Subgroups — C₂²×#11719175 #16769625

Order 3328.

8238 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719175	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88123	C₂	1	2	yes	maximal, complemented
3	C₂²×#88124	C₂	1	2	yes	maximal, complemented
4	C₂²×#88178	C₂	1	4	yes	maximal
5	C₂²×C₄×Dic₂₆	C₂	1	4	yes	maximal, complemented
6	C₂²×#88180	C₂	1	2	yes	maximal
7	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×Dic₂₆	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₃	C₂²	1	24	yes	—
15	C₂²×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
16	C₂×#88180	C₂²	1	48	yes	—
17	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
19	C₂×#88178	C₂²	1	96	yes	—
20	C₂²×#6101	C₂²	1	8	yes	—
21	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
22	#11719175	C₂²	1	256	yes	direct product factor
23	C₂×#88124	C₂²	1	48	yes	complemented
24	C₂×#88123	C₂²	1	48	yes	complemented
25	C₂×C₄×Dic₂₆	C₂²	1	96	yes	complemented
26	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	—	2	16	no	—
28	C₂³×C₄×C₁₃	Q₈	1	16	yes	complemented
29	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	24	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×Dic₁₃	C₂³	1	8	yes	—
36	C₂²×Dic₂₆	—	2	32	no	—
37	C₂²×Dic₂₆	C₂³	1	48	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	24	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	64	yes	—
42	C₂×C₂²⋊Dic₁₃	C₂³	1	48	yes	—
43	#88180	C₂³	1	128	yes	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	—
46	#88178	C₂³	1	256	yes	—
47	C₂×#6101	C₂³	1	96	yes	—
48	C₂×C₄×Dic₁₃	C₂³	1	48	yes	—
49	#88124	C₂³	1	128	yes	complemented
50	#88123	C₂³	1	128	yes	complemented
51	C₄×Dic₂₆	C₂³	1	256	yes	complemented
52	C₂²×#533	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	—	2	8	no	—
54	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	—
55	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
56	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	—	2	96	no	—
59	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	96	yes	complemented
62	C₂²×C₄×C₁₃	#41	1	16	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	32	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
69	C₂²×Dic₁₃	—	2	32	no	—
70	C₂²×Dic₁₃	C₂⁴	1	48	yes	—
71	C₂×Dic₂₆	—	2	192	no	—
72	C₂×Dic₂₆	C₂⁴	1	64	yes	—
73	C₄²×C₁₃	C₂⁴	1	32	yes	—
74	C₄²×C₁₃	C₂⁴	1	16	yes	—
75	C₄²×C₁₃	C₂⁴	1	16	yes	—
76	C₂²⋊Dic₁₃	C₂⁴	1	64	yes	—
77	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
78	C₄⋊Dic₁₃	C₂⁴	1	64	yes	—
79	#6101	C₂⁴	1	128	yes	—
80	C₄×Dic₁₃	C₂⁴	1	64	yes	—
81	C₂×#533	—	13	48	no	—
82	C₂²×C₄⋊Q₈	—	13	2	no	—
83	C₂²×#125	—	13	2	no	—
84	C₂²×C₂²⋊Q₈	—	13	4	no	—
85	C₂²×C₄×Q₈	—	13	4	no	—
86	C₂²×#117	—	13	2	no	—
87	C₂³×C₄²	D₁₃	1	1	yes	characteristic
88	C₂³×C₁₃	—	2	32	no	—
89	C₂³×C₁₃	—	2	24	no	—
90	C₂³×C₁₃	C₂²×Q₈	1	24	yes	—
91	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
92	C₂³×C₁₃	C₂×#41	1	8	yes	—
93	C₂³×C₁₃	C₂⁵	1	4	yes	—
94	C₂³×C₁₃	C₂⁵	1	3	yes	—
95	C₂×C₄×C₁₃	—	2	24	no	—
96	C₂×C₄×C₁₃	—	2	64	no	—
97	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	—
98	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	—
99	C₂×C₄×C₁₃	C₂²×Q₈	1	64	yes	complemented
100	C₂×C₄×C₁₃	C₂×#41	1	48	yes	—
101	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
104	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
105	C₂×Dic₁₃	—	2	96	no	—
106	C₂×Dic₁₃	C₂⁵	1	32	yes	—
107	Dic₂₆	—	2	128	no	—
108	C₂³×Q₈	—	13	4	no	—
109	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
110	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
111	#533	—	13	256	no	—
112	C₂×C₄⋊Q₈	—	13	48	no	—
113	C₂×#125	—	13	48	no	—
114	C₂×C₂²⋊Q₈	—	13	96	no	—
115	C₂×C₄×Q₈	—	13	96	no	—
116	C₂×#117	—	13	48	no	—
117	C₂²×C₄⋊C₄	—	13	8	no	—
118	C₂²×C₄⋊C₄	—	13	4	no	—
119	C₂²×C₄⋊C₄	—	13	4	no	—
120	C₂²×C₂²⋊C₄	—	13	4	no	—
121	C₂²×C₄²	—	13	4	no	—
122	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
124	C₂²×C₄²	C₂×D₁₃	1	24	yes	—
125	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
126	C₂²×C₁₃	—	2	8	no	—
127	C₂²×C₁₃	—	2	48	no	—
128	C₂²×C₁₃	#533	1	16	yes	complemented
129	C₂²×C₁₃	C₂³×Q₈	1	8	yes	—
130	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
131	C₂²×C₁₃	C₂²×#41	1	12	yes	—
132	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
133	C₄×C₁₃	—	2	8	no	—
134	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
135	C₄×C₁₃	C₂³×Q₈	1	8	yes	—
136	C₄×C₁₃	C₂²×#41	1	16	yes	—
137	Dic₁₃	—	2	32	no	—
138	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
139	C₂²×Q₈	—	13	48	no	—
140	C₂²×Q₈	—	26	32	no	—
141	C₂³×C₄	—	2	16	no	—
142	C₂³×C₄	—	13	8	no	—
143	C₂³×C₄	Dic₂₆	1	16	yes	complemented
144	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
145	C₂³×C₄	C₂²×D₁₃	1	24	yes	—
146	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
147	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
148	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
149	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	13	128	no	—
151	#125	—	13	128	no	—
152	C₂²⋊Q₈	—	13	256	no	—
153	C₄×Q₈	—	13	256	no	—
154	#117	—	13	128	no	—
155	C₂×C₄⋊C₄	—	13	48	no	—
156	C₂×C₄⋊C₄	—	13	96	no	—
157	C₂×C₄⋊C₄	—	13	48	no	—
158	C₂×C₂²⋊C₄	—	13	48	no	—
159	C₂×C₄²	—	13	48	no	—
160	C₂×C₄²	C₂²×D₁₃	1	24	yes	—
161	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
162	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
163	C₂×C₄²	C₂²×D₁₃	1	64	yes	—
164	C₂×C₁₃	—	2	8	no	—
165	C₂×C₁₃	C₂×#533	1	12	yes	complemented
166	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
167	C₂×C₁₃	C₂³×#41	1	2	yes	—
168	C₂⁴	—	2	8	no	—
169	C₂⁴	C₂×Dic₂₆	1	8	yes	—
170	C₂⁴	C₂³×D₁₃	1	6	yes	—
171	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
172	C₂×Q₈	—	13	64	no	—
173	C₂×Q₈	—	26	192	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	96	no	—
176	C₂²×C₄	—	13	48	no	—
177	C₂²×C₄	—	26	32	no	—
178	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
179	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
180	C₂²×C₄	C₂×Dic₂₆	1	96	yes	complemented
181	C₂²×C₄	#6127	1	16	yes	—
182	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
183	C₂²×C₄	C₂³×D₁₃	1	32	yes	—
184	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
187	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
188	C₄⋊C₄	—	13	64	no	—
189	C₄⋊C₄	—	13	128	no	—
190	C₄⋊C₄	—	13	64	no	—
191	C₂²⋊C₄	—	13	64	no	—
192	C₄²	—	13	64	no	—
193	C₄²	C₂³×D₁₃	1	32	yes	—
194	C₄²	C₂³×D₁₃	1	16	yes	—
195	C₄²	C₂³×D₁₃	1	16	yes	—
196	C₁₃	C₂²×#533	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	2	32	no	—
198	C₂³	—	2	24	no	—
199	C₂³	C₂²×Dic₂₆	1	24	yes	—
200	C₂³	C₂²×Dic₂₆	1	4	yes	—
201	C₂³	C₂×#6127	1	8	yes	—
202	C₂³	C₂⁴×D₁₃	1	4	yes	—
203	C₂³	C₂⁴×D₁₃	1	3	yes	—
204	Q₈	—	26	128	no	—
205	C₂×C₄	—	2	24	no	—
206	C₂×C₄	—	2	64	no	—
207	C₂×C₄	—	13	32	no	—
208	C₂×C₄	—	26	96	no	—
209	C₂×C₄	C₂²×Dic₂₆	1	24	yes	—
210	C₂×C₄	C₂²×Dic₂₆	1	24	yes	—
211	C₂×C₄	C₂²×Dic₂₆	1	64	yes	complemented
212	C₂×C₄	C₂×#6127	1	48	yes	—
213	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
214	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
215	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
216	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	48	no	—
219	C₂²	#11719175	1	16	yes	direct product factor
220	C₂²	C₂³×Dic₂₆	1	8	yes	—
221	C₂²	C₂³×Dic₂₆	1	6	yes	—
222	C₂²	C₂²×#6127	1	12	yes	—
223	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	26	32	no	—
226	C₄	C₂³×Dic₂₆	1	8	yes	—
227	C₄	C₂³×Dic₂₆	1	8	yes	—
228	C₄	C₂²×#6127	1	16	yes	—
229	C₂	—	2	8	no	—
230	C₂	C₂×#11719175	1	12	yes	direct product factor
231	C₂	C₂⁴×Dic₂₆	1	1	yes	characteristic
232	C₂	C₂³×#6127	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₄²×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁵×C₁₃	C₂³
hypercenter	C₂³×C₄²	D₁₃
hypocenter	C₁₃	C₂²×#533
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#11719175	C₂	direct (N × H)
2. C₂²×#88123	C₂	semidirect
3. C₂²×#88124	C₂	semidirect
5. C₂²×C₄×Dic₂₆	C₂	semidirect
22. #11719175	C₂²	direct (N × H)
23. C₂×#88124	C₂²	semidirect
24. C₂×#88123	C₂²	semidirect
25. C₂×C₄×Dic₂₆	C₂²	semidirect
28. C₂³×C₄×C₁₃	Q₈	semidirect
49. #88124	C₂³	semidirect
50. #88123	C₂³	semidirect
51. C₄×Dic₂₆	C₂³	semidirect
61. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
99. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
128. C₂²×C₁₃	#533	semidirect
143. C₂³×C₄	Dic₂₆	semidirect
165. C₂×C₁₃	C₂×#533	semidirect
180. C₂²×C₄	C₂×Dic₂₆	semidirect
196. C₁₃	C₂²×#533	semidirect
211. C₂×C₄	C₂²×Dic₂₆	semidirect
219. C₂²	#11719175	direct (N × H)
230. C₂	C₂×#11719175	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719175	2
C₂²×#88123	2
C₂²×#88124	2
C₂²×#88178	2
C₂²×C₄×Dic₂₆	2
C₂²×#88180	2
C₂³×C₄²×C₁₃	2
C₂²×#533	13

Subgroups — C22×#11719175 #16769625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719175 #16769625