Subgroups — C₂×D₁₇⋊#108 #14545441

Order 2176.

480 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×D₁₇⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#10587536	C₂	1	2	yes	maximal, complemented
3	D₁₇⋊#108	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10587535	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
7	C₂×#63247	—	2	2	no	—
8	D₁₇⋊M₁₆	—	2	4	no	—
9	D₁₇⋊M₁₆	C₂²	1	8	yes	—
10	C₂×#63243	C₂²	1	4	yes	—
11	C₂×#63242	—	2	2	no	—
12	C₂²×C₈×C₁₇	C₄	1	1	yes	characteristic
13	C₂²×#3622	C₄	1	1	yes	characteristic
14	C₂×C₈×D₁₇	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₇	C₄	1	4	yes	—
16	C₂×C₈×D₁₇	C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₇	C₂²	1	4	yes	—
19	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
20	#10587536	C₂²	1	8	yes	complemented
21	#10587535	C₂²	1	8	yes	complemented
22	C₂³×D₁₇	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₇	D₄	1	2	yes	—
29	C₂×C₄×D₁₇	Q₈	1	2	yes	—
30	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₂³	1	2	yes	—
33	#63247	—	2	8	no	—
34	C₂×#3624	—	2	4	no	—
35	#63243	—	2	4	no	—
36	#63243	C₂³	1	8	yes	—
37	#63242	—	2	8	no	—
38	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×#3622	C₂×C₄	1	1	yes	characteristic
42	C₂×#3622	C₂×C₄	1	4	yes	—
43	C₂×#3622	C₂×C₄	1	1	yes	characteristic
44	C₈×D₁₇	C₂×C₄	1	2	yes	—
45	C₈×D₁₇	C₂×C₄	1	4	yes	—
46	C₈×D₁₇	C₂×C₄	1	4	yes	—
47	C₈×D₁₇	C₂×C₄	1	2	yes	—
48	C₈×D₁₇	C₂³	1	2	yes	—
49	C₈×D₁₇	C₂³	1	2	yes	—
50	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×D₁₇	—	2	4	no	—
52	C₂²×D₁₇	C₄⋊C₄	1	2	yes	—
53	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×D₁₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×D₁₇	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
61	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
64	C₄×D₁₇	C₄⋊C₄	1	2	yes	—
65	C₄×D₁₇	C₄⋊C₄	1	2	yes	—
66	C₄×D₁₇	C₄⋊C₄	1	2	yes	—
67	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄×D₁₇	C₂²×C₄	1	2	yes	—
69	C₄×D₁₇	C₂×D₄	1	2	yes	—
70	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
71	C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
72	C₄×D₁₇	C₂×Q₈	1	2	yes	—
73	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
74	#3624	—	2	8	no	—
75	C₈×C₁₇	C₂²×C₄	1	2	yes	—
76	C₈×C₁₇	C₂²×C₄	1	2	yes	—
77	#3622	C₂²×C₄	1	2	yes	—
78	#3622	C₂²×C₄	1	2	yes	—
79	C₂²×#108	—	17	1	no	maximal, Sylow
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×D₁₇	—	2	2	no	—
84	C₂×D₁₇	—	2	4	no	—
85	C₂×D₁₇	—	2	4	no	—
86	C₂×D₁₇	#108	1	4	yes	complemented
87	C₂×D₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×D₁₇	C₂²×Q₈	1	1	yes	characteristic
90	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
93	Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	Dic₁₇	C₂×C₄⋊C₄	1	2	yes	—
95	Dic₁₇	C₂²×D₄	1	1	yes	characteristic
96	C₂²×M₁₆	—	17	2	no	—
97	C₂³×C₈	—	17	1	no	—
98	C₂×#108	—	17	2	no	—
99	C₂×#108	—	17	8	no	—
100	C₂×#108	—	17	2	no	—
101	C₂×C₁₇	—	2	2	no	—
102	C₂×C₁₇	C₂×#108	1	2	yes	complemented
103	C₂×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	D₁₇	—	2	1	no	—
105	D₁₇	—	2	1	no	—
106	D₁₇	C₂×#108	1	4	yes	complemented
107	C₂³×C₄	—	17	1	no	—
108	C₂×M₁₆	—	17	4	no	—
109	C₂×M₁₆	—	17	8	no	—
110	C₂×M₁₆	—	34	4	no	—
111	C₂×M₁₆	—	34	2	no	—
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	17	4	no	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	—	17	4	no	—
119	C₂²×C₈	—	34	2	no	—
120	C₂²×C₈	#595	1	1	yes	characteristic
121	#108	—	17	8	no	—
122	#108	—	17	8	no	—
123	C₁₇	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
124	C₂⁴	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	—	17	2	no	—
127	C₂²×C₄	—	17	2	no	—
128	C₂²×C₄	—	17	1	no	—
129	C₂²×C₄	—	17	1	no	—
130	C₂²×C₄	—	17	1	no	—
131	C₂²×C₄	—	17	2	no	—
132	C₂²×C₄	—	17	1	no	—
133	C₂²×C₄	—	17	2	no	—
134	C₂²×C₄	C₂×#595	1	1	yes	characteristic
135	M₁₆	—	17	8	no	—
136	M₁₆	—	34	4	no	—
137	M₁₆	—	34	8	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	4	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	4	no	—
144	C₂×C₈	—	17	4	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	2	no	—
147	C₂×C₈	—	34	8	no	—
148	C₂×C₈	—	34	4	no	—
149	C₂×C₈	C₂×#595	1	1	yes	characteristic
150	C₂×C₈	C₂×#595	1	1	yes	characteristic
151	C₂×C₈	C₂×#595	1	4	yes	—
152	C₂³	—	17	1	no	—
153	C₂³	—	17	2	no	—
154	C₂³	—	17	2	no	—
155	C₂³	—	17	1	no	—
156	C₂³	—	34	4	no	—
157	C₂³	C₄⋊#595	1	1	yes	characteristic
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	17	2	no	—
160	C₂×C₄	—	17	2	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	17	2	no	—
165	C₂×C₄	—	17	2	no	—
166	C₂×C₄	—	17	2	no	—
167	C₂×C₄	—	17	1	no	—
168	C₂×C₄	—	17	2	no	—
169	C₂×C₄	—	17	1	no	—
170	C₂×C₄	—	17	2	no	—
171	C₂×C₄	—	17	1	no	—
172	C₂×C₄	C₄⋊#595	1	2	yes	—
173	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
174	C₂×C₄	C₂²×#595	1	1	yes	characteristic
175	C₂×C₄	C₂²×#595	1	2	yes	—
176	C₈	—	17	2	no	—
177	C₈	—	17	2	no	—
178	C₈	—	34	8	no	—
179	C₈	C₂²×#595	1	2	yes	—
180	C₈	C₂²×#595	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	—	17	4	no	—
183	C₂²	—	17	2	no	—
184	C₂²	—	17	1	no	—
185	C₂²	—	17	1	no	—
186	C₂²	—	34	4	no	—
187	C₂²	—	34	4	no	—
188	C₂²	—	34	2	no	—
189	C₂²	C₂×C₄⋊#595	1	2	yes	—
190	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
191	C₄	—	17	1	no	—
192	C₄	—	17	1	no	—
193	C₄	—	17	2	no	—
194	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
195	C₄	C₂×C₄⋊#595	1	2	yes	—
196	C₄	C₂³×#595	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	17	4	no	—
199	C₂	—	34	1	no	—
200	C₂	—	34	1	no	—
201	C₂	D₁₇⋊#108	1	2	yes	direct product factor
202	C₂	C₂²×C₄⋊#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#595
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×#595
Fitting	C₂²×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₇	Q₈
cosocle	C₄×D₁₇	C₂⁴
hypercenter	C₂²×C₈	#595
hypocenter	C₁₇	C₂²×#108
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10587536	C₂	semidirect
3. D₁₇⋊#108	C₂	direct (N × H)
4. C₂×#10587535	C₂	semidirect
20. #10587536	C₂²	semidirect
21. #10587535	C₂²	semidirect
86. C₂×D₁₇	#108	semidirect
102. C₂×C₁₇	C₂×#108	semidirect
106. D₁₇	C₂×#108	semidirect
123. C₁₇	C₂²×#108	semidirect
201. C₂	D₁₇⋊#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₇⋊M₁₆	2
C₂×#10587536	2
D₁₇⋊#108	2
C₂×#10587535	2
C₂²×C₈×D₁₇	2
C₂²×#108	17

Subgroups — C2×D17⋊#108 #14545441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₇⋊#108 #14545441