Subgroups — #16806999

Order 3328.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13131171	C₂	1	8	yes	maximal, complemented
3	#13131168	C₂	1	4	yes	maximal, complemented
4	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	—	2	8	no	—
9	C₂×#88106	C₂²	1	4	yes	—
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	C₂²	1	4	yes	—
12	#11718658	C₄	1	4	yes	—
13	#11718658	C₂²	1	4	yes	—
14	C₂×#88043	C₄	1	2	yes	—
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	C₄⋊#6092	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₁₃×#117	C₈	1	4	yes	—
19	C₁₃×#117	C₂×C₄	1	4	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×#6099	—	2	4	no	—
24	C₂×#6099	C₂×C₄	1	4	yes	—
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	D₄	1	4	yes	—
29	C₂²×#1055	Q₈	1	4	yes	—
30	C₂²×#1055	C₂³	1	2	yes	—
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	#88106	—	2	8	no	—
33	#88106	—	2	8	no	—
34	C₂×#6092	—	2	8	no	—
35	C₂×#6092	C₂³	1	8	yes	—
36	#88043	C₂×C₄	1	8	yes	—
37	C₄×#1055	C₂×C₄	1	4	yes	—
38	C₄×#1055	C₂³	1	4	yes	—
39	#33397	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
49	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
50	C₄²×C₁₃	C₂×C₈	1	4	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
52	#6099	—	2	8	no	—
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	C₄⋊C₄	1	8	yes	—
55	C₂×#1055	C₄⋊C₄	1	4	yes	—
56	C₂×#1055	C₂²×C₄	1	4	yes	—
57	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
58	C₂×#1055	C₂²×C₄	1	2	yes	—
59	C₂×#1055	C₂×D₄	1	4	yes	—
60	C₂×#1055	C₂×D₄	1	2	yes	—
61	C₂×#1055	C₂×Q₈	1	4	yes	—
62	C₂×#1055	C₂×Q₈	1	2	yes	—
63	C₂×#1055	C₂⁴	1	1	yes	characteristic
64	#6092	—	2	8	no	—
65	C₂²×M₃₂	—	13	2	no	—
66	C₂×#405	—	13	1	no	—
67	#2151	—	13	8	no	—
68	#2150	—	13	4	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
71	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
72	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
75	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	#1055	—	2	4	no	—
85	#1055	C₂×C₄⋊C₄	1	2	yes	—
86	#1055	C₂×C₄⋊C₄	1	4	yes	—
87	#1055	C₂²×D₄	1	1	yes	characteristic
88	#1055	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×M₁₆	—	13	1	no	—
90	C₂³×C₈	—	13	1	no	—
91	C₂×#117	#261	1	1	yes	characteristic
92	C₂×M₃₂	—	13	4	no	—
93	C₂×M₃₂	—	13	4	no	—
94	C₂×M₃₂	—	26	8	no	—
95	C₂²×C₁₆	—	13	4	no	—
96	#405	—	13	4	no	—
97	#405	—	13	4	no	—
98	C₂×#97	—	13	2	no	—
99	C₂×C₄×C₈	—	13	2	no	—
100	C₄⋊C₁₆	—	13	16	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	4	4	no	—
104	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
105	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
106	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₁₃	—	2	8	no	—
108	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
109	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
110	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
112	C₂³×C₄	C₂×#261	1	1	yes	characteristic
113	C₂×M₁₆	—	13	4	no	—
114	C₂×M₁₆	—	26	4	no	—
115	C₂²×C₈	—	13	4	no	—
116	C₂²×C₈	—	13	4	no	—
117	C₂²×C₈	—	13	1	no	—
118	C₂²×C₈	—	13	2	no	—
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	2	no	—
121	C₂²×C₈	—	13	2	no	—
122	#117	#1057	1	4	yes	—
123	#117	C₂×#261	1	4	yes	—
124	C₂×C₄⋊C₄	#1057	1	2	yes	—
125	C₂×C₂²⋊C₄	#1057	1	2	yes	—
126	C₂×C₄²	C₂×#261	1	2	yes	—
127	M₃₂	—	26	8	no	—
128	M₃₂	—	26	8	no	—
129	C₂×C₁₆	—	13	8	no	—
130	C₂×C₁₆	—	26	8	no	—
131	#97	—	13	8	no	—
132	C₄×C₈	—	13	4	no	—
133	C₄×C₈	—	13	4	no	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	2	4	no	—
136	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₁₃	#3407	1	2	yes	—
138	C₂⁴	C₂×#1057	1	1	yes	characteristic
139	C₂²×C₄	#6118	1	4	yes	—
140	C₂²×C₄	C₄⋊#261	1	4	yes	—
141	C₂²×C₄	C₂×#1057	1	4	yes	—
142	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
143	C₂²×C₄	C₂×#1057	1	2	yes	—
144	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
145	C₂²×C₄	C₂²×#261	1	2	yes	—
146	M₁₆	—	26	8	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	13	2	no	—
149	C₂×C₈	—	13	4	no	—
150	C₂×C₈	—	13	8	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	4	no	—
153	C₂×C₈	—	13	1	no	—
154	C₂×C₈	—	13	1	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	4	no	—
157	C₂×C₈	—	26	4	no	—
158	C₄⋊C₄	C₂×#1057	1	8	yes	—
159	C₂²⋊C₄	C₂×#1057	1	8	yes	—
160	C₄²	C₂×#1057	1	4	yes	—
161	C₄²	C₂²×#261	1	4	yes	—
162	C₁₆	—	26	8	no	—
163	C₁₃	#33397	1	1	yes	characteristic, complemented, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	C₄⋊#1057	1	4	yes	—
166	C₂³	C₂²×#1057	1	2	yes	—
167	C₂³	C₂²×#1057	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₄⋊#1057	1	4	yes	—
170	C₂×C₄	C₄⋊#1057	1	8	yes	—
171	C₂×C₄	C₂×#6118	1	2	yes	—
172	C₂×C₄	C₂×#6118	1	4	yes	—
173	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
175	C₂×C₄	C₂²×#1057	1	8	yes	—
176	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
177	C₂×C₄	C₂²×#1057	1	2	yes	—
178	C₂×C₄	C₂³×#261	1	1	yes	characteristic
179	C₈	—	13	1	no	—
180	C₈	—	13	4	no	—
181	C₈	—	13	2	no	—
182	C₈	—	13	1	no	—
183	C₈	—	26	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₄⋊#1057	1	2	yes	—
188	C₂²	C₂×C₄⋊#1057	1	4	yes	—
189	C₂²	C₂³×#1057	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₄⋊#1057	1	2	yes	—
192	C₄	C₂×C₄⋊#1057	1	4	yes	—
193	C₄	C₂²×#6118	1	1	yes	characteristic
194	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₄⋊#1057	1	1	yes	characteristic
198	C₂	#13131803	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×#117	C₄
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#33397
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13131171	C₂	semidirect
3. #13131168	C₂	semidirect
16. C₄⋊#6092	C₂²	semidirect
163. C₁₃	#33397	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13131171	2
#13131168	2
C₂×#11718658	2
#33397	13