Subgroups — #14537702

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587590	C₂	1	1	yes	maximal, characteristic
2	#11763575	C₂	1	2	yes	maximal, complemented
3	#11762932	C₂	1	4	yes	maximal, complemented
4	#11762927	C₂	1	4	yes	maximal, complemented
5	C₂×#10587490	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762827	C₂	1	2	yes	maximal, complemented
7	C₁₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#63252	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₁₇×#117	C₂²	1	1	yes	characteristic
12	C₂×#63223	C₂²	1	2	yes	—
13	#10587593	—	2	4	no	—
14	M₁₆×D₁₇	—	2	4	no	—
15	C₂×Dic₆₈	C₄	1	1	yes	characteristic, complemented
16	#10587590	C₄	1	2	yes	complemented
17	#10587590	C₄	1	2	yes	complemented
18	#10587590	C₂²	1	2	yes	—
19	#10587590	C₂²	1	2	yes	—
20	C₂×D₁₃₆	C₄	1	1	yes	characteristic, complemented
21	C₂×#63219	C₄	1	2	yes	complemented
22	#10587587	C₂²	1	8	yes	complemented
23	C₁₇×#97	C₂²	1	2	yes	complemented
24	C₄×C₈×C₁₇	C₂²	1	2	yes	complemented
25	#10587490	C₂²	1	2	yes	complemented
26	#10587490	C₂²	1	2	yes	complemented
27	#10587476	C₂²	1	8	yes	complemented
28	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊D₁₇	—	2	2	no	—
30	#63252	—	2	2	no	—
31	#63252	—	2	2	no	—
32	#63252	C₂×C₄	1	4	yes	—
33	#63252	C₂³	1	4	yes	—
34	C₂×D₆₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₇	—	2	2	no	—
36	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
37	M₁₆×C₁₇	D₄	1	2	yes	—
38	M₁₆×C₁₇	D₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₁₇	—	2	1	no	—
47	C₂²⋊C₄×C₁₇	—	2	1	no	—
48	C₄²×C₁₇	C₂³	1	2	yes	—
49	#63223	—	2	2	no	—
50	#63223	C₂³	1	4	yes	—
51	C₂×#3622	—	2	2	no	—
52	Dic₆₈	C₂×C₄	1	2	yes	—
53	Dic₆₈	C₂×C₄	1	2	yes	—
54	D₁₃₆	C₂×C₄	1	2	yes	—
55	D₁₃₆	C₂×C₄	1	2	yes	—
56	#63219	C₂×C₄	1	4	yes	—
57	#63219	C₂×C₄	1	4	yes	—
58	#63218	—	2	8	no	—
59	C₈×D₁₇	—	2	8	no	—
60	C₂³×C₁₇	#41	1	1	yes	characteristic
61	C₂²×D₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
69	C₂²⋊D₁₇	—	2	4	no	—
70	C₂²⋊D₁₇	—	2	4	no	—
71	C₂×Dic₁₇	—	2	2	no	—
72	D₆₈	—	2	2	no	—
73	D₆₈	C₂²×C₄	1	4	yes	—
74	C₄×D₁₇	—	2	4	no	—
75	C₄×D₁₇	—	2	4	no	—
76	Dic₃₄	—	2	2	no	—
77	Dic₃₄	C₂²×C₄	1	4	yes	—
78	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₇	C₂×D₄	1	4	yes	—
83	#3622	—	2	4	no	—
84	#2957	—	17	1	no	maximal, Sylow
85	C₂²×C₁₇	—	4	1	no	—
86	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
89	C₂×D₁₇	—	2	4	no	—
90	C₂×D₁₇	—	8	2	no	—
91	C₄×C₁₇	—	4	2	no	—
92	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
93	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
94	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
95	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
96	Dic₁₇	—	2	4	no	—
97	C₂×#135	—	17	1	no	—
98	#568	—	17	2	no	—
99	#444	—	17	4	no	—
100	#443	—	17	4	no	—
101	C₂×#108	—	17	1	no	—
102	#421	—	17	2	no	—
103	#405	D₁₇	1	1	yes	characteristic, complemented
104	C₂×C₁₇	—	2	1	no	—
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	—	2	1	no	—
107	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
108	D₁₇	—	4	4	no	—
109	C₂×#41	—	17	2	no	—
110	#135	—	17	2	no	—
111	#135	—	17	2	no	—
112	#135	—	17	2	no	—
113	#135	—	17	2	no	—
114	C₂×Q₁₆	—	17	1	no	—
115	C₂×SD₁₆	—	17	2	no	—
116	C₂×D₈	—	17	1	no	—
117	#131	—	17	8	no	—
118	#131	—	34	4	no	—
119	C₂×M₁₆	—	17	2	no	—
120	C₂×M₁₆	—	34	4	no	—
121	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
123	#117	C₂×D₁₇	1	1	yes	characteristic
124	#108	—	17	2	no	—
125	#108	—	17	2	no	—
126	#104	—	17	8	no	—
127	#97	C₂×D₁₇	1	2	yes	complemented
128	C₄×C₈	C₂×D₁₇	1	2	yes	complemented
129	C₁₇	#2957	1	1	yes	characteristic, complemented, Sylow
130	#41	—	17	4	no	—
131	#41	—	17	4	no	—
132	#41	—	34	2	no	—
133	#41	—	34	2	no	—
134	C₂×Q₈	—	17	2	no	—
135	C₂×D₄	—	17	2	no	—
136	C₂×D₄	—	34	2	no	—
137	C₂²×C₄	—	34	2	no	—
138	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
139	Q₁₆	—	17	2	no	—
140	Q₁₆	—	17	2	no	—
141	SD₁₆	—	17	4	no	—
142	SD₁₆	—	17	4	no	—
143	D₈	—	17	2	no	—
144	D₈	—	17	2	no	—
145	M₁₆	—	17	4	no	—
146	M₁₆	—	34	2	no	—
147	M₁₆	—	34	8	no	—
148	M₁₆	D₆₈	1	2	yes	—
149	M₁₆	D₆₈	1	2	yes	—
150	C₂×C₈	—	34	2	no	—
151	C₂×C₈	—	34	8	no	—
152	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
153	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
154	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₁₇	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂²×D₁₇	1	2	yes	—
162	C₂³	—	34	2	no	—
163	C₂³	#63252	1	1	yes	characteristic
164	Q₈	—	17	4	no	—
165	Q₈	—	34	2	no	—
166	D₄	—	17	4	no	—
167	D₄	—	34	4	no	—
168	D₄	—	34	4	no	—
169	D₄	—	34	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	34	2	no	—
172	C₂×C₄	—	34	4	no	—
173	C₂×C₄	—	34	4	no	—
174	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
176	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
177	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
178	C₂×C₄	#63252	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
180	C₈	—	34	4	no	—
181	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
182	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
183	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
184	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
185	C₈	C₂×D₆₈	1	4	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	34	4	no	—
188	C₂²	—	136	2	no	—
189	C₂²	C₄×D₆₈	1	1	yes	characteristic
190	C₂²	C₄×D₆₈	1	1	yes	characteristic
191	C₂²	C₂×#63252	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	34	4	no	—
194	C₄	C₄×D₆₈	1	1	yes	characteristic
195	C₄	C₄×D₆₈	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
197	C₄	C₂²×D₆₈	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	68	4	no	—
202	C₂	C₂×C₄×D₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₆₈
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#405	C₂
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	C₂×#10587590	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#405	D₁₇
hypocenter	C₁₇	#2957
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #11763575	C₂	semidirect
3. #11762932	C₂	semidirect
4. #11762927	C₂	semidirect
5. C₂×#10587490	C₂	semidirect
6. #11762827	C₂	semidirect
7. C₁₇×#405	C₂	semidirect
15. C₂×Dic₆₈	C₄	semidirect
16. #10587590	C₄	semidirect
17. #10587590	C₄	semidirect
20. C₂×D₁₃₆	C₄	semidirect
21. C₂×#63219	C₄	semidirect
22. #10587587	C₂²	semidirect
23. C₁₇×#97	C₂²	semidirect
24. C₄×C₈×C₁₇	C₂²	semidirect
25. #10587490	C₂²	semidirect
26. #10587490	C₂²	semidirect
27. #10587476	C₂²	semidirect
103. #405	D₁₇	semidirect
127. #97	C₂×D₁₇	semidirect
128. C₄×C₈	C₂×D₁₇	semidirect
129. C₁₇	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587590	2
#11763575	2
#11762932	2
#11762927	2
C₂×#10587490	2
#11762827	2
C₁₇×#405	2
#2957	17