Subgroups — C₂²×#11718139 #15831202

Order 3328.

1334 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718139	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	—	2	1	no	—
6	C₂×C₁₃×C₄⋊Q₈	—	2	3	no	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	3	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₄	1	6	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	6	yes	—
10	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×#1055	—	2	1	no	—
14	C₂³×#1055	—	2	1	no	—
15	C₂×C₄⋊#1055	—	2	3	no	—
16	C₂×C₄⋊#1055	—	2	3	no	—
17	C₂×C₄⋊#1055	C₂²	1	6	yes	—
18	C₂×C₄⋊#1055	C₂²	1	6	yes	—
19	#11718139	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₁₃	—	2	6	no	—
21	C₂²×Q₈×C₁₃	—	2	6	no	—
22	C₂²×Q₈×C₁₃	—	4	4	no	—
23	C₂²×Q₈×C₁₃	—	4	4	no	—
24	C₂³×C₄×C₁₃	—	2	2	no	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₁₃×C₄⋊Q₈	—	2	12	no	—
29	C₁₃×C₄⋊Q₈	—	2	6	no	—
30	C₁₃×C₄⋊Q₈	—	2	6	no	—
31	C₁₃×C₄⋊Q₈	C₂×C₄	1	12	yes	—
32	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
38	C₂×C₄²×C₁₃	D₄	1	3	yes	—
39	C₂×C₄²×C₁₃	D₄	1	3	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
41	C₂²×#1055	—	2	6	no	—
42	C₂²×#1055	—	2	6	no	—
43	C₂²×#1055	—	4	4	no	—
44	C₂²×#1055	—	4	4	no	—
45	C₄⋊#1055	—	2	6	no	—
46	C₄⋊#1055	—	2	6	no	—
47	C₄⋊#1055	C₂³	1	4	yes	—
48	C₄⋊#1055	C₂³	1	4	yes	—
49	C₂²×#332	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃	—	2	12	no	—
52	C₂×Q₈×C₁₃	—	2	4	no	—
53	C₂×Q₈×C₁₃	—	4	24	no	—
54	C₂×Q₈×C₁₃	—	4	12	no	—
55	C₂×Q₈×C₁₃	—	4	12	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	12	no	—
58	C₂²×C₄×C₁₃	—	4	8	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	3	yes	—
62	C₂²×C₄×C₁₃	D₈	1	4	yes	—
63	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
64	C₂²×C₄×C₁₃	SD₁₆	1	4	yes	—
65	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
69	C₁₃×C₄⋊C₄	—	2	12	no	—
70	C₁₃×C₄⋊C₄	—	2	12	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
73	C₄²×C₁₃	C₂²⋊C₄	1	6	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
75	C₄²×C₁₃	C₂×D₄	1	3	yes	—
76	C₄²×C₁₃	C₂×D₄	1	3	yes	—
77	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	—	4	12	no	—
81	C₂×#1055	—	4	12	no	—
82	C₂²×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
83	C₂²×C₄⋊C₈	—	13	1	no	—
84	C₂²×C₄⋊C₈	—	13	1	no	—
85	C₂×#332	—	13	12	no	—
86	C₂³×C₁₃	#101	1	4	yes	—
87	C₂³×C₁₃	#102	1	4	yes	—
88	C₂³×C₁₃	#103	1	4	yes	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	4	24	no	—
93	C₂×C₄×C₁₃	—	2	6	no	—
94	C₂×C₄×C₁₃	—	2	6	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	—	4	24	no	—
97	C₂×C₄×C₁₃	#102	1	12	yes	—
98	C₂×C₄×C₁₃	#103	1	12	yes	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
102	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
104	C₂×C₄×C₁₃	C₂×SD₁₆	1	6	yes	—
105	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	#1055	—	4	4	no	—
110	#1055	—	4	4	no	—
111	C₂³×Q₈	—	2	1	no	—
112	C₂³×C₈	—	26	1	no	—
113	C₂³×C₈	—	26	1	no	—
114	C₂×C₄⋊Q₈	—	2	3	no	—
115	C₂×C₄⋊Q₈	—	2	3	no	—
116	C₂×C₄⋊Q₈	Dic₁₃	1	6	yes	—
117	C₂×C₄⋊Q₈	C₂×D₁₃	1	6	yes	—
118	C₂²×C₄⋊C₄	—	2	1	no	—
119	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
120	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
121	C₂×C₄⋊C₈	—	13	6	no	—
122	C₂×C₄⋊C₈	—	13	6	no	—
123	C₂×C₄⋊C₈	—	26	3	no	—
124	C₂×C₄⋊C₈	—	26	3	no	—
125	#332	—	13	16	no	—
126	C₂²×C₁₃	#332	1	16	yes	complemented
127	C₂²×C₁₃	C₂×#101	1	6	yes	—
128	C₂²×C₁₃	C₂×#102	1	6	yes	—
129	C₂²×C₁₃	C₂×#103	1	6	yes	—
130	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₁₃	—	2	3	no	—
132	C₄×C₁₃	—	2	1	no	—
133	C₄×C₁₃	—	4	8	no	—
134	C₄×C₁₃	C₂×#102	1	6	yes	—
135	C₄×C₁₃	C₂×#103	1	6	yes	—
136	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
139	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×Q₈	—	2	6	no	—
141	C₂²×Q₈	—	2	6	no	—
142	C₂²×Q₈	—	4	4	no	—
143	C₂²×Q₈	—	4	4	no	—
144	C₂³×C₄	—	2	2	no	—
145	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂²×C₈	—	26	6	no	—
149	C₂²×C₈	—	26	6	no	—
150	C₂²×C₈	—	52	4	no	—
151	C₂²×C₈	—	52	4	no	—
152	C₄⋊Q₈	—	2	12	no	—
153	C₄⋊Q₈	—	2	6	no	—
154	C₄⋊Q₈	—	2	6	no	—
155	C₄⋊Q₈	C₂×Dic₁₃	1	12	yes	—
156	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
157	C₂×C₄⋊C₄	—	2	6	no	—
158	C₂×C₄⋊C₄	—	2	6	no	—
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	C₂×Dic₁₃	1	12	yes	—
161	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
162	C₂×C₄²	C₂²⋊D₁₃	1	3	yes	—
163	C₂×C₄²	C₂²⋊D₁₃	1	3	yes	—
164	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
165	C₄⋊C₈	—	13	4	no	—
166	C₄⋊C₈	—	13	4	no	—
167	C₄⋊C₈	—	26	6	no	—
168	C₄⋊C₈	—	26	6	no	—
169	C₂×C₁₃	C₂×#332	1	12	yes	complemented
170	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
171	C₂×C₁₃	C₂²×#102	1	1	yes	characteristic
172	C₂×C₁₃	C₂²×#103	1	1	yes	characteristic
173	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
174	C₂×Q₈	—	2	12	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	24	no	—
177	C₂×Q₈	—	4	12	no	—
178	C₂×Q₈	—	4	12	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	4	8	no	—
182	C₂²×C₄	#6104	1	4	yes	—
183	C₂²×C₄	#6105	1	4	yes	—
184	C₂²×C₄	#6106	1	4	yes	—
185	C₂²×C₄	#6107	1	4	yes	—
186	C₂²×C₄	C₂²⋊Dic₁₃	1	3	yes	—
187	C₂²×C₄	C₂²⋊Dic₁₃	1	3	yes	—
188	C₂²×C₄	C₂²⋊Dic₁₃	1	3	yes	—
189	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
192	C₂×C₈	—	26	4	no	—
193	C₂×C₈	—	26	4	no	—
194	C₂×C₈	—	52	12	no	—
195	C₂×C₈	—	52	12	no	—
196	C₄⋊C₄	—	2	12	no	—
197	C₄⋊C₄	—	2	12	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
200	C₄²	C₂²⋊Dic₁₃	1	6	yes	—
201	C₄²	C₂²×Dic₁₃	1	3	yes	—
202	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
203	C₄²	C₂×C₂²⋊D₁₃	1	3	yes	—
204	C₄²	C₂³×D₁₃	1	1	yes	characteristic
205	C₁₃	C₂²×#332	1	1	yes	characteristic, complemented, Sylow
206	C₂³	#88072	1	4	yes	—
207	C₂³	#88075	1	4	yes	—
208	C₂³	#88076	1	4	yes	—
209	C₂³	C₂×C₂²⋊Dic₁₃	1	3	yes	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	24	no	—
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	2	6	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	24	no	—
217	C₂×C₄	#88072	1	12	yes	—
218	C₂×C₄	#88075	1	12	yes	—
219	C₂×C₄	C₂×#6104	1	6	yes	—
220	C₂×C₄	C₂×#6105	1	6	yes	—
221	C₂×C₄	C₂×#6106	1	6	yes	—
222	C₂×C₄	C₂×#6107	1	6	yes	—
223	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
225	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
226	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₈	—	52	4	no	—
230	C₈	—	52	4	no	—
231	C₂²	#11718139	1	16	yes	direct product factor
232	C₂²	C₂×#88072	1	6	yes	—
233	C₂²	C₂×#88075	1	6	yes	—
234	C₂²	C₂×#88076	1	6	yes	—
235	C₂²	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	8	no	—
239	C₄	C₂×#88072	1	6	yes	—
240	C₄	C₂×#88075	1	6	yes	—
241	C₄	C₂²×#6104	1	1	yes	characteristic
242	C₄	C₂²×#6105	1	1	yes	characteristic
243	C₄	C₂²×#6106	1	1	yes	characteristic
244	C₄	C₂²×#6107	1	1	yes	characteristic
245	C₂	C₂×#11718139	1	12	yes	direct product factor
246	C₂	C₂²×#88076	1	1	yes	characteristic
247	C₂	C₂²×#88072	1	1	yes	characteristic
248	C₂	C₂²×#88075	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂²×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	C₂²×#332
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×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
3. C₂×#11718139	C₂	direct (N × H)
19. #11718139	C₂²	direct (N × H)
126. C₂²×C₁₃	#332	semidirect
169. C₂×C₁₃	C₂×#332	semidirect
205. C₁₃	C₂²×#332	semidirect
231. C₂²	#11718139	direct (N × H)
245. C₂	C₂×#11718139	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×C₄⋊#1055	2
C₂×#11718139	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×#332	13

Subgroups — C22×#11718139 #15831202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718139 #15831202