Subgroups — C₂²×#11762587 #17186978

Order 4352.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587483	C₂	1	2	yes	maximal
2	C₂×#11762587	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587483	C₄	1	12	yes	complemented
5	C₂×#10587483	C₂²	1	12	yes	—
6	C₂²×#63214	C₄	1	4	yes	complemented
7	#11762587	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₁₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
11	C₂×C₁₇×#97	C₄	1	6	yes	—
12	C₂×C₁₇×#97	C₂²	1	6	yes	—
13	C₂³×C₄×C₁₇	—	2	1	no	—
14	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	—	2	3	no	—
16	C₂²×C₈×C₁₇	—	2	1	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	6	yes	—
21	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
22	C₂×C₄²×C₁₇	—	2	3	no	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
24	C₁₇×#97	C₂×C₄	1	12	yes	—
25	C₁₇×#97	C₂³	1	4	yes	—
26	#10587483	C₂×C₄	1	24	yes	complemented
27	#10587483	C₂³	1	8	yes	—
28	C₂×#63214	C₂×C₄	1	24	yes	complemented
29	C₂⁴×C₁₇	M₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	—	2	6	no	—
31	C₂²×C₄×C₁₇	—	4	4	no	—
32	C₂²×C₄×C₁₇	C₄²	1	3	yes	—
33	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	M₁₆	1	3	yes	—
35	C₂²×C₄×C₁₇	M₁₆	1	3	yes	—
36	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
38	C₂×C₈×C₁₇	—	2	6	no	—
39	C₂×C₈×C₁₇	—	2	6	no	—
40	C₂×C₈×C₁₇	C₄²	1	6	yes	—
41	C₂×C₈×C₁₇	C₄²	1	6	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₄²×C₁₇	—	2	6	no	—
48	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
49	#63214	C₂²×C₄	1	16	yes	complemented
50	C₂²×#347	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	—	2	4	no	—
52	C₂³×C₁₇	#97	1	3	yes	—
53	C₂³×C₁₇	#97	1	1	yes	characteristic
54	C₂³×C₁₇	C₂×M₁₆	1	3	yes	—
55	C₂×C₄×C₁₇	—	2	4	no	—
56	C₂×C₄×C₁₇	—	4	12	no	—
57	C₂×C₄×C₁₇	#97	1	6	yes	—
58	C₂×C₄×C₁₇	#97	1	3	yes	—
59	C₂×C₄×C₁₇	#97	1	3	yes	—
60	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
62	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
63	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₁₇	C₂×M₁₆	1	3	yes	—
65	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₁₇	—	2	4	no	—
67	C₈×C₁₇	C₂×C₄²	1	6	yes	—
68	C₈×C₁₇	C₂³×C₄	1	2	yes	—
69	C₂²×M₃₂	—	17	2	no	—
70	C₂²×#97	D₁₇	1	1	yes	characteristic
71	C₂×#347	—	17	12	no	—
72	C₂²×C₁₇	—	2	6	no	—
73	C₂²×C₁₇	—	2	6	no	—
74	C₂²×C₁₇	#347	1	4	yes	complemented
75	C₂²×C₁₇	C₂×#97	1	3	yes	—
76	C₂²×C₁₇	C₂×#97	1	3	yes	—
77	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
78	C₄×C₁₇	—	4	4	no	—
79	C₄×C₁₇	C₂×#97	1	3	yes	—
80	C₄×C₁₇	C₂×#97	1	3	yes	—
81	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
82	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
83	C₂³×C₈	Dic₁₇	1	1	yes	characteristic
84	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
85	C₂²×C₄²	Dic₁₇	1	1	yes	characteristic
86	C₂×M₃₂	—	17	12	no	—
87	C₂×M₃₂	—	17	12	no	—
88	C₂²×C₁₆	—	17	4	no	—
89	C₂×#97	Dic₁₇	1	6	yes	—
90	C₂×#97	C₂×D₁₇	1	6	yes	—
91	#347	—	17	16	no	—
92	C₂×C₁₇	—	2	1	no	—
93	C₂×C₁₇	—	2	3	no	—
94	C₂×C₁₇	C₂×#347	1	6	yes	complemented
95	C₂×C₁₇	C₂²×#97	1	1	yes	characteristic
96	C₂³×C₄	—	2	1	no	—
97	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	3	no	—
100	C₂²×C₈	C₄×D₁₇	1	6	yes	—
101	C₂²×C₈	C₄×D₁₇	1	2	yes	—
102	C₂²×C₈	C₂×Dic₁₇	1	6	yes	—
103	C₂²×C₈	C₂×Dic₁₇	1	3	yes	—
104	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
105	C₂×C₄²	—	2	3	no	—
106	C₂×C₄²	C₂×Dic₁₇	1	6	yes	—
107	M₃₂	—	17	24	no	—
108	M₃₂	—	17	8	no	—
109	C₂×C₁₆	—	17	24	no	—
110	#97	C₂×Dic₁₇	1	12	yes	—
111	#97	C₂²×D₁₇	1	4	yes	—
112	C₁₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴	#63223	1	1	yes	characteristic
114	C₂²×C₄	—	2	6	no	—
115	C₂²×C₄	—	4	4	no	—
116	C₂²×C₄	#63223	1	3	yes	—
117	C₂²×C₄	#63223	1	1	yes	characteristic
118	C₂²×C₄	#63223	1	3	yes	—
119	C₂²×C₄	C₄×Dic₁₇	1	3	yes	—
120	C₂²×C₄	C₄×Dic₁₇	1	1	yes	characteristic
121	C₂²×C₄	C₂²×Dic₁₇	1	3	yes	—
122	C₂×C₈	—	2	6	no	—
123	C₂×C₈	—	2	6	no	—
124	C₂×C₈	C₄×Dic₁₇	1	6	yes	—
125	C₂×C₈	C₄×Dic₁₇	1	6	yes	—
126	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
127	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
128	C₂×C₈	C₂²×Dic₁₇	1	4	yes	—
129	C₂×C₈	C₂²×Dic₁₇	1	3	yes	—
130	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
131	C₄²	—	2	6	no	—
132	C₄²	C₂²×Dic₁₇	1	4	yes	—
133	C₁₆	—	17	16	no	—
134	C₂³	—	2	4	no	—
135	C₂³	#10587474	1	3	yes	—
136	C₂³	#10587474	1	1	yes	characteristic
137	C₂³	C₂×#63223	1	3	yes	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	#10587474	1	6	yes	—
141	C₂×C₄	#10587474	1	3	yes	—
142	C₂×C₄	#10587474	1	3	yes	—
143	C₂×C₄	C₂×#63223	1	3	yes	—
144	C₂×C₄	C₂×#63223	1	3	yes	—
145	C₂×C₄	C₂×#63223	1	3	yes	—
146	C₂×C₄	C₂×C₄×Dic₁₇	1	3	yes	—
147	C₂×C₄	C₂×C₄×Dic₁₇	1	3	yes	—
148	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
149	C₈	—	2	4	no	—
150	C₈	C₂×C₄×Dic₁₇	1	6	yes	—
151	C₈	C₂²×C₄×D₁₇	1	2	yes	—
152	C₂²	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	#11762587	1	4	yes	direct product factor
155	C₂²	C₂×#10587474	1	3	yes	—
156	C₂²	C₂×#10587474	1	3	yes	—
157	C₂²	C₂²×#63223	1	1	yes	characteristic
158	C₄	—	4	4	no	—
159	C₄	C₂×#10587474	1	3	yes	—
160	C₄	C₂×#10587474	1	3	yes	—
161	C₄	C₂²×C₄×Dic₁₇	1	1	yes	characteristic
162	C₄	C₂²×#63223	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#11762587	1	6	yes	direct product factor
166	C₂	C₂²×#10587474	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂²×C₁₇×#97	C₂
socle	C₂³×C₁₇	#97
prime radical	C₂⁴×C₁₇	M₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂²×#97	D₁₇
hypocenter	C₁₇	C₂²×#347
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. C₂×#11762587	C₂	direct (N × H)
4. C₂×#10587483	C₄	semidirect
6. C₂²×#63214	C₄	semidirect
7. #11762587	C₂²	direct (N × H)
26. #10587483	C₂×C₄	semidirect
28. C₂×#63214	C₂×C₄	semidirect
49. #63214	C₂²×C₄	semidirect
74. C₂²×C₁₇	#347	semidirect
94. C₂×C₁₇	C₂×#347	semidirect
112. C₁₇	C₂²×#347	semidirect
154. C₂²	#11762587	direct (N × H)
165. C₂	C₂×#11762587	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10587483	2
C₂×#11762587	2
C₂²×C₁₇×#97	2
C₂²×#347	17

Subgroups — C22×#11762587 #17186978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11762587 #17186978