Subgroups — C₂²×#11718157 #16015874

Order 3328.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718157	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	—	2	3	no	—
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	3	no	—
9	C₂²×C₁₃×Q₁₆	C₄	1	6	yes	—
10	C₂²×C₁₃×Q₁₆	C₄	1	2	yes	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	6	yes	—
12	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#6099	—	2	1	no	—
14	C₂×#88059	—	2	3	no	—
15	C₂×#88059	C₂²	1	6	yes	—
16	C₂×#88052	—	2	3	no	—
17	C₂×#88052	C₂²	1	6	yes	—
18	C₂²×#6090	—	2	1	no	—
19	#11718157	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₁₃	—	2	6	no	—
21	C₂²×Q₈×C₁₃	—	2	2	no	—
22	C₂²×Q₈×C₁₃	—	2	6	no	—
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×Q₈×C₁₃	—	4	3	no	—
25	C₂²×Q₈×C₁₃	—	4	3	no	—
26	C₂³×C₄×C₁₃	—	4	1	no	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₁₃×Q₁₆	—	2	6	no	—
29	C₂×C₁₃×Q₁₆	—	2	12	no	—
30	C₂×C₁₃×Q₁₆	—	2	6	no	—
31	C₂×C₁₃×Q₁₆	—	2	6	no	—
32	C₂×C₁₃×Q₁₆	—	2	6	no	—
33	C₂×C₁₃×Q₁₆	—	2	6	no	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	12	yes	—
35	C₂×C₁₃×Q₁₆	C₂×C₄	1	12	yes	—
36	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	3	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	3	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
43	C₂×#6099	—	2	6	no	—
44	C₂×#6099	—	4	3	no	—
45	C₂²×#1055	—	4	1	no	—
46	#88059	—	2	6	no	—
47	#88059	C₂³	1	4	yes	—
48	#88052	—	2	6	no	—
49	#88052	C₂³	1	4	yes	—
50	C₂×#6090	—	2	6	no	—
51	C₂²×#362	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×Q₈×C₁₃	—	2	12	no	—
54	C₂×Q₈×C₁₃	—	2	12	no	—
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×Q₈×C₁₃	—	4	12	no	—
57	C₂×Q₈×C₁₃	—	4	6	no	—
58	C₂×Q₈×C₁₃	—	4	6	no	—
59	C₂×Q₈×C₁₃	—	4	6	no	—
60	C₂×Q₈×C₁₃	—	4	6	no	—
61	C₂×Q₈×C₁₃	—	4	6	no	—
62	C₂²×C₄×C₁₃	—	4	6	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	6	no	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	D₈	1	3	yes	—
69	C₂²×C₄×C₁₃	SD₁₆	1	3	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
71	C₁₃×Q₁₆	—	2	12	no	—
72	C₁₃×Q₁₆	—	2	12	no	—
73	C₁₃×Q₁₆	—	2	4	no	—
74	C₁₃×Q₁₆	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	6	yes	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	3	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	#6099	—	2	4	no	—
85	#6099	—	4	6	no	—
86	C₂×#1055	—	4	6	no	—
87	#6090	—	2	4	no	—
88	C₂³×Q₁₆	D₁₃	1	1	yes	characteristic
89	C₂²×M₃₂	—	13	1	no	—
90	C₂²×#108	—	13	1	no	—
91	C₂×#362	—	13	12	no	—
92	C₂³×C₁₃	—	2	4	no	—
93	C₂³×C₁₃	#102	1	3	yes	—
94	C₂³×C₁₃	#102	1	1	yes	characteristic
95	C₂³×C₁₃	C₂×SD₁₆	1	3	yes	—
96	Q₈×C₁₃	—	2	8	no	—
97	Q₈×C₁₃	—	4	4	no	—
98	Q₈×C₁₃	—	4	12	no	—
99	Q₈×C₁₃	—	4	12	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	4	12	no	—
102	C₂×C₄×C₁₃	—	4	12	no	—
103	C₂×C₄×C₁₃	#102	1	6	yes	—
104	C₂×C₄×C₁₃	#102	1	3	yes	—
105	C₂×C₄×C₁₃	#102	1	3	yes	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
109	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	3	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
113	C₈×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
114	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	#1055	—	4	4	no	—
117	C₂³×Q₈	—	2	1	no	—
118	C₂²×Q₁₆	—	2	3	no	—
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂²×Q₁₆	—	2	3	no	—
121	C₂²×Q₁₆	Dic₁₃	1	6	yes	—
122	C₂²×Q₁₆	Dic₁₃	1	2	yes	—
123	C₂²×Q₁₆	C₂×D₁₃	1	6	yes	—
124	C₂²×M₁₆	—	26	1	no	—
125	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
126	C₂×M₃₂	—	13	6	no	—
127	C₂×M₃₂	—	26	3	no	—
128	C₂²×C₁₆	—	26	1	no	—
129	C₂×#108	—	13	6	no	—
130	C₂×#108	—	26	3	no	—
131	#362	—	13	16	no	—
132	C₂²×C₁₃	—	2	6	no	—
133	C₂²×C₁₃	—	2	6	no	—
134	C₂²×C₁₃	#362	1	4	yes	complemented
135	C₂²×C₁₃	C₂×#102	1	3	yes	—
136	C₂²×C₁₃	C₂×#102	1	3	yes	—
137	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₁₃	—	4	8	no	—
139	C₄×C₁₃	C₂×#102	1	3	yes	—
140	C₄×C₁₃	C₂×#102	1	3	yes	—
141	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
143	C₂²×Q₈	—	2	6	no	—
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	—	4	3	no	—
147	C₂²×Q₈	—	4	1	no	—
148	C₂²×Q₈	—	4	3	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
151	C₂×Q₁₆	—	2	6	no	—
152	C₂×Q₁₆	—	2	12	no	—
153	C₂×Q₁₆	—	2	6	no	—
154	C₂×Q₁₆	—	2	6	no	—
155	C₂×Q₁₆	—	2	6	no	—
156	C₂×Q₁₆	—	2	6	no	—
157	C₂×Q₁₆	C₂×Dic₁₃	1	12	yes	—
158	C₂×Q₁₆	C₂×Dic₁₃	1	12	yes	—
159	C₂×Q₁₆	C₂²×D₁₃	1	4	yes	—
160	C₂×M₁₆	—	26	6	no	—
161	C₂×M₁₆	—	52	3	no	—
162	C₂²×C₈	—	52	1	no	—
163	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
164	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₁₃	1	3	yes	—
166	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₁₃	1	3	yes	—
168	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
169	M₃₂	—	13	4	no	—
170	M₃₂	—	26	6	no	—
171	C₂×C₁₆	—	26	6	no	—
172	#108	—	13	4	no	—
173	#108	—	26	6	no	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	—	2	3	no	—
176	C₂×C₁₃	C₂×#362	1	6	yes	complemented
177	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
178	C₂⁴	#6105	1	1	yes	characteristic
179	C₂×Q₈	—	2	12	no	—
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	12	no	—
183	C₂×Q₈	—	4	6	no	—
184	C₂×Q₈	—	4	6	no	—
185	C₂×Q₈	—	4	6	no	—
186	C₂×Q₈	—	4	6	no	—
187	C₂×Q₈	—	4	6	no	—
188	C₂²×C₄	—	4	6	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	6	no	—
191	C₂²×C₄	#6104	1	3	yes	—
192	C₂²×C₄	#6104	1	1	yes	characteristic
193	C₂²×C₄	#6105	1	3	yes	—
194	C₂²×C₄	C₂²⋊Dic₁₃	1	3	yes	—
195	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
197	Q₁₆	—	2	12	no	—
198	Q₁₆	—	2	12	no	—
199	Q₁₆	—	2	4	no	—
200	Q₁₆	C₂²×Dic₁₃	1	8	yes	—
201	M₁₆	—	26	4	no	—
202	M₁₆	—	52	6	no	—
203	C₂×C₈	—	52	6	no	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	3	yes	—
205	C₂×C₈	C₂²⋊Dic₁₃	1	6	yes	—
206	C₂×C₈	C₂²⋊Dic₁₃	1	3	yes	—
207	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
208	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
209	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
210	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	3	yes	—
212	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
213	C₁₆	—	26	4	no	—
214	C₁₃	C₂²×#362	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	4	no	—
216	C₂³	#88072	1	3	yes	—
217	C₂³	#88072	1	1	yes	characteristic
218	C₂³	C₂×#6105	1	3	yes	—
219	Q₈	—	2	8	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	4	12	no	—
222	Q₈	—	4	12	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	12	no	—
226	C₂×C₄	#88072	1	6	yes	—
227	C₂×C₄	#88072	1	3	yes	—
228	C₂×C₄	#88072	1	3	yes	—
229	C₂×C₄	C₂×#6104	1	3	yes	—
230	C₂×C₄	C₂×#6104	1	3	yes	—
231	C₂×C₄	C₂×#6105	1	3	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
233	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₈	—	52	4	no	—
236	C₈	C₂×C₂²⋊Dic₁₃	1	3	yes	—
237	C₈	C₂×C₂²⋊Dic₁₃	1	3	yes	—
238	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
239	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	#11718157	1	4	yes	direct product factor
243	C₂²	C₂×#88072	1	3	yes	—
244	C₂²	C₂×#88072	1	3	yes	—
245	C₂²	C₂²×#6105	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×#88072	1	3	yes	—
248	C₄	C₂×#88072	1	3	yes	—
249	C₄	C₂²×#6104	1	1	yes	characteristic
250	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	3	no	—
253	C₂	C₂×#11718157	1	6	yes	direct product factor
254	C₂	C₂²×#88072	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88072
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂³×C₁₃×Q₁₆	C₂
socle	C₂³×C₁₃	#102
prime radical	C₂⁴×C₁₃	SD₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂³×Q₁₆	D₁₃
hypocenter	C₁₃	C₂²×#362
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#11718157	C₂	direct (N × H)
19. #11718157	C₂²	direct (N × H)
134. C₂²×C₁₃	#362	semidirect
176. C₂×C₁₃	C₂×#362	semidirect
214. C₁₃	C₂²×#362	semidirect
242. C₂²	#11718157	direct (N × H)
253. C₂	C₂×#11718157	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×#88052	2
C₂²×#88059	2
C₂×#11718157	2
C₂³×C₁₃×Q₁₆	2
C₂²×#362	13

Subgroups — C22×#11718157 #16015874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718157 #16015874