Subgroups — #14537735

Order 2176.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₄	C₂	1	1	yes	maximal, characteristic
2	#11762938	C₂	1	4	yes	maximal, complemented
3	#11762936	C₂	1	2	yes	maximal, complemented
4	#11762933	C₂	1	2	yes	maximal, complemented
5	#11762832	C₂	1	2	yes	maximal
6	C₄×#63223	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₁₇	C₂	1	2	yes	maximal
8	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₄	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₇	C₄	1	1	yes	characteristic
13	C₂×#63225	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
15	C₂×#63223	C₂²	1	1	yes	characteristic
16	C₂×#63223	C₂²	1	1	yes	characteristic
17	C₄×Dic₃₄	C₄	1	2	yes	—
18	C₄×Dic₃₄	C₄	1	1	yes	characteristic
19	C₄×Dic₃₄	C₄	1	1	yes	characteristic
20	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₄	C₂²	1	2	yes	—
23	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
25	#10587486	C₂²	1	4	yes	—
26	#10587485	C₂²	1	4	yes	—
27	#10587485	C₂²	1	4	yes	—
28	C₈×Dic₁₇	C₂²	1	4	yes	—
29	#10587474	C₂²	1	2	yes	—
30	C₄×#3622	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂²×Dic₁₇	C₂×C₄	1	2	yes	—
36	C₂×Dic₃₄	—	2	4	no	—
37	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
39	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₁₇	—	2	4	no	—
41	C₂×C₈×C₁₇	C₂³	1	4	yes	—
42	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₂³	1	1	yes	characteristic
45	C₄²×C₁₇	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
49	#63225	C₂×C₄	1	2	yes	—
50	#63225	C₂×C₄	1	2	yes	—
51	#63225	C₂×C₄	1	2	yes	—
52	#63225	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₇	C₂³	1	2	yes	—
56	C₄×Dic₁₇	C₂³	1	2	yes	—
57	#63223	—	2	2	no	—
58	#63223	Q₈	1	4	yes	complemented
59	C₂×#3622	C₂³	1	2	yes	—
60	C₂×#3622	C₂³	1	2	yes	—
61	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	4	1	no	—
64	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₇	#41	1	2	yes	—
71	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₁₇	—	2	4	no	—
73	C₂×Dic₁₇	—	4	2	no	—
74	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
78	Dic₃₄	—	2	2	no	—
79	Dic₃₄	—	2	4	no	—
80	Dic₃₄	M₁₆	1	4	yes	complemented
81	C₈×C₁₇	—	2	4	no	—
82	#3622	—	2	2	no	—
83	#3622	C₂×Q₈	1	4	yes	complemented
84	Q₈×M₁₆	—	17	1	no	maximal, Sylow
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	C₄×Q₈	1	2	yes	—
88	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	—	2	1	no	—
92	C₄×C₁₇	C₄×Q₈	1	2	yes	—
93	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
94	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
95	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
96	Dic₁₇	—	2	2	no	—
97	Dic₁₇	—	2	2	no	—
98	Dic₁₇	—	2	2	no	—
99	Dic₁₇	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×Q₈	—	17	1	no	—
101	#446	—	17	2	no	—
102	#446	—	17	4	no	—
103	C₈×Q₈	—	17	2	no	—
104	#423	—	17	2	no	—
105	#423	D₁₇	1	1	yes	characteristic
106	C₄×M₁₆	—	17	2	no	—
107	C₄×M₁₆	—	17	1	no	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₁₇	#568	1	1	yes	characteristic
112	C₂²×Q₈	—	17	1	no	—
113	C₂×M₁₆	—	17	1	no	—
114	C₂×M₁₆	—	17	1	no	—
115	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
116	C₄×Q₈	—	17	1	no	—
117	C₄×Q₈	—	17	2	no	—
118	C₄×Q₈	—	17	1	no	—
119	C₄×Q₈	—	17	1	no	—
120	C₄×Q₈	—	17	2	no	—
121	C₄×Q₈	—	17	1	no	—
122	C₂×C₄⋊C₄	—	17	2	no	—
123	C₂×C₄⋊C₄	—	17	1	no	—
124	C₂×C₄²	—	17	2	no	—
125	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
126	C₄⋊C₈	—	17	4	no	—
127	C₄⋊C₈	—	17	4	no	—
128	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
129	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
130	#97	—	17	4	no	—
131	#97	—	17	2	no	—
132	C₄×C₈	—	17	4	no	—
133	C₄×C₈	—	17	2	no	—
134	C₁₇	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	17	1	no	—
136	C₂×Q₈	—	17	2	no	—
137	C₂×Q₈	—	17	1	no	—
138	C₂×Q₈	—	34	4	no	—
139	C₂²×C₄	—	17	2	no	—
140	C₂²×C₄	—	17	2	no	—
141	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	17	4	no	—
146	M₁₆	—	34	2	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	C₂²×D₁₇	1	4	yes	—
150	C₄⋊C₄	—	17	2	no	—
151	C₄⋊C₄	—	17	2	no	—
152	C₄⋊C₄	—	17	2	no	—
153	C₄⋊C₄	—	17	2	no	—
154	C₄⋊C₄	—	17	2	no	—
155	C₄⋊C₄	—	17	1	no	—
156	C₄⋊C₄	—	17	1	no	—
157	C₄²	—	17	2	no	—
158	C₄²	—	17	2	no	—
159	C₄²	—	17	2	no	—
160	C₄²	—	17	2	no	—
161	C₄²	C₄×D₁₇	1	1	yes	characteristic
162	C₄²	C₄×D₁₇	1	1	yes	characteristic
163	C₄²	C₂²×D₁₇	1	1	yes	characteristic
164	C₄²	C₂²×D₁₇	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
166	Q₈	—	17	4	no	—
167	Q₈	—	34	4	no	—
168	Q₈	—	34	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	17	2	no	—
172	C₂×C₄	—	17	2	no	—
173	C₂×C₄	—	17	2	no	—
174	C₂×C₄	—	17	2	no	—
175	C₂×C₄	—	34	4	no	—
176	C₂×C₄	—	68	2	no	—
177	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
182	C₂×C₄	#63254	1	2	yes	—
183	C₂×C₄	Q₈×D₁₇	1	2	yes	—
184	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	17	4	no	—
187	C₈	—	34	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#10587573	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	17	4	no	—
196	C₄	—	34	2	no	—
197	C₄	—	34	2	no	—
198	C₄	—	34	2	no	—
199	C₄	#10587573	1	2	yes	—
200	C₄	M₁₆×D₁₇	1	2	yes	—
201	C₄	C₂×#63254	1	1	yes	characteristic
202	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#10587573	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
206	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#423	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#423	D₁₇
hypocenter	C₁₇	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #11762938	C₂	semidirect
3. #11762936	C₂	semidirect
4. #11762933	C₂	semidirect
58. #63223	Q₈	semidirect
80. Dic₃₄	M₁₆	semidirect
83. #3622	C₂×Q₈	semidirect
134. C₁₇	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₄	2
#11762938	2
#11762936	2
#11762933	2
#11762832	2
C₄×#63223	2
M₁₆×Dic₁₇	2
C₁₇×#423	2
Q₈×M₁₆	17