Subgroups — #17053326

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#63223	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
3	#14529559	C₂	1	3	yes	maximal
4	#14529557	C₂	1	3	yes	maximal
5	#14528921	C₂	1	3	yes	maximal
6	C₄×#10587493	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
8	#11762940	—	2	3	no	—
9	Q₈×#3622	—	2	1	no	—
10	Q₈×#3622	C₄	1	6	yes	—
11	Q₈×#3622	C₄	1	2	yes	—
12	C₂×#10587493	C₂²	1	3	yes	—
13	C₂×#10587493	C₂²	1	3	yes	—
14	#11762833	C₂²	1	3	yes	—
15	C₂×C₄⋊#3622	C₂²	1	3	yes	—
16	C₄×#63223	C₂²	1	3	yes	—
17	C₂×C₄×#3622	C₂²	1	3	yes	—
18	#11762573	C₂²	1	6	yes	—
19	#11762573	C₂²	1	3	yes	—
20	#11762573	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#446	—	2	3	no	—
23	C₈×Q₈×C₁₇	—	2	1	no	—
24	C₁₇×#423	C₂²	1	3	yes	—
25	C₄×M₁₆×C₁₇	C₂²	1	3	yes	—
26	C₂²×Q₈×C₁₇	Q₈	1	1	yes	characteristic
27	C₂×M₁₆×C₁₇	C₂³	1	3	yes	—
28	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₇	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	D₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₁₇	Q₈	1	3	yes	—
34	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
35	C₂×#63223	C₂³	1	3	yes	—
36	C₂×#63223	C₂³	1	1	yes	characteristic
37	C₂²×#3622	C₂³	1	1	yes	characteristic
38	C₂²×#3622	C₂³	1	3	yes	—
39	C₁₇×C₄⋊C₈	—	2	3	no	—
40	C₁₇×C₄⋊C₈	—	2	3	no	—
41	C₁₇×#97	—	2	3	no	—
42	C₄×C₈×C₁₇	—	2	3	no	—
43	#10587493	—	2	6	no	—
44	#10587493	—	2	6	no	—
45	C₄⋊#3622	—	2	3	no	—
46	C₄⋊#3622	—	2	3	no	—
47	C₄⋊#3622	C₂×C₄	1	6	yes	—
48	C₄⋊#3622	C₂×C₄	1	6	yes	—
49	#10587474	—	2	3	no	—
50	C₄×#3622	—	2	3	no	—
51	C₄×#3622	C₂×C₄	1	6	yes	—
52	C₄×#3622	C₂×C₄	1	6	yes	—
53	C₂×Q₈×C₁₇	—	2	4	no	—
54	C₂×Q₈×C₁₇	C₄⋊C₄	1	3	yes	—
55	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
57	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₁₇	—	2	1	no	—
60	M₁₆×C₁₇	—	2	3	no	—
61	M₁₆×C₁₇	—	2	3	no	—
62	M₁₆×C₁₇	#41	1	2	yes	—
63	C₂×C₈×C₁₇	—	2	1	no	—
64	C₂×C₈×C₁₇	—	2	3	no	—
65	C₄⋊C₄×C₁₇	C₄⋊C₄	1	6	yes	—
66	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
67	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
68	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
69	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₁₇	C₂×D₄	1	3	yes	—
71	C₄²×C₁₇	C₂×D₄	1	3	yes	—
72	#63223	—	2	1	no	—
73	#63223	—	2	3	no	—
74	#63223	—	2	3	no	—
75	#63223	#41	1	2	yes	—
76	C₂×#3622	—	2	1	no	—
77	C₂×#3622	—	2	3	no	—
78	C₂×#3622	—	2	3	no	—
79	C₂×#3622	—	2	1	no	—
80	C₂×#3622	—	2	1	no	—
81	C₂×#3622	—	2	3	no	—
82	C₂×#3622	C₂²×C₄	1	2	yes	—
83	C₂×#3622	C₂²×C₄	1	6	yes	—
84	#22746	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	C₂²×Q₈	1	1	yes	characteristic
86	Q₈×C₁₇	—	2	6	no	—
87	Q₈×C₁₇	#108	1	4	yes	complemented
88	C₂×C₄×C₁₇	—	2	6	no	—
89	C₂×C₄×C₁₇	—	4	3	no	—
90	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₈×C₁₇	—	2	2	no	—
101	C₈×C₁₇	—	4	3	no	—
102	#3622	—	2	6	no	—
103	#3622	—	2	2	no	—
104	#3622	—	2	2	no	—
105	#3622	—	4	3	no	—
106	Q₈×M₁₆	—	17	1	no	—
107	Q₈×M₁₆	D₁₇	1	1	yes	characteristic
108	C₂×C₈×Q₈	—	17	1	no	—
109	#1950	—	17	3	no	—
110	#1950	—	17	3	no	—
111	#1837	—	17	3	no	—
112	C₄×#108	—	17	3	no	—
113	C₂²×C₁₇	—	2	1	no	—
114	C₂²×C₁₇	—	2	1	no	—
115	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₇	#557	1	1	yes	characteristic
117	C₂²×C₁₇	#557	1	1	yes	characteristic
118	C₄×C₁₇	—	2	3	no	—
119	C₄×C₁₇	—	2	3	no	—
120	C₄×C₁₇	—	2	3	no	—
121	C₄×C₁₇	C₂×#108	1	6	yes	—
122	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
123	C₄×C₁₇	#519	1	1	yes	characteristic
124	C₄×C₁₇	#548	1	1	yes	characteristic
125	C₄×C₁₇	#548	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
127	#446	—	2	3	no	—
128	#446	—	34	3	no	—
129	C₈×Q₈	—	2	1	no	—
130	C₈×Q₈	—	17	6	no	—
131	C₈×Q₈	—	17	2	no	—
132	C₈×Q₈	—	34	1	no	—
133	C₂×#108	—	17	3	no	—
134	C₂×#108	—	17	3	no	—
135	#423	—	17	3	no	—
136	#423	C₂×D₁₇	1	3	yes	—
137	C₂×C₄⋊C₈	—	17	3	no	—
138	C₄×M₁₆	—	17	3	no	—
139	C₄×M₁₆	C₂×D₁₇	1	3	yes	—
140	C₂×C₄×C₈	—	17	3	no	—
141	#341	—	17	3	no	—
142	#341	—	17	6	no	—
143	#341	—	17	1	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	—	2	1	no	—
146	C₂×C₁₇	#2351	1	1	yes	characteristic
147	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
148	C₂×C₁₇	#2916	1	1	yes	characteristic
149	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
150	C₂×M₁₆	—	17	1	no	—
151	C₂×M₁₆	—	17	3	no	—
152	C₂×M₁₆	C₂²×D₁₇	1	3	yes	—
153	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
154	C₂²×C₈	—	17	1	no	—
155	C₂²×C₈	—	17	3	no	—
156	C₄×Q₈	C₄×D₁₇	1	3	yes	—
157	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
158	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
159	C₄×Q₈	C₂²⋊D₁₇	1	3	yes	—
160	C₂×C₄⋊C₄	Dic₃₄	1	3	yes	—
161	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
162	#108	—	34	6	no	—
163	#108	—	34	6	no	—
164	C₄⋊C₈	—	2	3	no	—
165	C₄⋊C₈	—	2	3	no	—
166	C₄⋊C₈	—	17	6	no	—
167	C₄⋊C₈	—	17	6	no	—
168	C₄⋊C₈	—	34	3	no	—
169	C₄⋊C₈	—	34	3	no	—
170	#97	—	2	3	no	—
171	#97	—	34	3	no	—
172	C₄×C₈	—	2	3	no	—
173	C₄×C₈	—	17	6	no	—
174	C₄×C₈	—	17	6	no	—
175	C₄×C₈	—	34	3	no	—
176	C₁₇	#22746	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	#63225	1	3	yes	—
179	C₂×Q₈	#63225	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
181	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
182	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	3	no	—
185	M₁₆	—	2	3	no	—
186	M₁₆	—	17	2	no	—
187	M₁₆	—	34	1	no	—
188	M₁₆	—	34	3	no	—
189	M₁₆	—	34	3	no	—
190	M₁₆	#63256	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	3	no	—
193	C₂×C₈	—	17	2	no	—
194	C₂×C₈	—	17	6	no	—
195	C₂×C₈	—	34	3	no	—
196	C₂×C₈	—	34	1	no	—
197	C₂×C₈	—	34	3	no	—
198	C₂×C₈	—	34	1	no	—
199	C₂×C₈	—	34	1	no	—
200	C₂×C₈	—	34	3	no	—
201	C₄⋊C₄	#63225	1	6	yes	—
202	C₄⋊C₄	#63225	1	3	yes	—
203	C₄⋊C₄	#63225	1	3	yes	—
204	C₄²	C₂×C₄×D₁₇	1	3	yes	—
205	C₄²	C₂×C₄×D₁₇	1	3	yes	—
206	C₄²	C₂×C₂²⋊D₁₇	1	3	yes	—
207	C₄²	C₂×C₂²⋊D₁₇	1	3	yes	—
208	C₂³	C₂²×Dic₃₄	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	#10587493	1	4	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×#63225	1	3	yes	—
214	C₂×C₄	C₂×#63225	1	3	yes	—
215	C₂×C₄	C₂×#63225	1	3	yes	—
216	C₂×C₄	C₂×#63225	1	3	yes	—
217	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
218	C₂×C₄	C₂×#63256	1	1	yes	characteristic
219	C₂×C₄	#10587694	1	1	yes	characteristic
220	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
221	C₂×C₄	#10587697	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	3	no	—
225	C₈	—	34	2	no	—
226	C₈	—	34	6	no	—
227	C₈	—	34	2	no	—
228	C₈	—	68	3	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×#63225	1	1	yes	characteristic
232	C₂²	#11763535	1	1	yes	characteristic
233	C₂²	#11763538	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#10587493	1	6	yes	—
238	C₄	C₄×#63256	1	1	yes	characteristic
239	C₄	#11763285	1	1	yes	characteristic
240	C₄	#11763466	1	1	yes	characteristic
241	C₄	#11763470	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#14531702	1	1	yes	characteristic
245	C₂	C₂²×#10587493	1	1	yes	characteristic
246	C₂	#14537393	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	Q₈×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₇	C₂²×Q₈
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	Q₈×M₁₆	D₁₇
hypocenter	C₁₇	#22746
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
87. Q₈×C₁₇	#108	semidirect
176. C₁₇	#22746	semidirect
210. Q₈	#10587493	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#63223	2
C₂×Q₈×#3622	2
#14529559	2
#14529557	2
#14528921	2
C₄×#10587493	2
Q₈×M₁₆×C₁₇	2
#22746	17