Subgroups — #17948232

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3624	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#3624	C₄	1	4	yes	—
4	C₂×C₄×#3624	—	2	2	no	—
5	#11763675	—	2	2	no	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	—	2	2	no	—
8	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×#3624	—	2	4	no	—
11	C₂²×C₈×C₁₇	—	2	1	no	—
12	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
16	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
17	C₂²×#3622	—	2	1	no	—
18	C₄⋊#3624	—	2	4	no	—
19	C₄⋊#3624	C₈	1	8	yes	complemented
20	C₄×#3624	—	4	2	no	—
21	C₄×#3624	—	4	2	no	—
22	C₄×C₈×C₁₇	—	2	1	no	—
23	C₄×C₈×C₁₇	C₈	1	2	yes	complemented
24	C₈×Dic₁₇	—	2	8	no	—
25	C₄×#3622	—	2	1	no	—
26	C₄×#3622	C₈	1	2	yes	complemented
27	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₁₇	C₂²⋊C₄	1	2	yes	—
31	C₂²×Dic₁₇	C₄⋊C₄	1	2	yes	—
32	C₂×#3624	—	2	8	no	—
33	C₂×#3624	—	4	4	no	—
34	C₂×#3624	—	4	4	no	—
35	C₂×C₈×C₁₇	—	2	4	no	—
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
40	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
41	C₄×Dic₁₇	—	2	1	no	—
42	C₄×Dic₁₇	—	2	1	no	—
43	C₄×Dic₁₇	—	2	4	no	—
44	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
45	C₄×Dic₁₇	M₁₆	1	4	yes	—
46	C₄×Dic₁₇	D₈	1	1	yes	characteristic
47	C₄×Dic₁₇	SD₁₆	1	2	yes	—
48	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
49	C₂×#3622	—	2	4	no	—
50	C₂×#3622	—	2	2	no	—
51	#7015	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
55	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#97	1	2	yes	—
57	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
66	C₂×Dic₁₇	—	2	2	no	—
67	C₂×Dic₁₇	—	2	2	no	—
68	C₂×Dic₁₇	C₂²⋊C₈	1	4	yes	—
69	C₂×Dic₁₇	#102	1	2	yes	—
70	C₂×Dic₁₇	#103	1	2	yes	—
71	C₂×Dic₁₇	#104	1	4	yes	—
72	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
73	#3624	—	4	8	no	—
74	C₈×C₁₇	—	2	4	no	—
75	#3622	—	2	4	no	—
76	C₄×C₄⋊C₈	—	17	2	no	—
77	C₄²×C₈	—	17	1	no	—
78	C₂²×C₁₇	#336	1	2	yes	—
79	C₂²×C₁₇	#336	1	1	yes	characteristic
80	C₂²×C₁₇	#339	1	2	yes	—
81	C₂²×C₁₇	#340	1	1	yes	characteristic
82	C₂²×C₁₇	#341	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	C₈²	1	1	yes	characteristic
86	C₄×C₁₇	#322	1	1	yes	characteristic
87	C₄×C₁₇	#322	1	2	yes	—
88	C₄×C₁₇	#334	1	1	yes	characteristic
89	C₄×C₁₇	#334	1	1	yes	characteristic
90	C₄×C₁₇	#335	1	1	yes	characteristic
91	C₄×C₁₇	#335	1	1	yes	characteristic
92	Dic₁₇	—	2	4	no	—
93	Dic₁₇	D₄⋊C₈	1	4	yes	—
94	Dic₁₇	Q₈⋊C₈	1	4	yes	—
95	C₂×C₄⋊C₈	—	17	4	no	—
96	C₂×C₄×C₈	—	17	1	no	—
97	C₂×C₄×C₈	—	34	2	no	—
98	C₂×C₄×C₈	—	34	2	no	—
99	C₂×C₄×C₈	#595	1	1	yes	characteristic
100	C₄³	—	17	1	no	—
101	#336	—	34	2	no	—
102	C₂×C₁₇	#1273	1	1	yes	characteristic
103	C₂×C₁₇	#1276	1	1	yes	characteristic
104	C₂×C₁₇	#1277	1	1	yes	characteristic
105	C₂×C₁₇	#1278	1	2	yes	—
106	C₂×C₁₇	#1279	1	2	yes	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	34	1	no	—
109	C₂²×C₈	—	34	4	no	—
110	C₂×C₄²	—	17	2	no	—
111	C₂×C₄²	—	17	1	no	—
112	C₂×C₄²	—	17	1	no	—
113	C₂×C₄²	—	17	2	no	—
114	C₂×C₄²	C₂×#595	1	1	yes	characteristic
115	C₄⋊C₈	—	17	8	no	—
116	C₄⋊C₈	—	34	4	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	—	17	2	no	—
119	C₄×C₈	—	34	8	no	—
120	C₄×C₈	—	34	1	no	—
121	C₄×C₈	—	68	2	no	—
122	C₄×C₈	—	68	2	no	—
123	C₄×C₈	#3624	1	2	yes	complemented
124	C₁₇	#7015	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	17	2	no	—
126	C₂²×C₄	—	17	2	no	—
127	C₂²×C₄	C₄×#595	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	34	4	no	—
133	C₂×C₈	—	34	2	no	—
134	C₂×C₈	—	34	8	no	—
135	C₂×C₈	—	68	4	no	—
136	C₂×C₈	—	68	4	no	—
137	C₄²	—	17	4	no	—
138	C₄²	—	17	4	no	—
139	C₄²	—	17	2	no	—
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₄²	—	34	4	no	—
143	C₄²	—	34	1	no	—
144	C₄²	—	34	1	no	—
145	C₄²	#63242	1	1	yes	characteristic
146	C₄²	#63243	1	1	yes	characteristic
147	C₄²	C₂×#3624	1	1	yes	characteristic
148	C₄²	#63247	1	1	yes	characteristic
149	C₂³	(C₂×#595)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	17	4	no	—
152	C₂×C₄	—	17	4	no	—
153	C₂×C₄	—	17	2	no	—
154	C₂×C₄	—	17	2	no	—
155	C₂×C₄	—	17	4	no	—
156	C₂×C₄	—	34	2	no	—
157	C₂×C₄	—	34	2	no	—
158	C₂×C₄	C₈×#595	1	2	yes	—
159	C₂×C₄	#10587532	1	2	yes	—
160	C₂×C₄	#10587533	1	1	yes	characteristic
161	C₂×C₄	#10587534	1	1	yes	characteristic
162	C₂×C₄	#10587535	1	1	yes	characteristic
163	C₂×C₄	#10587536	1	1	yes	characteristic
164	C₂×C₄	C₄×#3624	1	1	yes	characteristic
165	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
166	C₂×C₄	#10587542	1	1	yes	characteristic
167	C₂×C₄	#10587543	1	1	yes	characteristic
168	C₂×C₄	#10587544	1	1	yes	characteristic
169	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	34	4	no	—
172	C₈	—	68	8	no	—
173	C₂²	#11763674	1	2	yes	—
174	C₂²	#11763675	1	1	yes	characteristic
175	C₂²	#11763680	1	2	yes	—
176	C₂²	#11763683	1	1	yes	characteristic
177	C₂²	#11763685	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	17	4	no	—
181	C₄	—	17	4	no	—
182	C₄	—	34	4	no	—
183	C₄	C₈×#3624	1	1	yes	characteristic
184	C₄	#11763650	1	1	yes	characteristic
185	C₄	#11763651	1	2	yes	—
186	C₄	#11763670	1	1	yes	characteristic
187	C₄	#11763671	1	1	yes	characteristic
188	C₄	#11763672	1	1	yes	characteristic
189	C₄	#11763673	1	1	yes	characteristic
190	C₂	#14543606	1	1	yes	characteristic
191	C₂	#14543612	1	1	yes	characteristic
192	C₂	#14543614	1	1	yes	characteristic
193	C₂	#14543616	1	2	yes	—
194	C₂	#14543619	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄⋊#595
commutator	C₄×C₁₇	C₈²
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	#95
cosocle	C₄²×Dic₁₇	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7015
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
19. C₄⋊#3624	C₈	semidirect
23. C₄×C₈×C₁₇	C₈	semidirect
26. C₄×#3622	C₈	semidirect
123. C₄×C₈	#3624	semidirect
124. C₁₇	#7015	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3624	2
C₄×C₈×Dic₁₇	2
#7015	17