Subgroups — #14537882

Order 2176.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763363	C₂	1	1	yes	maximal, characteristic
2	#11762937	C₂	1	2	yes	maximal, complemented
3	#11762936	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762933	C₂	1	2	yes	maximal, complemented
7	#11762871	C₂	1	2	yes	maximal
8	#11762869	C₂	1	2	yes	maximal
9	#11762861	C₂	1	1	yes	maximal, characteristic
10	#11762860	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587632	C₄	1	1	yes	characteristic
14	#10587630	C₄	1	1	yes	characteristic
15	#10587583	C₄	1	1	yes	characteristic
16	#10587582	C₄	1	2	yes	—
17	#10587575	C₄	1	2	yes	—
18	#10587574	C₄	1	1	yes	characteristic
19	#10587558	C₂²	1	2	yes	—
20	C₄²×D₁₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₄	C₂²	1	2	yes	—
22	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
25	C₁₇×#97	C₂²	1	2	yes	—
26	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	#10587491	C₂²	1	1	yes	characteristic
28	#10587491	C₂²	1	1	yes	characteristic
29	#10587491	C₂²	1	2	yes	—
30	#10587486	C₂²	1	2	yes	—
31	#10587485	C₂²	1	1	yes	characteristic
32	#10587485	C₂²	1	1	yes	characteristic
33	#10587485	C₂²	1	2	yes	—
34	#10587485	C₂²	1	2	yes	—
35	C₈×Dic₁₇	C₂²	1	2	yes	—
36	C₈×Dic₁₇	C₂²	1	2	yes	—
37	C₄⋊#3622	C₂²	1	1	yes	characteristic
38	C₄⋊#3622	C₂²	1	2	yes	—
39	C₄×#3622	C₂²	1	1	yes	characteristic
40	C₄×#3622	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂³	1	2	yes	—
49	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
52	C₄²×C₁₇	C₂³	1	1	yes	characteristic
53	C₄²×C₁₇	C₂³	1	2	yes	—
54	#63227	C₂×C₄	1	2	yes	—
55	#63227	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
58	#63225	C₂×C₄	1	2	yes	—
59	#63225	C₂×C₄	1	2	yes	—
60	#63225	C₂×C₄	1	2	yes	—
61	#63225	C₂×C₄	1	2	yes	—
62	#63225	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂³	1	2	yes	—
67	C₂×#3622	C₂³	1	1	yes	characteristic
68	C₂×#3622	C₂³	1	2	yes	—
69	C₂×#3622	C₂³	1	1	yes	characteristic
70	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₇	—	2	2	no	—
83	C₄×D₁₇	—	2	1	no	—
84	C₄×D₁₇	—	2	1	no	—
85	C₄×D₁₇	—	2	2	no	—
86	Dic₃₄	—	2	2	no	—
87	Dic₃₄	—	2	4	no	—
88	C₈×C₁₇	—	2	1	no	—
89	C₈×C₁₇	—	2	1	no	—
90	C₈×C₁₇	—	2	2	no	—
91	#3622	—	2	2	no	—
92	#3622	—	2	1	no	—
93	#3622	—	2	1	no	—
94	#2986	—	17	1	no	maximal, Sylow
95	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₇	—	2	1	no	—
97	C₂×D₁₇	—	2	1	no	—
98	C₂×D₁₇	—	2	1	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	2	no	—
102	C₄×C₁₇	#131	1	2	yes	—
103	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	—	2	2	no	—
107	Dic₁₇	—	2	2	no	—
108	Dic₁₇	—	2	1	no	—
109	Dic₁₇	#131	1	2	yes	—
110	#533	—	17	1	no	—
111	#446	—	17	1	no	—
112	#446	—	17	2	no	—
113	#446	D₁₇	1	1	yes	characteristic, complemented
114	C₈×Q₈	—	17	1	no	—
115	C₈×Q₈	—	17	2	no	—
116	C₈×Q₈	—	17	1	no	—
117	#433	—	17	2	no	—
118	#433	—	17	2	no	—
119	#431	—	17	1	no	—
120	#431	—	17	1	no	—
121	C₂×C₁₇	#519	1	1	yes	characteristic
122	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
123	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
124	D₁₇	—	4	1	no	—
125	C₄⋊Q₈	—	17	1	no	—
126	C₄⋊Q₈	—	17	1	no	—
127	#125	—	17	2	no	—
128	C₂²⋊Q₈	—	17	1	no	—
129	C₂²⋊Q₈	—	17	2	no	—
130	C₂²⋊Q₈	—	17	1	no	—
131	C₄×Q₈	—	17	2	no	—
132	C₄×Q₈	—	17	1	no	—
133	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
134	#117	—	17	2	no	—
135	C₂×C₄²	—	17	1	no	—
136	C₄⋊C₈	—	17	1	no	—
137	C₄⋊C₈	—	17	1	no	—
138	C₄⋊C₈	—	17	2	no	—
139	C₄⋊C₈	—	17	2	no	—
140	C₄⋊C₈	—	17	1	no	—
141	C₄⋊C₈	—	17	2	no	—
142	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
143	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
144	C₂²⋊C₈	—	17	2	no	—
145	C₂²⋊C₈	—	17	1	no	—
146	C₂²⋊C₈	—	17	1	no	—
147	#97	—	17	2	no	—
148	#97	C₂×D₁₇	1	2	yes	—
149	C₄×C₈	—	17	2	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	1	no	—
152	C₄×C₈	—	17	2	no	—
153	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
154	C₁₇	#2986	1	1	yes	characteristic, complemented, Sylow
155	C₂×Q₈	—	17	2	no	—
156	C₂×Q₈	—	17	1	no	—
157	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂×C₈	—	17	2	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	1	no	—
164	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₇	1	2	yes	—
166	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
167	C₄⋊C₄	—	17	2	no	—
168	C₄⋊C₄	—	17	2	no	—
169	C₄⋊C₄	—	17	2	no	—
170	C₄⋊C₄	—	17	2	no	—
171	C₄⋊C₄	—	17	2	no	—
172	C₄⋊C₄	—	17	2	no	—
173	C₄⋊C₄	—	17	1	no	—
174	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
175	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
176	C₂²⋊C₄	—	17	2	no	—
177	C₂²⋊C₄	—	17	2	no	—
178	C₄²	—	17	1	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	2	no	—
182	C₄²	C₂²×D₁₇	1	1	yes	characteristic
183	C₄²	C₂²×D₁₇	1	2	yes	—
184	C₂³	—	17	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	34	4	no	—
187	Q₈	—	34	2	no	—
188	C₂×C₄	—	17	1	no	—
189	C₂×C₄	—	17	1	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	2	no	—
192	C₂×C₄	—	17	2	no	—
193	C₂×C₄	—	34	2	no	—
194	C₂×C₄	—	34	1	no	—
195	C₂×C₄	—	34	1	no	—
196	C₂×C₄	—	34	2	no	—
197	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	2	no	—
205	C₈	—	34	2	no	—
206	C₈	—	34	1	no	—
207	C₈	—	34	1	no	—
208	C₂²	—	34	1	no	—
209	C₂²	—	34	1	no	—
210	C₂²	—	34	1	no	—
211	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	17	2	no	—
216	C₄	—	34	1	no	—
217	C₄	—	34	2	no	—
218	C₄	—	34	1	no	—
219	C₄	—	34	2	no	—
220	C₄	#10587593	1	2	yes	—
221	C₄	#10587694	1	1	yes	characteristic
222	C₄	#10587697	1	1	yes	characteristic
223	C₂	—	68	1	no	—
224	C₂	#11763285	1	1	yes	characteristic
225	C₂	#131×D₁₇	1	1	yes	characteristic
226	C₂	C₂×#10587593	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2986
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11762937	C₂	semidirect
3. #11762936	C₂	semidirect
4. Q₈×#3622	C₂	semidirect
5. C₈×Dic₃₄	C₂	semidirect
6. #11762933	C₂	semidirect
11. C₁₇×#446	C₂	semidirect
113. #446	D₁₇	semidirect
154. C₁₇	#2986	semidirect

Maximal subgroups

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

subgroup	index
#11763363	2
#11762937	2
#11762936	2
Q₈×#3622	2
C₈×Dic₃₄	2
#11762933	2
#11762871	2
#11762869	2
#11762861	2
#11762860	2
C₁₇×#446	2
#2986	17