Subgroups — C₂²×#11718739 #16732515

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718739	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88047	C₂	1	4	yes	maximal
4	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₃	—	2	1	no	—
7	C₂³×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×C₄⋊Q₈	—	2	6	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	12	yes	—
10	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊Dic₁₃	—	2	1	no	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂³×#1055	C₂²	1	2	yes	—
16	C₂×#88123	—	2	6	no	—
17	C₂×#88123	C₂²	1	12	yes	—
18	#11718739	C₂²	1	64	yes	direct product factor
19	C₂×#88047	C₂²	1	48	yes	—
20	C₂×C₄×#1055	C₂²	1	12	yes	—
21	C₂²×Q₈×C₁₃	—	2	12	no	—
22	C₂²×Q₈×C₁₃	—	4	8	no	—
23	C₂³×C₄×C₁₃	—	2	2	no	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	—	2	2	no	—
28	C₂²×Dic₂₆	—	2	12	no	—
29	C₂²×Dic₂₆	—	4	8	no	—
30	C₁₃×C₄⋊Q₈	—	2	24	no	—
31	C₁₃×C₄⋊Q₈	C₂³	1	16	yes	—
32	C₂×C₁₃×C₄⋊C₄	—	2	12	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₄²×C₁₃	D₄	1	6	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
37	C₂×C₄⋊Dic₁₃	—	2	6	no	—
38	C₂×C₄⋊Dic₁₃	—	2	12	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
40	C₂²×#1055	Q₈	1	16	yes	complemented
41	C₂²×#1055	C₂³	1	12	yes	—
42	#88123	—	2	24	no	—
43	#88123	C₂³	1	16	yes	—
44	#88047	C₂³	1	64	yes	—
45	C₄×#1055	C₂³	1	16	yes	—
46	C₂²×#500	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	—	2	16	no	—
49	C₂×Q₈×C₁₃	—	4	48	no	—
50	C₂²×C₄×C₁₃	—	2	12	no	—
51	C₂²×C₄×C₁₃	—	4	8	no	—
52	C₂²×C₄×C₁₃	D₈	1	8	yes	—
53	C₂²×C₄×C₁₃	Q₁₆	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
59	C₂²×Dic₁₃	—	2	12	no	—
60	C₂²×Dic₁₃	—	4	8	no	—
61	C₂×Dic₂₆	—	2	16	no	—
62	C₂×Dic₂₆	—	4	48	no	—
63	C₁₃×C₄⋊C₄	—	2	16	no	—
64	C₁₃×C₄⋊C₄	—	2	12	no	—
65	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
66	C₄²×C₁₃	C₂×D₄	1	12	yes	—
67	C₄²×C₁₃	C₂⁴	1	4	yes	—
68	C₄⋊Dic₁₃	—	2	12	no	—
69	C₄⋊Dic₁₃	—	2	16	no	—
70	C₄⋊Dic₁₃	C₂⁴	1	8	yes	—
71	C₂×#1055	C₂×Q₈	1	48	yes	complemented
72	C₂×#1055	C₂⁴	1	8	yes	—
73	C₂²×C₄⋊Q₈	—	13	1	no	—
74	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
75	C₂×#500	—	13	24	no	—
76	C₂²×#107	—	13	4	no	—
77	C₂²×C₄×C₈	—	13	1	no	—
78	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
79	C₂³×C₁₃	C₂×D₈	1	4	yes	—
80	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
81	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
82	Q₈×C₁₃	—	4	32	no	—
83	C₂×C₄×C₁₃	—	2	8	no	—
84	C₂×C₄×C₁₃	—	4	24	no	—
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
86	C₂×C₄×C₁₃	C₂×D₈	1	24	yes	—
87	C₂×C₄×C₁₃	C₂×Q₁₆	1	24	yes	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
89	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
92	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₁₃	—	2	8	no	—
94	C₂×Dic₁₃	—	4	24	no	—
95	Dic₂₆	—	4	32	no	—
96	#1055	C₂²×Q₈	1	16	yes	complemented
97	C₂³×Q₈	—	2	1	no	—
98	C₂³×Q₈	—	26	1	no	—
99	C₂³×C₈	—	13	2	no	—
100	C₂×C₄⋊Q₈	—	2	6	no	—
101	C₂×C₄⋊Q₈	—	13	12	no	—
102	C₂×C₄⋊Q₈	—	26	6	no	—
103	C₂×C₄⋊Q₈	C₂×D₁₃	1	12	yes	—
104	C₂²×C₄⋊C₄	—	2	1	no	—
105	C₂²×C₄⋊C₄	—	13	2	no	—
106	C₂²×C₄⋊C₄	—	26	1	no	—
107	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	#500	—	13	64	no	—
110	C₂×#107	—	13	48	no	—
111	C₂×C₄×C₈	—	13	12	no	—
112	C₂²×C₁₃	#500	1	16	yes	complemented
113	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
114	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
115	C₂²×C₁₃	C₂²×Q₁₆	1	6	yes	—
116	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₁₃	—	4	8	no	—
118	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
119	C₄×C₁₃	C₂²×D₈	1	8	yes	—
120	C₄×C₁₃	C₂²×Q₁₆	1	8	yes	—
121	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
122	Dic₁₃	—	4	8	no	—
123	C₂²×Q₈	—	2	12	no	—
124	C₂²×Q₈	—	4	8	no	—
125	C₂²×Q₈	—	26	12	no	—
126	C₂²×Q₈	—	52	8	no	—
127	C₂³×C₄	—	2	2	no	—
128	C₂³×C₄	—	26	2	no	—
129	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
130	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
132	C₂²×C₈	—	13	16	no	—
133	C₂²×C₈	—	13	12	no	—
134	C₄⋊Q₈	—	2	24	no	—
135	C₄⋊Q₈	—	13	16	no	—
136	C₄⋊Q₈	—	26	24	no	—
137	C₄⋊Q₈	C₂²×D₁₃	1	16	yes	—
138	C₂×C₄⋊C₄	—	2	12	no	—
139	C₂×C₄⋊C₄	—	2	6	no	—
140	C₂×C₄⋊C₄	—	13	12	no	—
141	C₂×C₄⋊C₄	—	26	6	no	—
142	C₂×C₄⋊C₄	—	26	12	no	—
143	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
144	C₂×C₄²	C₂²⋊D₁₃	1	6	yes	—
145	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
146	#107	—	13	64	no	—
147	C₄×C₈	—	13	16	no	—
148	C₂×C₁₃	C₂×#500	1	12	yes	complemented
149	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
151	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
152	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×Q₈	—	2	16	no	—
154	C₂×Q₈	—	4	48	no	—
155	C₂×Q₈	—	26	16	no	—
156	C₂×Q₈	—	52	48	no	—
157	C₂²×C₄	—	2	12	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	26	12	no	—
160	C₂²×C₄	—	52	8	no	—
161	C₂²×C₄	#6104	1	8	yes	—
162	C₂²×C₄	#6107	1	8	yes	—
163	C₂²×C₄	Q₈×D₁₃	1	8	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
167	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
168	C₂×C₈	—	13	48	no	—
169	C₂×C₈	—	13	8	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄⋊C₄	—	2	12	no	—
172	C₄⋊C₄	—	13	8	no	—
173	C₄⋊C₄	—	26	12	no	—
174	C₄⋊C₄	—	26	16	no	—
175	C₄⋊C₄	C₂³×D₁₃	1	8	yes	—
176	C₄²	C₂×C₂²⋊D₁₃	1	12	yes	—
177	C₄²	C₂³×D₁₃	1	4	yes	—
178	C₁₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂×#6104	1	4	yes	—
180	C₂³	C₂×#6107	1	4	yes	—
181	C₂³	#88198	1	4	yes	—
182	C₂³	C₂²×C₂²⋊D₁₃	1	3	yes	—
183	Q₈	—	4	32	no	—
184	Q₈	—	52	32	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	24	no	—
187	C₂×C₄	—	26	8	no	—
188	C₂×C₄	—	52	24	no	—
189	C₂×C₄	C₂×#6104	1	24	yes	—
190	C₂×C₄	C₂×#6107	1	24	yes	—
191	C₂×C₄	#88198	1	12	yes	—
192	C₂×C₄	C₂×Q₈×D₁₃	1	12	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₁₃	1	4	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₁₃	1	3	yes	—
196	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
197	C₈	—	13	16	no	—
198	C₂²	#11718739	1	16	yes	direct product factor
199	C₂²	C₂²×#6104	1	6	yes	—
200	C₂²	C₂²×#6107	1	6	yes	—
201	C₂²	C₂×#88198	1	6	yes	—
202	C₂²	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
203	C₄	—	4	8	no	—
204	C₄	—	52	8	no	—
205	C₄	C₂²×#6104	1	8	yes	—
206	C₄	C₂²×#6107	1	8	yes	—
207	C₄	C₂×#88198	1	6	yes	—
208	C₄	C₂²×Q₈×D₁₃	1	2	yes	—
209	C₂	C₂×#11718739	1	12	yes	direct product factor
210	C₂	C₂²×#88198	1	1	yes	characteristic
211	C₂	C₂³×#6104	1	1	yes	characteristic
212	C₂	C₂³×#6107	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂²×#500
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718739	C₂	direct (N × H)
18. #11718739	C₂²	direct (N × H)
40. C₂²×#1055	Q₈	semidirect
71. C₂×#1055	C₂×Q₈	semidirect
96. #1055	C₂²×Q₈	semidirect
112. C₂²×C₁₃	#500	semidirect
148. C₂×C₁₃	C₂×#500	semidirect
178. C₁₃	C₂²×#500	semidirect
198. C₂²	#11718739	direct (N × H)
209. C₂	C₂×#11718739	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88123	2
C₂×#11718739	2
C₂²×#88047	2
C₂²×C₄×#1055	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×#500	13

Subgroups — C22×#11718739 #16732515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718739 #16732515