Subgroups — #14672143

Order 2816.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10622147	C₂	1	2	yes	maximal, complemented
4	#13069482	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈₈	C₂²	1	2	yes	—
7	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×#66885	—	2	2	no	—
11	C₂×#66882	C₂²	1	2	yes	—
12	#10622552	—	2	2	no	—
13	C₂×D₁₇₆	—	2	2	no	—
14	C₂×D₁₇₆	—	2	2	no	—
15	C₂×D₁₇₆	C₄	1	4	yes	complemented
16	C₄×C₁₁×C₁₆	—	2	2	no	—
17	#10622147	—	2	4	no	—
18	#10622142	—	2	2	no	—
19	C₂²×D₄₄	—	2	2	no	—
20	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
23	C₂×(C₂×D₁₁)⋊C₄	—	4	4	no	—
24	C₂×C₄⋊Dic₁₁	—	2	2	no	—
25	C₂×D₈₈	—	2	2	no	—
26	C₂×D₈₈	—	2	2	no	—
27	C₂×D₈₈	—	2	2	no	—
28	C₂×D₈₈	—	2	2	no	—
29	C₂×D₈₈	—	2	2	no	—
30	C₂×D₈₈	C₂×C₄	1	4	yes	—
31	C₂×C₁₁×C₁₆	—	2	1	no	—
32	C₂×C₁₁×C₁₆	—	2	1	no	—
33	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
35	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
36	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
37	C₄×C₈×C₁₁	—	2	1	no	—
38	C₄×C₈×C₁₁	—	2	1	no	—
39	#66885	—	4	4	no	—
40	#66882	—	2	2	no	—
41	#66882	—	2	2	no	—
42	D₁₇₆	—	2	8	no	—
43	#13235	—	11	1	no	maximal, Sylow
44	C₂³×D₁₁	—	4	2	no	—
45	C₂²×C₄×C₁₁	D₈	1	2	yes	—
46	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₁	—	4	2	no	—
48	C₂×D₄₄	—	2	4	no	—
49	C₂×D₄₄	—	4	2	no	—
50	C₂×D₄₄	—	4	2	no	—
51	C₂×D₄₄	—	4	2	no	—
52	C₂×D₄₄	—	4	2	no	—
53	C₂×D₄₄	—	4	2	no	—
54	C₂×C₈×C₁₁	—	2	1	no	—
55	C₂×C₈×C₁₁	—	2	1	no	—
56	C₂×C₈×C₁₁	D₈	1	2	yes	—
57	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
62	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
63	C₄²×C₁₁	—	2	1	no	—
64	C₄²×C₁₁	—	2	1	no	—
65	(C₂×D₁₁)⋊C₄	—	8	8	no	—
66	C₄⋊Dic₁₁	—	4	2	no	—
67	C₄⋊Dic₁₁	—	4	2	no	—
68	D₈₈	—	2	4	no	—
69	D₈₈	—	2	4	no	—
70	D₈₈	—	4	2	no	—
71	D₈₈	—	4	2	no	—
72	C₁₁×C₁₆	—	2	2	no	—
73	C₁₁×C₁₆	C₂²⋊C₄	1	4	yes	complemented
74	C₂²×D₁₆	—	11	1	no	—
75	C₂×#366	—	11	1	no	—
76	C₂×#357	—	11	2	no	—
77	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
78	#1970	—	11	2	no	—
79	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
80	C₂²×D₁₁	—	4	4	no	—
81	C₂²×D₁₁	—	8	2	no	—
82	C₂²×D₁₁	—	8	2	no	—
83	C₂²×D₁₁	—	8	4	no	—
84	C₂²×D₁₁	—	8	4	no	—
85	C₂×C₄×C₁₁	—	2	2	no	—
86	C₂×C₄×C₁₁	—	2	2	no	—
87	C₂×C₄×C₁₁	D₁₆	1	4	yes	complemented
88	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
93	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
94	C₂×Dic₁₁	—	8	2	no	—
95	C₂×Dic₁₁	—	8	4	no	—
96	D₄₄	—	4	4	no	—
97	D₄₄	—	4	4	no	—
98	D₄₄	—	8	2	no	—
99	D₄₄	—	8	2	no	—
100	C₈×C₁₁	—	2	2	no	—
101	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
103	C₈×C₁₁	#121	1	1	yes	characteristic
104	C₈×C₁₁	#124	1	1	yes	characteristic
105	C₂²×D₈	—	11	2	no	—
106	C₂×D₁₆	—	11	4	no	—
107	C₂×D₁₆	—	22	2	no	—
108	C₂×D₁₆	—	22	2	no	—
109	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
111	C₂×#107	—	11	2	no	—
112	C₂×#102	—	22	2	no	—
113	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
114	#390	—	22	2	no	—
115	#366	—	22	2	no	—
116	#357	—	22	4	no	—
117	C₄×C₁₆	—	2	2	no	—
118	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
119	C₂²×C₁₁	#466	1	1	yes	characteristic
120	C₂²×C₁₁	#493	1	1	yes	characteristic
121	C₂²×C₁₁	C₂×D₁₆	1	2	yes	—
122	C₂²×C₁₁	#508	1	2	yes	—
123	C₂×D₁₁	—	8	8	no	—
124	C₂×D₁₁	—	8	4	no	—
125	C₂×D₁₁	—	16	4	no	—
126	C₂×D₁₁	—	16	2	no	—
127	C₂×D₁₁	—	16	2	no	—
128	C₄×C₁₁	—	2	4	no	—
129	C₄×C₁₁	#390	1	1	yes	characteristic
130	C₄×C₁₁	C₄×D₈	1	1	yes	characteristic
131	C₄×C₁₁	#466	1	1	yes	characteristic
132	C₄×C₁₁	#495	1	1	yes	characteristic
133	Dic₁₁	—	16	2	no	—
134	C₂²×D₄	—	22	2	no	—
135	C₂×D₈	—	11	4	no	—
136	C₂×D₈	—	22	2	no	—
137	C₂×D₈	—	22	2	no	—
138	C₂×D₈	—	22	2	no	—
139	C₂×D₈	—	22	2	no	—
140	C₂×D₈	—	22	2	no	—
141	C₂²×C₈	D₄₄	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	22	2	no	—
144	C₂×C₂²⋊C₄	—	44	4	no	—
145	C₂×C₄²	D₄₄	1	1	yes	characteristic
146	D₁₆	—	22	8	no	—
147	C₂×C₁₆	—	2	1	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
150	C₂×C₁₆	D₄₄	1	2	yes	—
151	C₂×C₁₆	D₄₄	1	2	yes	—
152	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
153	#107	—	22	2	no	—
154	#107	—	22	2	no	—
155	#102	—	44	4	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₂×C₁₁	#1970	1	1	yes	characteristic
159	C₂×C₁₁	C₄×D₁₆	1	2	yes	—
160	C₂×C₁₁	#2215	1	2	yes	—
161	C₂×C₁₁	#2246	1	1	yes	characteristic
162	C₂×C₁₁	#2249	1	1	yes	characteristic
163	D₁₁	—	16	4	no	—
164	C₂⁴	—	44	2	no	—
165	C₂×D₄	—	22	4	no	—
166	C₂×D₄	—	44	2	no	—
167	C₂×D₄	—	44	2	no	—
168	C₂×D₄	—	44	2	no	—
169	C₂×D₄	—	44	2	no	—
170	C₂×D₄	—	44	2	no	—
171	C₂²×C₄	—	44	2	no	—
172	C₂²×C₄	D₈₈	1	2	yes	—
173	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
174	D₈	—	22	4	no	—
175	D₈	—	22	4	no	—
176	D₈	—	44	2	no	—
177	D₈	—	44	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	D₈₈	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
184	C₂×C₈	#4339	1	1	yes	characteristic
185	C₂×C₈	#4339	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
187	C₄⋊C₄	—	44	2	no	—
188	C₄⋊C₄	—	44	2	no	—
189	C₂²⋊C₄	—	88	8	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	(C₂×D₁₁)⋊C₄	1	4	yes	complemented
194	C₁₁	#13235	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	44	4	no	—
196	C₂³	—	88	2	no	—
197	C₂³	—	88	4	no	—
198	C₂³	—	88	4	no	—
199	C₂³	—	88	2	no	—
200	C₂³	C₂×D₈₈	1	1	yes	characteristic
201	D₄	—	44	4	no	—
202	D₄	—	44	4	no	—
203	D₄	—	88	2	no	—
204	D₄	—	88	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	88	4	no	—
208	C₂×C₄	—	88	2	no	—
209	C₂×C₄	D₁₇₆	1	4	yes	complemented
210	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
211	C₂×C₄	#66927	1	1	yes	characteristic
212	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
213	C₂×C₄	#66957	1	1	yes	characteristic
214	C₂×C₄	#66957	1	1	yes	characteristic
215	C₂×C₄	#66983	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	C₄×D₄₄	1	1	yes	characteristic
218	C₈	#66928	1	1	yes	characteristic
219	C₈	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
220	C₈	#66983	1	1	yes	characteristic
221	C₂²	—	88	8	no	—
222	C₂²	—	88	4	no	—
223	C₂²	—	176	2	no	—
224	C₂²	—	176	2	no	—
225	C₂²	—	176	4	no	—
226	C₂²	C₄×D₈₈	1	1	yes	characteristic
227	C₂²	#10622308	1	1	yes	characteristic
228	C₂²	C₂×D₁₇₆	1	2	yes	—
229	C₂²	#10622517	1	2	yes	—
230	C₂²	#10622728	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	176	2	no	—
233	C₄	C₄×D₈₈	1	1	yes	characteristic
234	C₄	#10622309	1	1	yes	characteristic
235	C₄	#10622552	1	1	yes	characteristic
236	C₄	#10622728	1	1	yes	characteristic
237	C₂	—	176	4	no	—
238	C₂	#13069482	1	1	yes	characteristic
239	C₂	C₄×D₁₇₆	1	2	yes	—
240	C₂	#13070702	1	2	yes	—
241	C₂	#13070866	1	1	yes	characteristic
242	C₂	#13070877	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle	C₂³×C₁₁	C₂×D₈
prime radical	C₂²×D₁₇₆	C₂
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#13235
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#10622142	C₂	semidirect
3. C₂×#10622147	C₂	semidirect
4. #13069482	C₂	semidirect
5. C₂×C₄×C₁₁×C₁₆	C₂	semidirect
15. C₂×D₁₇₆	C₄	semidirect
73. C₁₁×C₁₆	C₂²⋊C₄	semidirect
77. C₂×C₄×C₁₆	D₁₁	semidirect
87. C₂×C₄×C₁₁	D₁₆	semidirect
193. C₁₆	(C₂×D₁₁)⋊C₄	semidirect
194. C₁₁	#13235	semidirect
209. C₂×C₄	D₁₇₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₇₆	2
C₂×#10622142	2
C₂×#10622147	2
#13069482	2
C₂×C₄×C₁₁×C₁₆	2
#13235	11