Subgroups — #14539318

Order 2176.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763437	C₂	1	1	yes	maximal, characteristic
2	#11763219	C₂	1	2	yes	maximal, complemented
3	#11763215	C₂	1	2	yes	maximal, complemented
4	#11763212	C₂	1	2	yes	maximal, complemented
5	#11763208	C₂	1	2	yes	maximal, complemented
6	#11762852	C₂	1	1	yes	maximal, characteristic
7	#11762849	C₂	1	1	yes	maximal, characteristic
8	#11762845	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587478	C₂	1	1	yes	maximal, characteristic
10	#11762765	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#544	C₂	1	1	yes	maximal, characteristic
12	C₁₇×C₄⋊Q₈	—	2	1	no	—
13	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₇×#125	—	2	1	no	—
15	C₁₇×#125	C₂²	1	2	yes	—
16	C₁₇×C₂²⋊Q₈	—	2	2	no	—
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	C₁₇×#117	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
20	#10587612	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63223	C₂²	1	1	yes	characteristic
23	C₂²×#3622	C₂²	1	1	yes	characteristic
24	#10587576	—	2	1	no	—
25	#10587574	—	2	1	no	—
26	#10587564	—	2	2	no	—
27	#10587556	C₂²	1	2	yes	—
28	#10587555	C₂²	1	2	yes	—
29	#10587479	C₂²	1	2	yes	—
30	#10587479	C₂²	1	2	yes	—
31	#10587479	C₂²	1	1	yes	characteristic
32	#10587479	C₂²	1	1	yes	characteristic
33	#10587478	C₂²	1	1	yes	characteristic
34	#10587478	C₂²	1	2	yes	—
35	#10587478	C₂²	1	1	yes	characteristic
36	#10587478	C₂²	1	2	yes	—
37	#10587478	C₂²	1	2	yes	—
38	#10587478	C₂²	1	2	yes	—
39	#10587474	C₂²	1	2	yes	—
40	C₄×#3622	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₇	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	2	1	no	—
45	C₂×Dic₃₄	—	2	2	no	—
46	C₄⋊C₄×C₁₇	—	2	2	no	—
47	C₄⋊C₄×C₁₇	—	2	2	no	—
48	C₄⋊C₄×C₁₇	—	2	2	no	—
49	C₄⋊C₄×C₁₇	—	2	1	no	—
50	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
55	C₂²⋊C₄×C₁₇	—	2	1	no	—
56	C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
57	C₄²×C₁₇	—	2	1	no	—
58	C₄²×C₁₇	C₂³	1	2	yes	—
59	C₂²⋊Dic₁₇	—	2	1	no	—
60	C₄⋊Dic₁₇	—	2	2	no	—
61	C₄⋊Dic₁₇	—	2	1	no	—
62	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₁₇	C₂³	1	2	yes	—
64	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
65	#63225	—	2	2	no	—
66	#63225	—	2	2	no	—
67	C₄×Dic₁₇	—	2	1	no	—
68	#63223	Q₈	1	4	yes	complemented
69	C₂×#3622	Q₈	1	4	yes	complemented
70	C₂×#3622	C₂³	1	1	yes	characteristic
71	C₂×#3622	C₂³	1	2	yes	—
72	C₂×#3622	C₂³	1	1	yes	characteristic
73	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₁₇	—	8	2	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	8	1	no	—
79	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₁₇	—	2	1	no	—
86	C₂×Dic₁₇	—	2	1	no	—
87	C₂×Dic₁₇	—	2	2	no	—
88	C₂×Dic₁₇	—	8	1	no	—
89	Dic₃₄	—	8	2	no	—
90	#3622	C₂×Q₈	1	2	yes	—
91	#3622	C₂×Q₈	1	2	yes	—
92	#3622	C₂×Q₈	1	4	yes	—
93	#3164	—	17	1	no	maximal, Sylow
94	C₂²×C₁₇	—	2	1	no	—
95	C₂²×C₁₇	—	2	1	no	—
96	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
97	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	4	2	no	—
101	C₄×C₁₇	—	4	1	no	—
102	C₄×C₁₇	—	4	1	no	—
103	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
104	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
106	Dic₁₇	—	4	1	no	—
107	Dic₁₇	—	4	2	no	—
108	Dic₁₇	—	4	1	no	—
109	#544	—	17	1	no	—
110	#544	D₁₇	1	1	yes	characteristic
111	#501	—	17	2	no	—
112	#501	—	17	2	no	—
113	#500	—	17	2	no	—
114	#499	—	17	2	no	—
115	#428	—	17	1	no	—
116	#428	—	17	1	no	—
117	#427	—	17	1	no	—
118	C₂×#107	—	17	1	no	—
119	#405	—	17	1	no	—
120	C₂×C₁₇	—	2	2	no	—
121	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₁₇	#576	1	1	yes	characteristic
123	C₂×C₁₇	#578	1	1	yes	characteristic
124	C₂×M₁₆	—	17	1	no	—
125	C₂²×C₈	—	17	1	no	—
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	—	17	2	no	—
128	C₄⋊Q₈	—	34	1	no	—
129	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
130	#125	—	2	1	no	—
131	#125	—	17	2	no	—
132	#125	—	34	1	no	—
133	#125	C₂×D₁₇	1	2	yes	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	34	2	no	—
136	#117	—	17	1	no	—
137	#117	C₂×D₁₇	1	1	yes	characteristic
138	#117	C₂×D₁₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	17	1	no	—
140	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
141	#107	—	17	2	no	—
142	#107	—	17	2	no	—
143	#107	—	17	1	no	—
144	#107	—	17	1	no	—
145	#107	—	17	2	no	—
146	#107	—	17	2	no	—
147	#106	—	17	1	no	—
148	#106	—	17	1	no	—
149	#106	—	17	2	no	—
150	#106	—	17	2	no	—
151	#97	—	17	2	no	—
152	C₄×C₈	—	17	2	no	—
153	C₁₇	#3164	1	1	yes	characteristic, complemented, Sylow
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	34	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	34	1	no	—
158	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
159	M₁₆	—	17	4	no	—
160	C₂×C₈	—	17	4	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	2	no	—
163	C₂×C₈	—	17	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	17	1	no	—
169	C₄⋊C₄	—	17	2	no	—
170	C₄⋊C₄	—	17	1	no	—
171	C₄⋊C₄	—	34	1	no	—
172	C₄⋊C₄	—	34	2	no	—
173	C₄⋊C₄	—	34	2	no	—
174	C₄⋊C₄	—	34	2	no	—
175	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
177	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
178	C₄⋊C₄	C₂²×D₁₇	1	2	yes	—
179	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	34	1	no	—
182	C₂²⋊C₄	C₂²⋊D₁₇	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	34	1	no	—
185	C₄²	C₂²×D₁₇	1	2	yes	—
186	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	136	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	34	1	no	—
194	C₂×C₄	—	34	1	no	—
195	C₂×C₄	—	34	2	no	—
196	C₂×C₄	—	136	1	no	—
197	C₂×C₄	Q₈×D₁₇	1	2	yes	—
198	C₂×C₄	Q₈×D₁₇	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
203	C₈	—	17	2	no	—
204	C₈	—	17	2	no	—
205	C₈	—	17	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#10587630	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	68	2	no	—
216	C₄	—	68	1	no	—
217	C₄	—	68	1	no	—
218	C₄	#10587630	1	2	yes	—
219	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#10587630	1	1	yes	characteristic
223	C₂	#11763612	1	1	yes	characteristic
224	C₂	#11763624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#544	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#544	D₁₇
hypocenter	C₁₇	#3164
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #11763219	C₂	semidirect
3. #11763215	C₂	semidirect
4. #11763212	C₂	semidirect
5. #11763208	C₂	semidirect
68. #63223	Q₈	semidirect
69. C₂×#3622	Q₈	semidirect
153. C₁₇	#3164	semidirect

Maximal subgroups

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

subgroup	index
#11763437	2
#11763219	2
#11763215	2
#11763212	2
#11763208	2
#11762852	2
#11762849	2
#11762845	2
C₂×#10587478	2
#11762765	2
C₁₇×#544	2
#3164	17