Subgroups — #16944267

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530583	C₂	1	1	yes	maximal, characteristic
2	#14530583	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#63214	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#63214	—	2	1	no	—
6	C₄⋊#63214	C₂²	1	2	yes	—
7	C₂²⋊#63214	—	2	1	no	—
8	C₂²⋊#63214	—	2	1	no	—
9	C₄×#63214	—	2	1	no	—
10	C₄×#63214	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₁₇	—	2	1	no	—
13	C₈×Q₈×C₁₇	—	2	1	no	—
14	C₈×Q₈×C₁₇	C₄	1	2	yes	—
15	C₈×Q₈×C₁₇	C₂²	1	2	yes	—
16	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
17	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₇	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₇	—	2	1	no	—
25	C₄×Q₈×C₁₇	—	2	1	no	—
26	C₄×Q₈×C₁₇	C₈	1	2	yes	—
27	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	—	2	1	no	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₁₇×C₄⋊C₈	—	2	2	no	—
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₇	—	2	2	no	—
37	C₄×C₈×C₁₇	—	2	2	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×#63214	—	2	2	no	—
43	C₂×#63214	—	2	2	no	—
44	C₂×Q₈×C₁₇	—	2	1	no	—
45	C₂×Q₈×C₁₇	—	2	2	no	—
46	C₂×Q₈×C₁₇	C₁₆	1	4	yes	complemented
47	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₇	—	2	1	no	—
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	4	1	no	—
56	C₂×C₈×C₁₇	—	4	2	no	—
57	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	—	2	1	no	—
63	C₄⋊C₄×C₁₇	—	2	2	no	—
64	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
65	C₄²×C₁₇	—	2	2	no	—
66	C₄²×C₁₇	—	2	2	no	—
67	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
70	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	#63214	—	4	2	no	—
72	#63214	—	4	2	no	—
73	#11637	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
75	Q₈×C₁₇	—	2	4	no	—
76	Q₈×C₁₇	—	2	2	no	—
77	Q₈×C₁₇	C₂×C₁₆	1	4	yes	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	—	2	1	no	—
82	C₂×C₄×C₁₇	—	2	2	no	—
83	C₂×C₄×C₁₇	—	4	1	no	—
84	C₂×C₄×C₁₇	—	4	2	no	—
85	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
89	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
90	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₁₇	—	2	1	no	—
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	—	4	2	no	—
97	C₈×C₁₇	#137	1	2	yes	—
98	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
99	#2162	—	17	1	no	—
100	#2162	—	17	1	no	—
101	Q₈⋊C₁₆	—	17	4	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	—	2	1	no	—
104	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
105	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₁₇	—	2	4	no	—
107	C₄×C₁₇	—	2	1	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	—	4	2	no	—
111	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
112	C₄×C₁₇	#419	1	1	yes	characteristic
113	C₄×C₁₇	#421	1	1	yes	characteristic
114	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
115	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
116	C₂×C₄×Q₈	Dic₁₇	1	1	yes	characteristic
117	C₈×Q₈	—	2	1	no	—
118	C₈×Q₈	—	2	1	no	—
119	C₈×Q₈	Dic₁₇	1	2	yes	—
120	C₈×Q₈	C₂×D₁₇	1	2	yes	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	Dic₁₇	1	1	yes	characteristic
123	C₂×C₄×C₈	—	2	1	no	—
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	C₄⋊C₁₆	—	17	2	no	—
126	C₄⋊C₁₆	—	34	1	no	—
127	C₂²⋊C₁₆	—	34	1	no	—
128	C₂²⋊C₁₆	—	34	1	no	—
129	C₄×C₁₆	—	17	2	no	—
130	C₄×C₁₆	—	34	1	no	—
131	C₂×C₁₇	—	2	2	no	—
132	C₂×C₁₇	#1479	1	1	yes	characteristic
133	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
134	C₂×C₁₇	#2117	1	1	yes	characteristic
135	C₂²×Q₈	#3622	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	#3622	1	2	yes	—
142	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	#3622	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
147	C₂×C₁₆	—	34	2	no	—
148	C₂×C₁₆	—	34	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
159	C₁₇	#11637	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	#63214	1	4	yes	complemented
163	C₂×Q₈	C₂×#3622	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
167	C₂²×C₄	#63223	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂×#3622	1	2	yes	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×#3622	1	1	yes	characteristic
184	C₄²	#63223	1	1	yes	characteristic
185	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
186	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
187	C₁₆	—	68	2	no	—
188	C₁₆	—	68	2	no	—
189	C₂³	C₂²⋊#3622	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	2	4	no	—
192	Q₈	C₂×#63214	1	4	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂×#63214	1	2	yes	—
201	C₂×C₄	#10587483	1	2	yes	—
202	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
205	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
206	C₂×C₄	C₂×#63223	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	#10587628	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂²⋊#63214	1	2	yes	—
216	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂²⋊#63214	1	2	yes	—
223	C₄	#11762818	1	1	yes	characteristic
224	C₄	#11762826	1	1	yes	characteristic
225	C₄	C₂²×#63214	1	1	yes	characteristic
226	C₄	C₂×#10587483	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	#14527179	1	1	yes	characteristic
229	C₂	C₂×C₂²⋊#63214	1	1	yes	characteristic
230	C₂	#14530412	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	#11637
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#63214	C₂	semidirect
46. C₂×Q₈×C₁₇	C₁₆	semidirect
159. C₁₇	#11637	semidirect
162. C₂×Q₈	#63214	semidirect

Maximal subgroups

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

subgroup	index
#14530583	2
Q₈⋊#63214	2
C₂×C₈×Q₈×C₁₇	2
#11637	17