Subgroups — #17978408

Order 4352.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 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	#14545119	C₂	1	6	yes	maximal, complemented
3	#14545116	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#63216	C₂	1	3	yes	maximal, complemented
5	C₄×#10587634	C₂	1	1	yes	maximal, characteristic
6	#11763867	—	2	8	no	—
7	C₂×#10587538	C₂²	1	6	yes	—
8	C₂²×#63216	C₂²	1	6	yes	—
9	C₄⋊#63216	C₂²	1	1	yes	characteristic
10	C₄⋊#63216	C₂²	1	3	yes	—
11	C₄⋊#63216	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	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
23	C₄×D₄×C₁₇	C₈	1	3	yes	—
24	C₄×D₄×C₁₇	C₈	1	3	yes	—
25	C₁₇×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
27	#10587634	—	2	4	no	—
28	#10587634	D₄	1	4	yes	—
29	#10587634	Q₈	1	4	yes	—
30	C₂×#63223	C₂×C₄	1	3	yes	—
31	C₂×#63223	C₂×C₄	1	3	yes	—
32	C₂²×#3622	C₂×C₄	1	3	yes	—
33	C₂²×#3622	C₂³	1	3	yes	—
34	#10587538	—	2	12	no	—
35	C₂×#63216	—	2	12	no	—
36	C₂×#63216	C₂³	1	2	yes	—
37	C₂×#63216	C₂³	1	6	yes	—
38	#10587474	C₂×C₄	1	6	yes	—
39	C₄×#3622	C₂×C₄	1	3	yes	—
40	C₄×#3622	C₂×C₄	1	3	yes	—
41	C₄×#3622	C₂³	1	1	yes	characteristic
42	C₄×#3622	C₂³	1	3	yes	—
43	#41×C₁₇	C₄⋊C₄	1	4	yes	—
44	#41×C₁₇	M₁₆	1	4	yes	—
45	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₁₇	C₂×C₈	1	3	yes	—
47	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
50	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
52	C₂²⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
53	C₄²×C₁₇	C₂×C₈	1	3	yes	—
54	C₄²×C₁₇	C₂×C₈	1	3	yes	—
55	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
57	#63223	—	2	6	no	—
58	#63223	C₄⋊C₄	1	12	yes	—
59	C₂×#3622	—	2	6	no	—
60	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
61	C₂×#3622	C₂²×C₄	1	3	yes	—
62	C₂×#3622	C₂²×C₄	1	3	yes	—
63	C₂×#3622	C₂×D₄	1	6	yes	—
64	C₂×#3622	C₂×Q₈	1	6	yes	—
65	C₂×#3622	C₂⁴	1	1	yes	characteristic
66	#63216	—	2	2	no	—
67	#63216	—	2	6	no	—
68	#33398	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
70	Q₈×C₁₇	C₄⋊C₈	1	4	yes	—
71	D₄×C₁₇	C₄⋊C₈	1	12	yes	—
72	C₂×C₄×C₁₇	—	2	6	no	—
73	C₂×C₄×C₁₇	—	4	3	no	—
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
79	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
81	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
82	#3622	—	2	1	no	—
83	#3622	—	2	3	no	—
84	#3622	C₂×C₄⋊C₄	1	6	yes	—
85	#3622	C₂²×D₄	1	1	yes	characteristic
86	#3622	C₂²×Q₈	1	1	yes	characteristic
87	C₂×#504	—	17	2	no	—
88	C₄×#131	—	17	1	no	—
89	#2151	—	17	6	no	—
90	#2150	—	17	3	no	—
91	C₂×C₄⋊C₁₆	—	17	3	no	—
92	C₂²×C₁₇	—	2	3	no	—
93	C₂²×C₁₇	—	2	3	no	—
94	C₂²×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
95	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₁₇	—	2	3	no	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	—	2	3	no	—
99	C₄×C₁₇	C₂×C₄⋊C₈	1	6	yes	—
100	C₄×C₁₇	#504	1	2	yes	complemented
101	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×#131	—	17	1	no	—
104	C₂×#131	—	17	1	no	—
105	C₄×#41	#595	1	1	yes	characteristic
106	#504	—	34	8	no	—
107	C₂×M₃₂	—	17	6	no	—
108	C₂²×C₁₆	—	17	6	no	—
109	#405	—	17	3	no	—
110	#405	—	17	3	no	—
111	C₄×M₁₆	—	17	3	no	—
112	C₂×C₄×C₈	—	17	3	no	—
113	C₄⋊C₁₆	—	17	1	no	—
114	C₄⋊C₁₆	—	17	3	no	—
115	C₄⋊C₁₆	—	17	6	no	—
116	C₄⋊C₁₆	—	17	6	no	—
117	C₂×C₁₇	—	2	6	no	—
118	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
120	C₂×C₁₇	#3407	1	1	yes	characteristic
121	C₂×#41	C₂×#595	1	1	yes	characteristic
122	#131	—	17	4	no	—
123	#131	—	17	4	no	—
124	#131	—	34	4	no	—
125	C₂×M₁₆	—	17	3	no	—
126	C₂×M₁₆	—	17	3	no	—
127	C₂²×C₈	—	17	3	no	—
128	C₂²×C₈	—	17	3	no	—
129	C₄×Q₈	#3624	1	1	yes	characteristic
130	C₄×Q₈	#3624	1	1	yes	characteristic
131	C₄×D₄	#3624	1	3	yes	—
132	C₄×D₄	#3624	1	3	yes	—
133	#117	C₂×#595	1	3	yes	—
134	C₂×C₄²	C₂×#595	1	3	yes	—
135	M₃₂	—	34	12	no	—
136	C₂×C₁₆	—	17	2	no	—
137	C₂×C₁₆	—	17	6	no	—
138	C₂×C₁₆	—	34	12	no	—
139	#97	—	17	6	no	—
140	C₄×C₈	—	17	3	no	—
141	C₄×C₈	—	17	3	no	—
142	C₄×C₈	—	17	1	no	—
143	C₄×C₈	—	17	3	no	—
144	C₁₇	#33398	1	1	yes	characteristic, complemented, Sylow
145	#41	#63243	1	4	yes	—
146	#41	C₄⋊#595	1	4	yes	—
147	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
148	C₂×D₄	C₂×#3624	1	3	yes	—
149	C₂²×C₄	C₂×#3624	1	3	yes	—
150	C₂²×C₄	C₂×#3624	1	3	yes	—
151	C₂²×C₄	C₂²×#595	1	3	yes	—
152	M₁₆	—	17	12	no	—
153	M₁₆	—	34	6	no	—
154	C₂×C₈	—	17	6	no	—
155	C₂×C₈	—	17	6	no	—
156	C₂×C₈	—	17	1	no	—
157	C₂×C₈	—	17	3	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	3	no	—
160	C₂×C₈	—	34	6	no	—
161	C₄⋊C₄	C₂×#3624	1	3	yes	—
162	C₄⋊C₄	C₂×#3624	1	3	yes	—
163	C₂²⋊C₄	C₂×#3624	1	6	yes	—
164	C₄²	C₂×#3624	1	3	yes	—
165	C₄²	C₂×#3624	1	3	yes	—
166	C₄²	C₂²×#595	1	1	yes	characteristic
167	C₄²	C₂²×#595	1	3	yes	—
168	C₁₆	—	34	2	no	—
169	C₁₆	—	34	6	no	—
170	C₂³	C₂²×#3624	1	3	yes	—
171	Q₈	C₄⋊#3624	1	4	yes	—
172	D₄	C₄⋊#3624	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×#63243	1	6	yes	—
176	C₂×C₄	C₂×C₄⋊#595	1	6	yes	—
177	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
178	C₂×C₄	C₂²×#3624	1	3	yes	—
179	C₂×C₄	C₂²×#3624	1	3	yes	—
180	C₂×C₄	C₂²×#3624	1	3	yes	—
181	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
182	C₂×C₄	C₂³×#595	1	1	yes	characteristic
183	C₈	—	17	1	no	—
184	C₈	—	17	1	no	—
185	C₈	—	17	6	no	—
186	C₈	—	34	1	no	—
187	C₈	—	34	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₄⋊#3624	1	6	yes	—
191	C₂²	C₂³×#3624	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₄⋊#3624	1	6	yes	—
196	C₄	#11763867	1	2	yes	complemented
197	C₄	C₂²×C₄⋊#595	1	1	yes	characteristic
198	C₄	C₂²×#63243	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₄⋊#3624	1	1	yes	characteristic
201	C₂	C₂×#11763867	1	1	yes	characteristic
202	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₄×#41×C₁₇	C₄
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33398
LCS	G ▷ C₂²×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. #14545119	C₂	semidirect
3. #14545116	C₂	semidirect
4. C₂×C₄⋊#63216	C₂	semidirect
100. C₄×C₁₇	#504	semidirect
144. C₁₇	#33398	semidirect
196. C₄	#11763867	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
#14545119	2
#14545116	2
C₂×C₄⋊#63216	2
C₄×#10587634	2
#33398	17