Subgroups — #17982656

Order 4352.

350 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	C₂×C₄⋊#63241	C₂	1	2	yes	maximal
2	C₂×C₄⋊#63241	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#63241	C₂	1	1	yes	maximal, characteristic
4	#14545854	C₂	1	2	yes	maximal
5	C₄⋊C₄×#3624	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63241	C₂²	1	2	yes	—
7	C₂²×#63241	C₂²	1	1	yes	characteristic
8	C₂²×#63241	C₂²	1	1	yes	characteristic
9	C₄⋊#63241	—	2	2	no	—
10	C₄⋊#63241	—	2	4	no	—
11	C₄×#63241	—	2	2	no	—
12	C₂×C₄⋊#3624	C₄	1	1	yes	characteristic
13	C₂×C₄×#3624	C₂²	1	2	yes	—
14	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
15	#11763675	C₄	1	2	yes	—
16	C₄⋊C₄×Dic₁₇	C₄	1	1	yes	characteristic
17	C₂²×#3624	C₂×C₄	1	2	yes	—
18	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
19	C₂²×#3624	C₂³	1	1	yes	characteristic
20	C₂×#63241	—	2	1	no	—
21	C₂×#63241	—	2	1	no	—
22	C₂×#63241	—	2	1	no	—
23	C₂×#63241	—	2	1	no	—
24	C₂×#63241	—	2	2	no	—
25	C₂×#63241	—	2	2	no	—
26	C₂×#63241	—	2	2	no	—
27	C₂×#63241	D₄	1	2	yes	—
28	C₂×#63241	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊#3624	—	2	2	no	—
34	C₄×#3624	—	2	2	no	—
35	C₄×#3624	—	2	2	no	—
36	C₄×#3624	D₄	1	2	yes	—
37	C₄×#3624	Q₈	1	2	yes	—
38	#10587502	C₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
42	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×#3624	—	2	1	no	—
45	C₂×#3624	—	2	1	no	—
46	C₂×#3624	—	2	2	no	—
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	2	2	no	—
49	C₂×#3624	C₄⋊C₄	1	2	yes	—
50	C₂×#3624	C₂×D₄	1	1	yes	characteristic
51	C₂×#3624	C₂×D₄	1	1	yes	characteristic
52	C₂×#3624	C₂×Q₈	1	1	yes	characteristic
53	C₂×#3624	C₂×Q₈	1	1	yes	characteristic
54	C₂×#3624	#41	1	1	yes	characteristic
55	C₂×#3624	#41	1	1	yes	characteristic
56	#63241	—	2	2	no	—
57	#63241	—	2	2	no	—
58	#63241	—	4	2	no	—
59	C₄⋊C₄×C₁₇	C₁₆	1	4	yes	complemented
60	C₄⋊Dic₁₇	—	2	2	no	—
61	C₄×Dic₁₇	—	2	2	no	—
62	C₄×Dic₁₇	—	2	2	no	—
63	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₁₇	M₁₆	1	2	yes	—
65	#13159	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
71	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
73	C₂×Dic₁₇	—	2	1	no	—
74	C₂×Dic₁₇	—	2	1	no	—
75	C₂×Dic₁₇	—	2	2	no	—
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	2	no	—
78	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
79	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₁₇	#131	1	1	yes	characteristic
84	C₂×Dic₁₇	#131	1	1	yes	characteristic
85	#3624	—	2	2	no	—
86	#3624	—	4	2	no	—
87	#3624	#121	1	1	yes	characteristic
88	#3624	C₂²⋊Q₈	1	1	yes	characteristic
89	#3624	#125	1	1	yes	characteristic
90	#3624	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	—	17	1	no	—
92	C₂×C₄⋊C₁₆	—	17	2	no	—
93	C₂×C₄×C₁₆	—	17	1	no	—
94	C₈×C₄⋊C₄	—	17	1	no	—
95	#1374	—	17	2	no	—
96	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
99	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
100	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
101	C₂²×C₁₇	#504	1	1	yes	characteristic
102	C₂²×C₁₇	#504	1	1	yes	characteristic
103	C₄×C₁₇	—	2	4	no	—
104	C₄×C₁₇	C₄⋊C₁₆	1	4	yes	complemented
105	Dic₁₇	—	2	2	no	—
106	Dic₁₇	—	4	2	no	—
107	Dic₁₇	#389	1	1	yes	characteristic
108	Dic₁₇	#424	1	1	yes	characteristic
109	Dic₁₇	#436	1	1	yes	characteristic
110	Dic₁₇	#446	1	1	yes	characteristic
111	C₂²×C₁₆	—	17	1	no	—
112	C₂²×C₁₆	—	17	1	no	—
113	C₂²×C₁₆	—	17	2	no	—
114	C₂×C₄⋊C₈	—	17	1	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	2	no	—
117	C₄×C₄⋊C₄	—	17	1	no	—
118	C₄⋊C₁₆	—	34	2	no	—
119	C₄⋊C₁₆	—	34	4	no	—
120	C₄×C₁₆	—	34	2	no	—
121	#336	—	17	2	no	—
122	C₂×C₁₇	#1939	1	1	yes	characteristic
123	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
124	C₂×C₁₇	#2151	1	1	yes	characteristic
125	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
126	C₂×C₁₇	#2169	1	1	yes	characteristic
127	C₂×C₁₇	Q₈×C₁₆	1	1	yes	characteristic
128	C₂×C₁₇	#2183	1	1	yes	characteristic
129	C₂²×C₈	—	17	1	no	—
130	C₂²×C₈	—	17	1	no	—
131	C₂²×C₈	—	17	2	no	—
132	C₂×C₄⋊C₄	—	17	1	no	—
133	C₂×C₄⋊C₄	#3633	1	1	yes	characteristic
134	C₂×C₄²	—	17	1	no	—
135	C₂×C₄²	—	17	2	no	—
136	C₂×C₁₆	—	17	2	no	—
137	C₂×C₁₆	—	17	2	no	—
138	C₂×C₁₆	—	34	1	no	—
139	C₂×C₁₆	—	34	1	no	—
140	C₂×C₁₆	—	34	1	no	—
141	C₂×C₁₆	—	34	1	no	—
142	C₂×C₁₆	—	34	2	no	—
143	C₂×C₁₆	—	34	2	no	—
144	C₂×C₁₆	—	34	2	no	—
145	C₄⋊C₈	—	34	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₄×C₈	—	34	2	no	—
150	#95	—	17	2	no	—
151	C₁₇	#13159	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₄	—	17	1	no	—
153	C₂²×C₄	—	17	1	no	—
154	C₂²×C₄	—	17	2	no	—
155	C₂²×C₄	C₂×#3633	1	2	yes	—
156	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
157	C₂×C₈	—	17	1	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	1	no	—
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	34	1	no	—
165	C₂×C₈	—	34	1	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	—	34	2	no	—
168	C₂×C₈	—	34	2	no	—
169	C₄⋊C₄	—	34	2	no	—
170	C₄⋊C₄	#63241	1	4	yes	complemented
171	C₄²	—	17	2	no	—
172	C₄²	—	17	2	no	—
173	C₄²	—	34	2	no	—
174	C₄²	—	34	2	no	—
175	C₁₆	—	34	2	no	—
176	C₁₆	—	34	2	no	—
177	C₁₆	—	68	2	no	—
178	C₂³	C₂²×#3633	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	17	1	no	—
183	C₂×C₄	—	17	1	no	—
184	C₂×C₄	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	17	2	no	—
188	C₂×C₄	—	34	1	no	—
189	C₂×C₄	—	34	1	no	—
190	C₂×C₄	—	34	2	no	—
191	C₂×C₄	—	34	2	no	—
192	C₂×C₄	—	34	2	no	—
193	C₂×C₄	#10587666	1	2	yes	—
194	C₂×C₄	C₄⋊#3633	1	2	yes	—
195	C₂×C₄	C₂×#63241	1	4	yes	—
196	C₂×C₄	C₂×#63241	1	2	yes	—
197	C₈	—	17	1	no	—
198	C₈	—	17	1	no	—
199	C₈	—	17	1	no	—
200	C₈	—	17	1	no	—
201	C₈	—	34	2	no	—
202	C₈	—	68	2	no	—
203	C₂²	C₂×C₄⋊#3633	1	1	yes	characteristic
204	C₂²	D₄×#3633	1	1	yes	characteristic
205	C₂²	Q₈×#3633	1	1	yes	characteristic
206	C₂²	C₂²×#63241	1	1	yes	characteristic
207	C₂²	C₂×#10587666	1	1	yes	characteristic
208	C₂²	#11763930	1	1	yes	characteristic
209	C₂²	#11763931	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	17	1	no	—
212	C₄	—	17	1	no	—
213	C₄	—	17	1	no	—
214	C₄	—	17	1	no	—
215	C₄	—	34	2	no	—
216	C₄	—	68	2	no	—
217	C₄	C₄⋊#63241	1	4	yes	complemented
218	C₂	#14545933	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#63241	1	1	yes	characteristic
220	C₂	#14545976	1	1	yes	characteristic
221	C₂	D₄×#63241	1	1	yes	characteristic
222	C₂	#14545999	1	1	yes	characteristic
223	C₂	Q₈×#63241	1	1	yes	characteristic
224	C₂	#14546009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3633
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₄⋊C₄×C₁₇	C₈
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3624	C₂³
hypercenter	C₂×C₄⋊C₄	#3633
hypocenter	C₁₇	#13159
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
59. C₄⋊C₄×C₁₇	C₁₆	semidirect
104. C₄×C₁₇	C₄⋊C₁₆	semidirect
151. C₁₇	#13159	semidirect
170. C₄⋊C₄	#63241	semidirect
217. C₄	C₄⋊#63241	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63241	2
C₂×C₄×#63241	2
#14545854	2
C₄⋊C₄×#3624	2
#13159	17