Subgroups — #16807112

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 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	C₂×#11718233	C₂	1	2	yes	maximal, complemented
4	#13131072	C₂	1	2	yes	maximal
5	#13131057	C₂	1	2	yes	maximal, complemented
6	C₄×#88202	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	2	yes	—
12	#11718658	C₄	1	1	yes	characteristic
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×C₄×#1055	C₄	1	2	yes	—
18	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
19	#11718284	C₂²	1	8	yes	—
20	#11718233	C₂²	1	4	yes	—
21	#11718233	C₂²	1	4	yes	—
22	#11718232	C₂²	1	8	yes	—
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	—	2	2	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	2	no	—
33	#88202	—	2	2	no	—
34	C₂×#6099	C₂×C₄	1	2	yes	—
35	C₂×#6099	C₂×C₄	1	1	yes	characteristic
36	C₂×#6099	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂³	1	2	yes	—
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂²×#1055	C₂³	1	1	yes	characteristic
42	#88106	—	2	4	no	—
43	#88106	C₂³	1	8	yes	—
44	C₂×#6092	—	2	4	no	—
45	#88043	C₂×C₄	1	2	yes	—
46	#88043	C₂×C₄	1	2	yes	—
47	#88043	C₂³	1	2	yes	—
48	C₄×#1055	C₂×C₄	1	1	yes	characteristic
49	C₄×#1055	C₂×C₄	1	2	yes	—
50	C₄×#1055	C₂×C₄	1	2	yes	—
51	C₄×#1055	C₂×C₄	1	1	yes	characteristic
52	C₄×#1055	C₂×C₄	1	2	yes	—
53	C₄×#1055	C₂³	1	1	yes	characteristic
54	C₄×#1055	C₂³	1	1	yes	characteristic
55	#33428	—	13	1	no	maximal, Sylow
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
59	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
60	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	#6099	—	2	2	no	—
75	#6099	—	2	2	no	—
76	#6099	—	2	4	no	—
77	#6099	—	2	4	no	—
78	C₂×#1055	—	2	2	no	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	M₁₆	1	4	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂²×C₄	1	2	yes	—
84	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
85	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
86	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
87	C₂×#1055	#41	1	2	yes	—
88	C₂×#1055	#41	1	2	yes	—
89	C₂×#1055	C₂⁴	1	1	yes	characteristic
90	#6092	—	2	8	no	—
91	C₄×#131	—	13	1	no	—
92	#2166	—	13	4	no	—
93	#2165	—	13	4	no	—
94	C₂×#364	—	13	2	no	—
95	#2115	—	13	2	no	—
96	#2110	—	13	2	no	—
97	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
98	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	2	no	—
100	D₄×C₁₃	—	2	4	no	—
101	D₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	#1055	—	2	2	no	—
115	#1055	—	2	2	no	—
116	#1055	#117	1	1	yes	characteristic
117	#1055	#117	1	1	yes	characteristic
118	#1055	C₂×M₁₆	1	2	yes	—
119	#1055	C₂×M₁₆	1	2	yes	—
120	#1055	C₂×#41	1	2	yes	—
121	C₂×#131	—	13	1	no	—
122	C₂×#131	—	13	1	no	—
123	C₄×#41	#261	1	1	yes	characteristic
124	C₂×M₃₂	—	13	4	no	—
125	#405	—	13	2	no	—
126	#405	—	13	2	no	—
127	#405	—	13	1	no	—
128	#405	—	13	1	no	—
129	C₄×M₁₆	—	13	2	no	—
130	C₄×M₁₆	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	2	no	—
133	#364	—	13	4	no	—
134	#364	—	13	4	no	—
135	#349	—	13	8	no	—
136	#347	—	13	8	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	4	1	no	—
139	C₂²×C₁₃	—	4	2	no	—
140	C₂²×C₁₃	#432	1	2	yes	—
141	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	4	2	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	#432	1	2	yes	—
147	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×#41	C₂×#261	1	1	yes	characteristic
150	#131	—	26	4	no	—
151	#131	—	26	2	no	—
152	#131	—	26	2	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	—	13	2	no	—
155	C₂×M₁₆	—	13	1	no	—
156	C₂×M₁₆	—	13	1	no	—
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	13	2	no	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	2	no	—
164	#117	C₂×#261	1	2	yes	—
165	#117	C₂×#261	1	1	yes	characteristic
166	C₂×C₄²	C₂×#261	1	1	yes	characteristic
167	C₂×C₄²	C₂×#261	1	2	yes	—
168	M₃₂	—	13	8	no	—
169	M₃₂	—	26	4	no	—
170	C₂×C₁₆	—	26	4	no	—
171	#97	—	13	2	no	—
172	#97	—	13	2	no	—
173	#97	—	13	2	no	—
174	C₄×C₈	—	13	1	no	—
175	C₄×C₈	—	13	1	no	—
176	C₄×C₈	—	13	1	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	13	1	no	—
179	C₄×C₈	—	13	2	no	—
180	C₄×C₈	—	13	2	no	—
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	4	2	no	—
184	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	C₂×Q₈	#6122	1	1	yes	characteristic
188	C₂×D₄	#6122	1	2	yes	—
189	C₂×D₄	#6122	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²×#261	1	2	yes	—
193	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
194	M₁₆	—	26	4	no	—
195	M₁₆	—	26	4	no	—
196	M₁₆	—	26	2	no	—
197	M₁₆	—	26	2	no	—
198	C₂×C₈	—	13	4	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	no	—
203	C₂×C₈	—	13	1	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	C₂²×#261	1	1	yes	characteristic
217	C₄²	C₂²×#261	1	1	yes	characteristic
218	C₄²	C₂²×#261	1	2	yes	—
219	C₁₆	—	26	8	no	—
220	C₁₃	#33428	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂×#6122	1	2	yes	—
222	C₂³	C₂×#6122	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
234	C₂×C₄	#88237	1	2	yes	—
235	C₂×C₄	C₂×#6122	1	2	yes	—
236	C₂×C₄	C₂×#6122	1	1	yes	characteristic
237	C₂×C₄	C₂³×#261	1	1	yes	characteristic
238	C₈	—	13	2	no	—
239	C₈	—	13	2	no	—
240	C₈	—	13	2	no	—
241	C₈	—	13	1	no	—
242	C₈	—	13	1	no	—
243	C₈	—	26	2	no	—
244	C₈	—	26	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#11719128	1	2	yes	—
249	C₂²	C₂²×#6122	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	2	no	—
254	C₄	#11719128	1	2	yes	—
255	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
256	C₄	C₂×#88237	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	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	#33428
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ 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. C₂×#11718233	C₂	semidirect
5. #13131057	C₂	semidirect
220. C₁₃	#33428	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
C₂×#11718233	2
#13131072	2
#13131057	2
C₄×#88202	2
#33428	13