Subgroups — #16944271

Order 4352.

300 conjugacy classes of subgroups.

The subgroup lattice has 218 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	C₁₇×#2960	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	C₁₇×#533	C₄	1	1	yes	characteristic
12	C₁₇×#446	—	2	1	no	—
13	C₁₇×#446	C₄	1	2	yes	—
14	C₈×Q₈×C₁₇	—	2	1	no	—
15	C₈×Q₈×C₁₇	C₂²	1	2	yes	—
16	C₁₇×#424	—	2	1	no	—
17	C₁₇×#423	C₄	1	1	yes	characteristic
18	C₁₇×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	—	2	1	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₁₇×#125	—	2	1	no	—
26	C₁₇×C₂²⋊Q₈	—	2	1	no	—
27	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
28	C₄×Q₈×C₁₇	—	2	1	no	—
29	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₁₇×#117	—	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	C₁₇×#97	—	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	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
49	M₁₆×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	—	4	1	no	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₄⋊C₄×C₁₇	—	2	2	no	—
59	C₄⋊C₄×C₁₇	—	2	1	no	—
60	C₄⋊C₄×C₁₇	—	2	1	no	—
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
63	C₂²⋊C₄×C₁₇	—	2	2	no	—
64	C₄²×C₁₇	—	2	2	no	—
65	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
68	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	#63214	—	4	2	no	—
70	#63214	—	4	2	no	—
71	#11638	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
73	Q₈×C₁₇	—	2	4	no	—
74	Q₈×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	—	2	2	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	1	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₁₇	—	2	1	no	—
88	C₈×C₁₇	—	2	1	no	—
89	C₈×C₁₇	—	2	1	no	—
90	C₈×C₁₇	—	4	2	no	—
91	C₈×C₁₇	#137	1	2	yes	—
92	#2960	D₁₇	1	1	yes	characteristic
93	#2162	—	17	1	no	—
94	#2162	—	17	1	no	—
95	Q₈⋊C₁₆	—	17	4	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	—	2	1	no	—
98	C₂²×C₁₇	—	2	1	no	—
99	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	—	4	2	no	—
104	C₄×C₁₇	—	4	2	no	—
105	C₄×C₁₇	#350	1	2	yes	—
106	C₄×C₁₇	#419	1	1	yes	characteristic
107	C₄×C₁₇	#421	1	1	yes	characteristic
108	C₄×C₁₇	#504	1	2	yes	—
109	#533	Dic₁₇	1	1	yes	characteristic
110	#446	—	2	1	no	—
111	#446	Dic₁₇	1	2	yes	—
112	C₈×Q₈	—	2	1	no	—
113	C₈×Q₈	C₂×D₁₇	1	2	yes	—
114	#424	—	2	1	no	—
115	#423	Dic₁₇	1	1	yes	characteristic
116	#405	—	2	1	no	—
117	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
118	C₄⋊C₁₆	—	17	2	no	—
119	C₄⋊C₁₆	—	34	1	no	—
120	C₂²⋊C₁₆	—	34	1	no	—
121	C₂²⋊C₁₆	—	34	1	no	—
122	C₄×C₁₆	—	17	2	no	—
123	C₄×C₁₆	—	34	1	no	—
124	C₂×C₁₇	—	4	1	no	—
125	C₂×C₁₇	#1479	1	1	yes	characteristic
126	C₂×C₁₇	#2113	1	1	yes	characteristic
127	C₂×C₁₇	C₂×#350	1	1	yes	characteristic
128	C₂×M₁₆	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
131	C₄⋊Q₈	#3622	1	1	yes	characteristic
132	C₄⋊Q₈	#3622	1	1	yes	characteristic
133	#125	—	2	1	no	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	#3622	1	2	yes	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
138	#117	—	2	1	no	—
139	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
140	C₂×C₁₆	—	34	2	no	—
141	C₂×C₁₆	—	34	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
146	#97	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
151	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
152	C₁₇	#11638	1	1	yes	characteristic, complemented, Sylow
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂×#3622	1	2	yes	—
155	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
156	C₂²×C₄	#63223	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂×#3622	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×#3622	1	1	yes	characteristic
175	C₄²	#63223	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
177	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
178	C₁₆	—	68	2	no	—
179	C₁₆	—	68	2	no	—
180	C₂³	C₂²⋊#3622	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	4	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
191	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
192	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
193	C₂×C₄	C₂×#63223	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	2	no	—
199	C₈	#10587628	1	2	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	#11762593	1	2	yes	—
210	C₄	#11762818	1	1	yes	characteristic
211	C₄	#11762826	1	1	yes	characteristic
212	C₄	#11763226	1	2	yes	—
213	C₂	—	4	1	no	—
214	C₂	#14527179	1	1	yes	characteristic
215	C₂	#14530398	1	1	yes	characteristic
216	C₂	C₂×#11762593	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#11638
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#63214	C₂	semidirect
152. C₁₇	#11638	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
C₁₇×#2960	2
#11638	17