Subgroups — #17978144

Order 4352.

818 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763867	C₂	1	2	yes	maximal
2	#14544993	C₂	1	6	yes	maximal, complemented
3	C₄×#10587538	C₂	1	3	yes	maximal, complemented
4	C₂×#11763697	C₂	1	3	yes	maximal, complemented
5	C₄×#10587634	C₂	1	1	yes	maximal, characteristic
6	#11763867	C₄	1	16	yes	complemented
7	C₂×#10587538	C₂²	1	6	yes	—
8	C₂²×#63216	C₂²	1	6	yes	—
9	#11763697	C₂²	1	1	yes	characteristic
10	#11763697	C₂²	1	3	yes	—
11	#11763697	C₂²	1	6	yes	—
12	C₄×#63216	C₂²	1	6	yes	—
13	C₂×#10587634	C₄	1	1	yes	characteristic
14	C₂×#10587634	C₂²	1	1	yes	characteristic
15	#11762765	C₄	1	3	yes	—
16	#11762765	C₂²	1	3	yes	—
17	C₄×#63223	C₄	1	3	yes	—
18	C₂×C₄×#3622	C₂²	1	3	yes	—
19	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
20	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	C₈	1	2	yes	—
22	C₄×D₄×C₁₇	C₈	1	6	yes	—
23	C₁₇×#117	C₂×C₄	1	3	yes	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
25	#10587634	C₈	1	8	yes	—
26	#10587634	C₂×C₄	1	8	yes	—
27	C₂×#63223	C₂×C₄	1	3	yes	—
28	C₂×#63223	C₂×C₄	1	3	yes	—
29	C₂²×#3622	C₂×C₄	1	3	yes	—
30	C₂²×#3622	C₂³	1	3	yes	—
31	#10587538	C₂×C₄	1	24	yes	complemented
32	C₂×#63216	C₂×C₄	1	24	yes	complemented
33	C₂×#63216	C₂³	1	2	yes	—
34	C₂×#63216	C₂³	1	6	yes	—
35	#10587474	C₂×C₄	1	6	yes	—
36	C₄×#3622	C₂×C₄	1	3	yes	—
37	C₄×#3622	C₂×C₄	1	3	yes	—
38	C₄×#3622	C₂³	1	1	yes	characteristic
39	C₄×#3622	C₂³	1	3	yes	—
40	#41×C₁₇	C₄²	1	4	yes	—
41	#41×C₁₇	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₁₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
46	C₄⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
47	C₂²⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
48	C₄²×C₁₇	C₂×C₈	1	6	yes	—
49	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
51	#63223	C₄²	1	12	yes	—
52	#63223	C₂×C₈	1	12	yes	—
53	C₂×#3622	C₂×C₈	1	12	yes	—
54	C₂×#3622	C₂²×C₄	1	3	yes	—
55	C₂×#3622	C₂²×C₄	1	12	yes	—
56	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
57	C₂×#3622	C₂²×C₄	1	3	yes	—
58	C₂×#3622	C₂⁴	1	1	yes	characteristic
59	#63216	C₂²×C₄	1	4	yes	—
60	#63216	C₂²×C₄	1	12	yes	—
61	#33326	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
63	Q₈×C₁₇	C₄×C₈	1	4	yes	—
64	D₄×C₁₇	C₄×C₈	1	12	yes	—
65	C₂×C₄×C₁₇	—	4	6	no	—
66	C₂×C₄×C₁₇	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
72	#3622	C₂×C₄²	1	6	yes	—
73	#3622	C₂²×C₈	1	2	yes	—
74	#3622	C₂²×C₈	1	6	yes	—
75	#3622	C₂³×C₄	1	2	yes	—
76	C₂×#504	—	17	2	no	—
77	C₄×#131	—	17	1	no	—
78	#2108	—	17	6	no	—
79	C₄×M₃₂	—	17	3	no	—
80	C₂×#346	—	17	3	no	—
81	C₂²×C₁₇	—	2	6	no	—
82	C₂²×C₁₇	C₂×C₄×C₈	1	6	yes	—
83	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₁₇	—	2	2	no	—
85	C₄×C₁₇	—	2	6	no	—
86	C₄×C₁₇	C₂×C₄×C₈	1	6	yes	—
87	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
88	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
89	C₂×#131	—	17	1	no	—
90	C₂×#131	—	17	1	no	—
91	C₄×#41	#595	1	1	yes	characteristic
92	#504	—	17	16	no	—
93	C₂×M₃₂	—	17	6	no	—
94	C₂²×C₁₆	—	17	6	no	—
95	#405	—	17	3	no	—
96	#405	—	17	3	no	—
97	C₄×M₁₆	—	17	3	no	—
98	C₂×C₄×C₈	—	17	3	no	—
99	#346	—	17	1	no	—
100	#346	—	17	3	no	—
101	#346	—	17	6	no	—
102	C₄×C₁₆	—	17	6	no	—
103	C₂×C₁₇	—	2	6	no	—
104	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
105	C₂×C₁₇	#3407	1	2	yes	—
106	C₂×#41	C₂×#595	1	1	yes	characteristic
107	#131	—	17	8	no	—
108	#131	—	17	8	no	—
109	C₂×M₁₆	—	17	3	no	—
110	C₂×M₁₆	—	17	3	no	—
111	C₂²×C₈	—	17	3	no	—
112	C₂²×C₈	—	17	3	no	—
113	C₄×Q₈	#3624	1	2	yes	—
114	C₄×D₄	#3624	1	6	yes	—
115	#117	C₂×#595	1	3	yes	—
116	C₂×C₄²	C₂×#595	1	3	yes	—
117	M₃₂	—	17	24	no	—
118	C₂×C₁₆	—	17	6	no	—
119	C₂×C₁₆	—	17	2	no	—
120	C₂×C₁₆	—	17	24	no	—
121	#97	—	17	6	no	—
122	C₄×C₈	—	17	3	no	—
123	C₄×C₈	—	17	3	no	—
124	C₄×C₈	—	17	3	no	—
125	C₄×C₈	—	17	1	no	—
126	C₁₇	#33326	1	1	yes	characteristic, complemented, Sylow
127	#41	#63242	1	4	yes	—
128	#41	C₄×#595	1	4	yes	—
129	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
130	C₂×D₄	C₂×#3624	1	3	yes	—
131	C₂²×C₄	C₂×#3624	1	6	yes	—
132	C₂²×C₄	C₂²×#595	1	3	yes	—
133	M₁₆	—	17	12	no	—
134	M₁₆	—	17	12	no	—
135	C₂×C₈	—	17	12	no	—
136	C₂×C₈	—	17	12	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	3	no	—
140	C₂×C₈	—	17	3	no	—
141	C₄⋊C₄	C₂×#3624	1	6	yes	—
142	C₂²⋊C₄	C₂×#3624	1	6	yes	—
143	C₄²	C₂×#3624	1	6	yes	—
144	C₄²	C₂²×#595	1	1	yes	characteristic
145	C₄²	C₂²×#595	1	3	yes	—
146	C₁₆	—	17	4	no	—
147	C₁₆	—	17	12	no	—
148	C₂³	C₂²×#3624	1	3	yes	—
149	Q₈	C₄×#3624	1	4	yes	—
150	D₄	C₄×#3624	1	12	yes	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×#63242	1	6	yes	—
153	C₂×C₄	C₂×C₄×#595	1	6	yes	—
154	C₂×C₄	C₂²×#3624	1	2	yes	—
155	C₂×C₄	C₂²×#3624	1	6	yes	—
156	C₂×C₄	C₂²×#3624	1	3	yes	—
157	C₂×C₄	C₂³×#595	1	1	yes	characteristic
158	C₈	—	17	2	no	—
159	C₈	—	17	6	no	—
160	C₈	—	17	2	no	—
161	C₈	—	17	6	no	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#3624	1	6	yes	—
164	C₂²	C₂³×#3624	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×#3624	1	6	yes	—
168	C₄	C₂²×C₄×#595	1	1	yes	characteristic
169	C₄	C₂²×#63242	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#3624	1	1	yes	characteristic
172	C₂	#14545751	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator	C₂×C₁₇	C₂²×C₄×C₈
Fitting	C₄×#41×C₁₇	C₄
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33326
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #14544993	C₂	semidirect
3. C₄×#10587538	C₂	semidirect
4. C₂×#11763697	C₂	semidirect
6. #11763867	C₄	semidirect
31. #10587538	C₂×C₄	semidirect
32. C₂×#63216	C₂×C₄	semidirect
126. C₁₇	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763867	2
#14544993	2
C₄×#10587538	2
C₂×#11763697	2
C₄×#10587634	2
#33326	17