Subgroups — Q₈×#10587493 #17053214

Order 4352.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 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	#14528955	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	6	yes	—
14	C₂×#10587493	C₂²	1	1	yes	characteristic
15	#11762833	C₂²	1	3	yes	—
16	C₂×C₄⋊#3622	C₂²	1	3	yes	—
17	C₄×#63223	C₂²	1	3	yes	—
18	C₂×C₄×#3622	C₂²	1	3	yes	—
19	#11762573	C₂²	1	3	yes	—
20	#11762573	C₂²	1	3	yes	—
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	3	yes	—
38	C₂²×#3622	C₂³	1	1	yes	characteristic
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	—	4	6	no	—
45	#10587493	Q₈	1	4	yes	direct product factor
46	C₄⋊#3622	—	2	3	no	—
47	C₄⋊#3622	—	2	3	no	—
48	C₄⋊#3622	C₂×C₄	1	6	yes	—
49	C₄⋊#3622	C₂×C₄	1	6	yes	—
50	#10587474	—	2	3	no	—
51	C₄×#3622	—	2	3	no	—
52	C₄×#3622	C₂×C₄	1	6	yes	—
53	C₄×#3622	C₂×C₄	1	6	yes	—
54	C₂×Q₈×C₁₇	—	2	4	no	—
55	C₂×Q₈×C₁₇	C₄⋊C₄	1	3	yes	—
56	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₁₇	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₁₇	—	2	1	no	—
61	M₁₆×C₁₇	—	2	3	no	—
62	M₁₆×C₁₇	—	2	3	no	—
63	M₁₆×C₁₇	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₁₇	—	2	1	no	—
65	C₂×C₈×C₁₇	—	2	3	no	—
66	C₄⋊C₄×C₁₇	C₄⋊C₄	1	6	yes	—
67	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
68	C₄⋊C₄×C₁₇	C₄⋊C₄	1	3	yes	—
69	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
70	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
71	C₄²×C₁₇	C₂×D₄	1	3	yes	—
72	C₄²×C₁₇	C₂×D₄	1	3	yes	—
73	#63223	—	2	1	no	—
74	#63223	—	2	3	no	—
75	#63223	—	2	3	no	—
76	#63223	C₂×Q₈	1	2	yes	—
77	C₂×#3622	—	2	1	no	—
78	C₂×#3622	—	2	3	no	—
79	C₂×#3622	—	2	3	no	—
80	C₂×#3622	—	2	3	no	—
81	C₂×#3622	C₂²×C₄	1	6	yes	—
82	C₂×#3622	C₂²×C₄	1	2	yes	—
83	C₂×#3622	C₂×Q₈	1	2	yes	—
84	C₂×#3622	#41	1	2	yes	—
85	Q₈×#108	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂²×Q₈	1	1	yes	characteristic
87	Q₈×C₁₇	—	2	6	no	—
88	Q₈×C₁₇	#108	1	4	yes	complemented
89	C₂×C₄×C₁₇	—	2	6	no	—
90	C₂×C₄×C₁₇	—	4	3	no	—
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	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
101	C₈×C₁₇	—	2	2	no	—
102	C₈×C₁₇	—	4	3	no	—
103	#3622	—	2	6	no	—
104	#3622	—	2	2	no	—
105	#3622	—	4	3	no	—
106	#3622	C₄×Q₈	1	4	yes	—
107	Q₈×M₁₆	—	17	1	no	—
108	Q₈×M₁₆	D₁₇	1	1	yes	characteristic
109	C₂×C₈×Q₈	—	17	1	no	—
110	#1950	—	17	3	no	—
111	#1950	—	17	3	no	—
112	#1848	—	17	3	no	—
113	C₄×#108	—	17	3	no	—
114	C₂²×C₁₇	—	2	1	no	—
115	C₂²×C₁₇	—	2	1	no	—
116	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₇	#557	1	1	yes	characteristic
118	C₂²×C₁₇	Q₈²	1	1	yes	characteristic
119	C₄×C₁₇	—	2	3	no	—
120	C₄×C₁₇	—	2	3	no	—
121	C₄×C₁₇	—	2	3	no	—
122	C₄×C₁₇	C₂×#108	1	6	yes	—
123	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₄×C₁₇	#520	1	1	yes	characteristic
125	C₄×C₁₇	D₄×Q₈	1	1	yes	characteristic
126	C₄×C₁₇	#550	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
128	#446	—	2	3	no	—
129	#446	—	34	3	no	—
130	C₈×Q₈	—	2	1	no	—
131	C₈×Q₈	—	17	6	no	—
132	C₈×Q₈	—	17	2	no	—
133	C₈×Q₈	—	34	1	no	—
134	C₂×#108	—	17	1	no	—
135	C₂×#108	—	17	3	no	—
136	C₂×#108	—	17	6	no	—
137	#423	—	17	3	no	—
138	#423	C₂×D₁₇	1	3	yes	—
139	C₂×C₄⋊C₈	—	17	3	no	—
140	C₄×M₁₆	—	17	3	no	—
141	C₄×M₁₆	C₂×D₁₇	1	3	yes	—
142	C₂×C₄×C₈	—	17	3	no	—
143	#341	—	17	3	no	—
144	#341	—	17	3	no	—
145	C₂×C₁₇	—	2	1	no	—
146	C₂×C₁₇	—	2	1	no	—
147	C₂×C₁₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₁₇	C₂²×#108	1	1	yes	characteristic
149	C₂×C₁₇	#2916	1	1	yes	characteristic
150	C₂²×Q₈	Dic₃₄	1	1	yes	characteristic
151	C₂×M₁₆	—	17	1	no	—
152	C₂×M₁₆	—	17	3	no	—
153	C₂×M₁₆	C₂²×D₁₇	1	3	yes	—
154	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
155	C₂²×C₈	—	17	1	no	—
156	C₂²×C₈	—	17	3	no	—
157	C₄×Q₈	C₄×D₁₇	1	3	yes	—
158	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
159	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₁₇	1	3	yes	—
161	C₂×C₄⋊C₄	Dic₃₄	1	3	yes	—
162	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
163	#108	—	17	4	no	—
164	#108	—	34	6	no	—
165	#108	—	68	6	no	—
166	C₄⋊C₈	—	2	3	no	—
167	C₄⋊C₈	—	2	3	no	—
168	C₄⋊C₈	—	17	6	no	—
169	C₄⋊C₈	—	17	6	no	—
170	C₄⋊C₈	—	34	3	no	—
171	C₄⋊C₈	—	34	3	no	—
172	#97	—	2	3	no	—
173	#97	—	34	3	no	—
174	C₄×C₈	—	2	3	no	—
175	C₄×C₈	—	17	6	no	—
176	C₄×C₈	—	17	6	no	—
177	C₄×C₈	—	34	3	no	—
178	C₁₇	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	#63225	1	3	yes	—
181	C₂×Q₈	#63225	1	1	yes	characteristic
182	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
183	C₂²×C₄	C₂×Dic₃₄	1	3	yes	—
184	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
185	M₁₆	—	2	1	no	—
186	M₁₆	—	2	3	no	—
187	M₁₆	—	2	3	no	—
188	M₁₆	—	17	2	no	—
189	M₁₆	—	34	1	no	—
190	M₁₆	—	34	3	no	—
191	M₁₆	—	34	3	no	—
192	M₁₆	Q₈×D₁₇	1	2	yes	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	3	no	—
195	C₂×C₈	—	17	2	no	—
196	C₂×C₈	—	17	2	no	—
197	C₂×C₈	—	17	2	no	—
198	C₂×C₈	—	17	6	no	—
199	C₂×C₈	—	34	3	no	—
200	C₂×C₈	—	34	3	no	—
201	C₂×C₈	—	34	1	no	—
202	C₂×C₈	—	34	3	no	—
203	C₄⋊C₄	#63225	1	6	yes	—
204	C₄⋊C₄	#63225	1	3	yes	—
205	C₄⋊C₄	#63225	1	3	yes	—
206	C₄²	C₂×C₄×D₁₇	1	3	yes	—
207	C₄²	C₂×C₄×D₁₇	1	3	yes	—
208	C₄²	C₂×C₂²⋊D₁₇	1	3	yes	—
209	C₄²	C₂×C₂²⋊D₁₇	1	3	yes	—
210	C₂³	C₂²×Dic₃₄	1	1	yes	characteristic
211	Q₈	—	2	6	no	—
212	Q₈	#10587493	1	4	yes	direct product factor
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	4	3	no	—
215	C₂×C₄	C₂×#63225	1	3	yes	—
216	C₂×C₄	C₂×#63225	1	3	yes	—
217	C₂×C₄	C₂×#63225	1	3	yes	—
218	C₂×C₄	C₂×#63225	1	3	yes	—
219	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
220	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
221	C₂×C₄	#10587694	1	1	yes	characteristic
222	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
223	C₂×C₄	#10587696	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	4	3	no	—
227	C₈	—	17	4	no	—
228	C₈	—	34	2	no	—
229	C₈	—	34	6	no	—
230	C₈	—	68	3	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	C₂²×#63225	1	1	yes	characteristic
234	C₂²	#11763535	1	1	yes	characteristic
235	C₂²	Q₈×Dic₃₄	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	3	no	—
238	C₄	—	2	3	no	—
239	C₄	C₂×#10587493	1	6	yes	—
240	C₄	C₄×Q₈×D₁₇	1	1	yes	characteristic
241	C₄	#11763291	1	1	yes	characteristic
242	C₄	Q₈×C₂²⋊D₁₇	1	1	yes	characteristic
243	C₄	#11763478	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	Q₈×#63225	1	1	yes	characteristic
247	C₂	C₂²×#10587493	1	1	yes	characteristic
248	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	Q₈×#108
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
45. #10587493	Q₈	direct (N × H)
88. Q₈×C₁₇	#108	semidirect
178. C₁₇	Q₈×#108	semidirect
212. Q₈	#10587493	direct (N × H)

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
#14528955	2
C₄×#10587493	2
Q₈×M₁₆×C₁₇	2
Q₈×#108	17

Subgroups — Q8×#10587493 #17053214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587493 #17053214