristic
113	#391	C₂×C₂₆₉	1	2	yes	complemented
114	#389	C₂×C₂₆₉	1	2	yes	complemented
115	#388	C₂×C₂₆₉	1	2	yes	—
116	#388	C₂×C₂₆₉	1	2	yes	—
117	#387	C₂×C₂₆₉	1	2	yes	complemented
118	#387	C₂×C₂₆₉	1	2	yes	complemented
119	#381	C₂×C₂₆₉	1	1	yes	characteristic
120	#380	C₂×C₂₆₉	1	1	yes	characteristic
121	C₂²×Q₈	C₂²×C₂₆₉	1	1	yes	characteristic
122	C₂³×C₄	C₂²×C₂₆₉	1	1	yes	characteristic
123	C₂²⋊Q₈	C₄×C₂₆₉	1	8	yes	complemented
124	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₂₆₉	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₂₆₉	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₂²×C₂₆₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₂₆₉	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₆₉	1	2	yes	—
133	C₂×C₄²	C₂²×C₂₆₉	1	2	yes	—
134	C₂×C₄²	C₂²×C₂₆₉	1	2	yes	—
135	#95	C₂²×C₂₆₉	1	2	yes	—
136	#95	C₂²×C₂₆₉	1	1	yes	characteristic
137	#95	C₂²×C₂₆₉	1	2	yes	—
138	#95	C₂²×C₂₆₉	1	1	yes	characteristic
139	#95	C₂²×C₂₆₉	1	2	yes	—
140	#95	C₂²×C₂₆₉	1	2	yes	—
141	#95	C₂²×C₂₆₉	1	2	yes	—
142	#95	C₂²×C₂₆₉	1	2	yes	—
143	C₂⁴	C₂³×C₂₆₉	1	1	yes	characteristic
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	C₂×C₄×C₂₆₉	1	4	yes	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×C₄×C₂₆₉	1	4	yes	—
148	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
149	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
151	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
154	C₂²×C₄	C₂³×C₂₆₉	1	2	yes	—
155	C₂²×C₄	C₂³×C₂₆₉	1	1	yes	characteristic
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	C₂×C₄×C₂₆₉	1	4	yes	—
160	C₄⋊C₄	C₂×C₄×C₂₆₉	1	8	yes	—
161	C₂²⋊C₄	—	4	1	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	C₂×C₄×C₂₆₉	1	8	yes	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	1	no	—
171	C₂³	C₂²×C₄×C₂₆₉	1	2	yes	—
172	C₂³	C₂⁴×C₂₆₉	1	1	yes	characteristic
173	Q₈	—	4	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂²×C₄×C₂₆₉	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₂₆₉	1	4	yes	—
195	C₂×C₄	C₂²×C₄×C₂₆₉	1	2	yes	—
196	C₂×C₄	C₂²×C₄×C₂₆₉	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂³×C₄×C₂₆₉	1	1	yes	characteristic
203	C₂²	H(5,𝔽₂)×C₂₆₉	1	2	yes	—
204	C₂²	H(5,𝔽₂)×C₂₆₉	1	1	yes	characteristic
205	C₂²	2₋¹⁺⁴×C₂₆₉	1	2	yes	—
206	C₂²	2₋¹⁺⁴×C₂₆₉	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₂	—	4	2	no	—
216	C₂	#518×C₂₆₉	1	1	yes	characteristic
217	C₂	#519×C₂₆₉	1	1	yes	characteristic
218	C₂	#520×C₂₆₉	1	1	yes	characteristic
219	C₂	#562×C₂₆₉	1	2	yes	—
220	C₂	#563×C₂₆₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₆₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₆₉
Frattini, cosocle	C₂³	C₂⁴×C₂₆₉
prime radical	C₂⁴×C₂₆₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₆₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #391×C₂₆₉	C₂	semidirect
3. #389×C₂₆₉	C₂	semidirect
6. #387×C₂₆₉	C₂	semidirect
7. #387×C₂₆₉	C₂	semidirect
12. C₂²⋊Q₈×C₂₆₉	C₄	semidirect
110. C₂₆₉	#2378	direct (N × H)
111. #2378	C₂₆₉	direct (N × H)
113. #391	C₂×C₂₆₉	semidirect
114. #389	C₂×C₂₆₉	semidirect
117. #387	C₂×C₂₆₉	semidirect
118. #387	C₂×C₂₆₉	semidirect
123. C₂²⋊Q₈	C₄×C₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₆₉	2
#391×C₂₆₉	2
#389×C₂₆₉	2
#388×C₂₆₉	2
#387×C₂₆₉	2
#381×C₂₆₉	2
#380×C₂₆₉	2
#2378	269

Subgroups — #2378×C269 #46807485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2378×C₂₆₉ #46807485