Subgroups — #16018436

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125375	C₂	1	2	yes	maximal
2	#13117122	C₂	1	4	yes	maximal, complemented
3	#13117118	C₂	1	2	yes	maximal, complemented
4	#13117115	C₂	1	2	yes	maximal, complemented
5	#13116624	C₂	1	2	yes	maximal
6	#13116620	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718097	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
9	#11719224	—	2	2	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#405	C₂²	1	1	yes	characteristic
13	#11718782	—	2	2	no	—
14	#11718767	C₂²	1	2	yes	—
15	C₂×#88058	C₂²	1	2	yes	—
16	#11718363	—	2	4	no	—
17	#11718344	C₂²	1	4	yes	—
18	#11718343	C₂²	1	4	yes	—
19	C₁₃×#346	C₂²	1	2	yes	—
20	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
21	#11718097	C₂²	1	1	yes	characteristic
22	#11718097	C₂²	1	2	yes	—
23	#11718097	C₂²	1	1	yes	characteristic
24	#11718097	C₂²	1	4	yes	—
25	#11718097	C₂²	1	2	yes	—
26	#11718096	C₂²	1	4	yes	—
27	#11718096	C₂²	1	2	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₁₃×#117	D₄	1	1	yes	characteristic
31	#88180	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	#88144	—	4	2	no	—
34	#88142	—	4	2	no	—
35	#88132	—	4	4	no	—
36	#88124	—	2	4	no	—
37	#88123	—	2	4	no	—
38	C₁₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
43	C₁₃×#97	D₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	#88058	—	2	2	no	—
46	#88058	—	2	4	no	—
47	#88058	C₂³	1	4	yes	—
48	#88058	C₂³	1	2	yes	—
49	#88058	C₂³	1	2	yes	—
50	#88057	—	2	2	no	—
51	#88057	—	2	4	no	—
52	#33730	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	4	2	no	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₁₃×M₁₆	Q₁₆	1	4	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
63	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄⋊Dic₁₃	—	2	4	no	—
68	C₄⋊Dic₁₃	—	2	2	no	—
69	C₄⋊Dic₁₃	—	4	4	no	—
70	C₄⋊Dic₁₃	—	4	2	no	—
71	#6101	—	4	4	no	—
72	#6101	—	4	4	no	—
73	C₄×Dic₁₃	—	4	2	no	—
74	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
75	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
77	#3164	—	13	2	no	—
78	#2255	—	13	4	no	—
79	#2254	—	13	2	no	—
80	#2253	—	13	2	no	—
81	#2159	—	13	2	no	—
82	#2158	—	13	1	no	—
83	C₂×#367	—	13	1	no	—
84	#2108	D₁₃	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	4	no	—
94	C₂×Dic₁₃	—	16	2	no	—
95	Dic₂₆	—	16	4	no	—
96	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
97	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
98	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
99	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
100	#544	—	26	2	no	—
101	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
103	#501	—	26	4	no	—
104	#500	—	13	4	no	—
105	#499	—	13	4	no	—
106	#428	—	26	2	no	—
107	#427	—	13	2	no	—
108	C₂×#107	—	13	2	no	—
109	#405	C₂×D₁₃	1	1	yes	characteristic
110	#367	—	13	2	no	—
111	#367	—	13	2	no	—
112	#367	—	13	1	no	—
113	#367	—	13	1	no	—
114	#367	—	13	4	no	—
115	#366	—	13	2	no	—
116	#366	—	13	4	no	—
117	#346	C₂×D₁₃	1	2	yes	—
118	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	#500	1	2	yes	—
121	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	#500	1	2	yes	—
124	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
126	Dic₁₃	—	8	2	no	—
127	Dic₁₃	—	8	4	no	—
128	Dic₁₃	—	8	2	no	—
129	C₂×M₁₆	D₅₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	26	4	no	—
132	C₄⋊Q₈	—	52	2	no	—
133	#125	—	26	4	no	—
134	#125	—	52	2	no	—
135	C₂²⋊Q₈	—	52	4	no	—
136	#117	—	26	2	no	—
137	#117	D₅₂	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	26	2	no	—
139	M₃₂	Dic₂₆	1	4	yes	complemented
140	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
141	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
142	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
143	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
144	#107	—	13	2	no	—
145	#107	—	13	4	no	—
146	#107	—	13	2	no	—
147	#107	—	26	2	no	—
148	#107	—	26	4	no	—
149	#106	—	26	2	no	—
150	#106	—	26	4	no	—
151	#97	D₅₂	1	2	yes	—
152	C₄×C₈	C₂²×D₁₃	1	2	yes	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
155	C₂×C₁₃	#3416	1	2	yes	—
156	C₂×Q₈	—	52	4	no	—
157	C₂²×C₄	—	52	2	no	—
158	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
159	M₁₆	Dic₅₂	1	4	yes	—
160	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
161	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
162	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
163	C₂×C₈	C₂×D₅₂	1	2	yes	—
164	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
165	C₄⋊C₄	—	26	2	no	—
166	C₄⋊C₄	—	26	4	no	—
167	C₄⋊C₄	—	26	2	no	—
168	C₄⋊C₄	—	52	2	no	—
169	C₄⋊C₄	—	52	4	no	—
170	C₄⋊C₄	—	52	4	no	—
171	C₄⋊C₄	—	52	4	no	—
172	C₄⋊C₄	D₁₀₄	1	2	yes	—
173	C₂²⋊C₄	—	52	2	no	—
174	C₂²⋊C₄	D₁₀₄	1	2	yes	—
175	C₄²	—	52	2	no	—
176	C₄²	C₂×D₅₂	1	2	yes	—
177	C₁₆	C₂×Dic₂₆	1	2	yes	—
178	C₁₆	C₂×Dic₂₆	1	2	yes	—
179	C₁₆	C₂×Dic₂₆	1	4	yes	—
180	C₁₃	#33730	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
182	Q₈	—	208	4	no	—
183	C₂×C₄	—	52	2	no	—
184	C₂×C₄	—	52	2	no	—
185	C₂×C₄	—	52	4	no	—
186	C₂×C₄	—	208	2	no	—
187	C₂×C₄	#88123	1	2	yes	—
188	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
189	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
191	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
192	C₈	#88123	1	2	yes	—
193	C₈	C₂×Dic₅₂	1	4	yes	—
194	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
195	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#11718343	1	2	yes	—
198	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	104	2	no	—
201	C₄	—	104	4	no	—
202	C₄	—	104	2	no	—
203	C₄	#11718343	1	2	yes	—
204	C₄	C₂×#88123	1	1	yes	characteristic
205	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#11718343	1	1	yes	characteristic
208	C₂	#13127984	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#33730
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13117122	C₂	semidirect
3. #13117118	C₂	semidirect
4. #13117115	C₂	semidirect
38. C₁₃×M₃₂	Q₈	semidirect
39. C₂×C₁₃×C₁₆	Q₈	semidirect
139. M₃₂	Dic₂₆	semidirect
140. C₂×C₁₆	Dic₂₆	semidirect
180. C₁₃	#33730	semidirect

Maximal subgroups

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

subgroup	index
#13125375	2
#13117122	2
#13117118	2
#13117115	2
#13116624	2
#13116620	2
C₂×#11718097	2
C₁₃×#2108	2
#33730	13