Subgroups — #16807100

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131209	C₂	1	4	yes	maximal, complemented
2	#13131205	C₂	1	4	yes	maximal, complemented
3	#13131179	C₂	1	2	yes	maximal, complemented
4	#13131072	C₂	1	2	yes	maximal
5	C₂×#11718232	C₂	1	2	yes	maximal, complemented
6	#13123264	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
8	C₂×#88202	C₄	1	1	yes	characteristic
9	C₂×#88202	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	#11718658	C₄	1	1	yes	characteristic
12	#11718658	C₄	1	2	yes	—
13	#11718658	C₂²	1	2	yes	—
14	#11718658	C₂²	1	1	yes	characteristic
15	C₄×#6099	C₄	1	1	yes	characteristic
16	C₄×#6099	C₂²	1	2	yes	—
17	C₂×#88043	C₄	1	2	yes	—
18	C₂×#88043	C₂²	1	1	yes	characteristic
19	#11718284	C₂²	1	8	yes	—
20	#11718233	C₂²	1	8	yes	—
21	#11718232	C₂²	1	4	yes	—
22	#11718232	C₂²	1	4	yes	—
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	—	2	2	no	—
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₁₃×#117	C₂×C₄	1	2	yes	—
28	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	#88202	—	2	4	no	—
32	#88202	—	2	4	no	—
33	C₂×#6099	C₂×C₄	1	2	yes	—
34	C₂×#6099	C₂×C₄	1	1	yes	characteristic
35	C₂×#6099	C₂×C₄	1	1	yes	characteristic
36	C₂×#6099	C₂³	1	2	yes	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂²×#1055	C₂³	1	1	yes	characteristic
41	#88106	—	2	4	no	—
42	#88106	C₂³	1	8	yes	—
43	C₂×#6092	—	2	4	no	—
44	#88043	C₂×C₄	1	1	yes	characteristic
45	#88043	C₂×C₄	1	2	yes	—
46	#88043	C₂×C₄	1	2	yes	—
47	#88043	C₂×C₄	1	1	yes	characteristic
48	#88043	C₂×C₄	1	2	yes	—
49	#88043	C₂³	1	1	yes	characteristic
50	#88043	C₂³	1	1	yes	characteristic
51	C₄×#1055	C₂×C₄	1	2	yes	—
52	C₄×#1055	C₂×C₄	1	2	yes	—
53	C₄×#1055	C₂³	1	2	yes	—
54	#33426	—	13	1	no	maximal, Sylow
55	C₁₃×#41	—	2	2	no	—
56	C₁₃×#41	—	2	2	no	—
57	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
59	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	#6099	—	2	2	no	—
74	#6099	—	2	4	no	—
75	#6099	—	2	4	no	—
76	#6099	M₁₆	1	4	yes	—
77	C₂×#1055	—	2	2	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₂²×C₄	1	2	yes	—
85	C₂×#1055	#41	1	4	yes	—
86	C₂×#1055	C₂⁴	1	1	yes	characteristic
87	#6092	—	2	8	no	—
88	#2875	—	13	1	no	—
89	#2166	—	13	4	no	—
90	#2165	—	13	4	no	—
91	#2153	—	13	2	no	—
92	#2115	—	13	2	no	—
93	C₂×#347	—	13	2	no	—
94	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
95	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₁₃	—	2	2	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	#1055	—	2	2	no	—
112	#1055	—	2	1	no	—
113	#1055	—	2	1	no	—
114	#1055	#117	1	2	yes	—
115	#1055	C₂×M₁₆	1	4	yes	—
116	#1055	C₂×#41	1	2	yes	—
117	C₂×#131	—	13	1	no	—
118	C₂×#131	—	13	1	no	—
119	C₄×#41	#261	1	1	yes	characteristic
120	C₂×M₃₂	—	13	4	no	—
121	#405	—	13	1	no	—
122	#405	—	13	2	no	—
123	#405	—	13	1	no	—
124	#405	—	13	2	no	—
125	C₄×M₁₆	—	13	2	no	—
126	C₄×M₁₆	—	13	1	no	—
127	C₂×#97	—	13	2	no	—
128	C₂×#97	—	13	1	no	—
129	#364	—	13	8	no	—
130	#349	—	13	8	no	—
131	#347	—	13	4	no	—
132	#347	—	13	4	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	#432	1	2	yes	—
137	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	#432	1	2	yes	—
143	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
145	C₂×#41	C₂×#261	1	1	yes	characteristic
146	#131	—	26	4	no	—
147	#131	—	26	4	no	—
148	C₂×M₁₆	—	13	1	no	—
149	C₂×M₁₆	—	13	2	no	—
150	C₂×M₁₆	—	13	2	no	—
151	C₂×M₁₆	—	13	1	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	2	no	—
154	C₂²×C₈	—	13	2	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	1	no	—
159	#117	C₂×#261	1	1	yes	characteristic
160	#117	C₂×#261	1	2	yes	—
161	C₂×C₄²	C₂×#261	1	2	yes	—
162	C₂×C₄²	C₂×#261	1	1	yes	characteristic
163	M₃₂	—	13	8	no	—
164	M₃₂	—	26	4	no	—
165	C₂×C₁₆	—	26	4	no	—
166	#97	—	13	1	no	—
167	#97	—	13	1	no	—
168	#97	—	13	1	no	—
169	#97	—	13	2	no	—
170	#97	—	13	2	no	—
171	#97	—	13	2	no	—
172	#97	—	13	1	no	—
173	C₄×C₈	—	13	2	no	—
174	C₄×C₈	—	13	2	no	—
175	C₄×C₈	—	13	2	no	—
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	4	2	no	—
179	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	#6122	1	1	yes	characteristic
183	C₂×D₄	#6122	1	2	yes	—
184	C₂×D₄	#6122	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂²×#261	1	2	yes	—
188	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
189	M₁₆	—	13	4	no	—
190	M₁₆	—	26	4	no	—
191	M₁₆	—	26	4	no	—
192	M₁₆	—	26	2	no	—
193	C₂×C₈	—	13	4	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	1	no	—
200	C₂×C₈	—	26	4	no	—
201	C₂×C₈	—	26	4	no	—
202	C₂×C₈	—	26	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₂²×#261	1	2	yes	—
210	C₄²	C₂²×#261	1	1	yes	characteristic
211	C₄²	C₂²×#261	1	1	yes	characteristic
212	C₁₆	—	26	8	no	—
213	C₁₃	#33426	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₂×#6122	1	2	yes	—
215	C₂³	C₂×#6122	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
227	C₂×C₄	#88237	1	2	yes	—
228	C₂×C₄	C₂×#6122	1	2	yes	—
229	C₂×C₄	C₂×#6122	1	1	yes	characteristic
230	C₂×C₄	C₂³×#261	1	1	yes	characteristic
231	C₈	—	13	4	no	—
232	C₈	—	13	2	no	—
233	C₈	—	13	2	no	—
234	C₈	—	26	1	no	—
235	C₈	—	26	1	no	—
236	C₈	—	26	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#11719128	1	2	yes	—
241	C₂²	C₂²×#6122	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	#11719128	1	2	yes	—
247	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
248	C₄	C₂×#88237	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×#11719128	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×#41	C₄
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#33426
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #13131209	C₂	semidirect
2. #13131205	C₂	semidirect
3. #13131179	C₂	semidirect
5. C₂×#11718232	C₂	semidirect
213. C₁₃	#33426	semidirect

Maximal subgroups

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

subgroup	index
#13131209	2
#13131205	2
#13131179	2
#13131072	2
C₂×#11718232	2
#13123264	2
#33426	13