Subgroups — #17978677

Order 4352.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#63216	C₂	1	2	yes	maximal, complemented
2	#14545163	C₂	1	6	yes	maximal, complemented
3	#14545144	C₂	1	3	yes	maximal, complemented
4	C₄×#10587529	C₂	1	3	yes	maximal, complemented
5	C₈×#63256	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	C₂²	1	6	yes	—
7	C₄⋊#63216	C₂²	1	6	yes	—
8	#11763704	C₂²	1	6	yes	—
9	#11763695	C₂²	1	1	yes	characteristic
10	#11763695	C₂²	1	3	yes	—
11	C₄×#63216	C₂²	1	6	yes	—
12	C₄×#63256	C₄	1	1	yes	characteristic
13	Q₈×#3622	C₂²	1	1	yes	characteristic
14	C₈×D₆₈	C₄	1	3	yes	—
15	#11762874	C₂²	1	3	yes	—
16	#11762861	C₄	1	3	yes	—
17	C₄×C₈×D₁₇	C₂²	1	3	yes	—
18	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic
19	C₂×#63256	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₁₇	C₂³	1	3	yes	—
24	#10587579	C₈	1	3	yes	—
25	#10587578	C₈	1	3	yes	—
26	C₄×D₆₈	C₂×C₄	1	3	yes	—
27	C₄²×D₁₇	C₂×C₄	1	3	yes	—
28	C₂×#63216	C₂³	1	6	yes	—
29	C₂×#63216	C₂³	1	2	yes	—
30	#10587529	—	2	12	no	—
31	C₁₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₄×C₈×C₁₇	C₂×C₄	1	3	yes	—
33	#10587491	C₂×C₄	1	6	yes	—
34	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
35	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
36	C₄⋊#3622	C₂×C₄	1	3	yes	—
37	C₄×#3622	C₂³	1	3	yes	—
38	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
39	#63256	C₁₆	1	8	yes	complemented
40	C₂×D₆₈	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₁₇	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
47	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
48	#63227	C₂×C₈	1	6	yes	—
49	C₄⋊Dic₁₇	C₂×C₈	1	3	yes	—
50	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
51	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
52	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×#3622	C₂²×C₄	1	3	yes	—
54	C₂×#3622	C₂⁴	1	1	yes	characteristic
55	C₈×D₁₇	—	2	6	no	—
56	C₈×D₁₇	—	2	6	no	—
57	#63216	—	2	6	no	—
58	#63216	#41	1	4	yes	complemented
59	C₁₆×#41	—	17	1	no	maximal, Sylow
60	C₂²×D₁₇	C₂²×C₈	1	3	yes	—
61	Q₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
62	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
63	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
64	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
66	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
68	D₆₈	C₂×C₁₆	1	12	yes	complemented
69	C₄×D₁₇	—	2	3	no	—
70	C₄×D₁₇	—	2	3	no	—
71	C₄×D₁₇	—	2	6	no	—
72	C₄×D₁₇	C₂×C₁₆	1	12	yes	complemented
73	C₈×C₁₇	—	2	3	no	—
74	C₈×C₁₇	#131	1	2	yes	—
75	#3622	—	2	3	no	—
76	#3622	C₂×#41	1	2	yes	—
77	C₈×#41	—	17	1	no	—
78	Q₈×C₁₆	—	17	2	no	—
79	D₄×C₁₆	—	17	6	no	—
80	#2162	—	17	3	no	—
81	C₂×C₄×C₁₆	—	17	3	no	—
82	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
83	C₂×D₁₇	—	2	3	no	—
84	C₂×D₁₇	—	2	3	no	—
85	C₂×D₁₇	C₂²×C₁₆	1	6	yes	—
86	C₄×C₁₇	—	2	3	no	—
87	C₄×C₁₇	C₂²×C₁₆	1	6	yes	—
88	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
89	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
90	Dic₁₇	—	2	3	no	—
91	Dic₁₇	C₂²×C₁₆	1	2	yes	—
92	Dic₁₇	#504	1	2	yes	—
93	C₄×#41	—	17	1	no	—
94	C₂²×C₁₆	—	17	6	no	—
95	C₈×Q₈	—	17	1	no	—
96	C₈×Q₈	#595	1	1	yes	characteristic
97	C₈×D₄	—	17	3	no	—
98	C₈×D₄	—	17	3	no	—
99	#431	—	17	3	no	—
100	C₂×C₄×C₈	—	17	3	no	—
101	C₄⋊C₁₆	—	17	6	no	—
102	C₂²⋊C₁₆	—	17	6	no	—
103	C₄×C₁₆	—	17	1	no	—
104	C₄×C₁₆	—	17	3	no	—
105	C₄×C₁₆	—	17	6	no	—
106	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
107	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
108	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
109	D₁₇	—	2	6	no	—
110	C₂×#41	—	17	1	no	—
111	C₂²×C₈	—	17	3	no	—
112	C₂²×C₈	—	17	3	no	—
113	C₄×Q₈	—	17	1	no	—
114	C₄×Q₈	C₂×#595	1	1	yes	characteristic
115	C₄×D₄	—	17	3	no	—
116	C₄×D₄	—	17	3	no	—
117	#117	—	17	3	no	—
118	C₂×C₄²	—	17	3	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	—	17	6	no	—
121	C₂×C₁₆	—	34	12	no	—
122	C₄⋊C₈	—	17	3	no	—
123	C₄⋊C₈	C₂×#595	1	3	yes	—
124	C₂²⋊C₈	—	17	6	no	—
125	C₄×C₈	—	17	3	no	—
126	C₄×C₈	—	17	1	no	—
127	C₄×C₈	—	17	3	no	—
128	C₄×C₈	C₂×#595	1	3	yes	—
129	C₁₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
130	#41	—	17	8	no	—
131	C₂×Q₈	#63242	1	1	yes	characteristic
132	C₂×D₄	—	17	3	no	—
133	C₂²×C₄	—	17	3	no	—
134	C₂²×C₄	—	17	3	no	—
135	C₂²×C₄	—	17	3	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	3	no	—
138	C₂×C₈	—	34	6	no	—
139	C₂×C₈	—	34	6	no	—
140	C₂×C₈	C₂²×#595	1	3	yes	—
141	C₂×C₈	C₂²×#595	1	1	yes	characteristic
142	C₄⋊C₄	—	17	3	no	—
143	C₄⋊C₄	#63242	1	3	yes	—
144	C₂²⋊C₄	—	17	6	no	—
145	C₄²	—	17	3	no	—
146	C₄²	—	17	3	no	—
147	C₄²	—	17	1	no	—
148	C₄²	C₂²×#595	1	3	yes	—
149	C₁₆	—	17	4	no	—
150	C₁₆	—	34	6	no	—
151	C₂³	—	17	3	no	—
152	Q₈	#10587529	1	4	yes	complemented
153	D₄	—	17	12	no	—
154	C₂×C₄	—	17	1	no	—
155	C₂×C₄	—	17	1	no	—
156	C₂×C₄	—	17	12	no	—
157	C₂×C₄	—	17	3	no	—
158	C₂×C₄	—	34	6	no	—
159	C₂×C₄	—	34	3	no	—
160	C₂×C₄	—	34	3	no	—
161	C₂×C₄	C₂×#63242	1	3	yes	—
162	C₂×C₄	C₂×#63242	1	3	yes	—
163	C₂×C₄	C₂³×#595	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	17	2	no	—
166	C₈	—	34	3	no	—
167	C₈	#10587680	1	2	yes	—
168	C₂²	—	17	6	no	—
169	C₂²	—	34	3	no	—
170	C₂²	—	34	3	no	—
171	C₂²	C₂²×#63242	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	17	2	no	—
174	C₄	—	17	2	no	—
175	C₄	—	34	3	no	—
176	C₄	C₂×#10587529	1	6	yes	—
177	C₄	#41×#595	1	1	yes	characteristic
178	C₄	C₂×#10587680	1	1	yes	characteristic
179	C₂	—	34	6	no	—
180	C₂	#14545590	1	1	yes	characteristic
181	C₂	C₂²×#10587529	1	1	yes	characteristic
182	C₂	#14545749	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#595
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#595
Fitting	C₈×Q₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#63256	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₈×Q₈	#595
hypocenter	C₁₇	C₁₆×#41
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#63216	C₂	semidirect
2. #14545163	C₂	semidirect
3. #14545144	C₂	semidirect
4. C₄×#10587529	C₂	semidirect
39. #63256	C₁₆	semidirect
58. #63216	#41	semidirect
61. Q₈×C₁₇	C₂×C₁₆	semidirect
68. D₆₈	C₂×C₁₆	semidirect
72. C₄×D₁₇	C₂×C₁₆	semidirect
129. C₁₇	C₁₆×#41	semidirect
152. Q₈	#10587529	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#63216	2
#14545163	2
#14545144	2
C₄×#10587529	2
C₈×#63256	2
C₁₆×#41	17