Subgroups — C₂×#11762869 #14537409

Order 2176.

412 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587558	C₂	1	1	yes	maximal, characteristic
2	#11762869	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10587491	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587491	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×#63227	C₄	1	1	yes	characteristic
14	C₂×#63227	C₄	1	1	yes	characteristic
15	C₂×#63225	C₄	1	1	yes	characteristic
16	C₂×#63225	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂²×#3622	C₂²	1	1	yes	characteristic
19	C₂²×#3622	C₂²	1	1	yes	characteristic
20	#10587558	C₄	1	4	yes	—
21	#10587558	C₂²	1	4	yes	—
22	C₁₇×C₄⋊C₈	C₂²	1	4	yes	complemented
23	#10587491	C₂²	1	4	yes	—
24	#10587491	C₂²	1	4	yes	—
25	#10587486	C₂²	1	4	yes	complemented
26	C₈×Dic₁₇	C₂²	1	4	yes	complemented
27	C₄⋊#3622	C₂²	1	4	yes	complemented
28	C₂³×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	—	2	2	no	—
36	C₂×C₄×D₁₇	—	2	2	no	—
37	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₇	C₂³	1	2	yes	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	C₂³	1	2	yes	—
42	C₂×C₈×C₁₇	C₂³	1	2	yes	—
43	C₄²×C₁₇	C₂×C₄	1	2	yes	—
44	C₄²×C₁₇	C₂³	1	2	yes	—
45	#63227	C₂×C₄	1	4	yes	—
46	#63227	C₂×C₄	1	4	yes	—
47	#63225	C₂×C₄	1	4	yes	—
48	#63225	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₇	C₂³	1	2	yes	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	C₂³	1	2	yes	—
54	C₂×#3622	C₂³	1	2	yes	—
55	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₁₇	—	2	2	no	—
57	C₂²×D₁₇	—	2	2	no	—
58	C₂²×D₁₇	—	2	2	no	—
59	C₂²×D₁₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#41	1	2	yes	—
66	C₂×C₄×C₁₇	#41	1	2	yes	—
67	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₇	—	2	2	no	—
69	C₂×Dic₁₇	—	2	2	no	—
70	C₂×Dic₁₇	—	2	2	no	—
71	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
74	C₄×D₁₇	—	2	4	no	—
75	C₄×D₁₇	—	2	2	no	—
76	C₄×D₁₇	—	2	2	no	—
77	C₈×C₁₇	—	2	2	no	—
78	C₈×C₁₇	—	2	2	no	—
79	#3622	—	2	2	no	—
80	#3622	—	2	2	no	—
81	C₂×#433	—	17	1	no	maximal, Sylow
82	C₂²×C₁₇	#117	1	2	yes	—
83	C₂²×C₁₇	#131	1	2	yes	—
84	C₂²×C₁₇	#131	1	2	yes	—
85	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×D₁₇	—	2	2	no	—
87	C₂×D₁₇	—	2	2	no	—
88	C₂×D₁₇	—	2	2	no	—
89	C₂×D₁₇	—	4	4	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	#117	1	2	yes	—
93	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
94	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
95	Dic₁₇	—	2	2	no	—
96	Dic₁₇	—	2	2	no	—
97	Dic₁₇	—	2	2	no	—
98	C₂×#117	—	17	1	no	—
99	#433	—	17	8	no	—
100	C₂×C₄⋊C₈	—	17	1	no	—
101	C₂×C₄⋊C₈	D₁₇	1	1	yes	characteristic, complemented
102	C₂×C₂²⋊C₈	—	17	1	no	—
103	C₂×C₂²⋊C₈	—	17	1	no	—
104	C₂×#97	—	17	1	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₂×C₁₇	#433	1	4	yes	complemented
107	C₂×C₁₇	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
110	D₁₇	—	4	2	no	—
111	C₂³×C₄	—	17	1	no	—
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
116	#117	—	17	4	no	—
117	#117	—	17	4	no	—
118	C₂×C₄⋊C₄	—	17	1	no	—
119	C₂×C₄⋊C₄	—	17	1	no	—
120	C₂×C₂²⋊C₄	—	17	1	no	—
121	C₂×C₂²⋊C₄	—	17	1	no	—
122	C₂×C₄²	—	17	1	no	—
123	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
124	C₄⋊C₈	—	17	4	no	—
125	C₄⋊C₈	C₂×D₁₇	1	4	yes	complemented
126	C₂²⋊C₈	—	17	4	no	—
127	C₂²⋊C₈	—	17	4	no	—
128	#97	—	17	4	no	—
129	C₄×C₈	—	17	4	no	—
130	C₁₇	C₂×#433	1	1	yes	characteristic, complemented, Sylow
131	C₂⁴	—	17	1	no	—
132	C₂²×C₄	—	17	1	no	—
133	C₂²×C₄	—	17	1	no	—
134	C₂²×C₄	—	17	1	no	—
135	C₂²×C₄	—	17	2	no	—
136	C₂²×C₄	—	17	2	no	—
137	C₂²×C₄	—	34	2	no	—
138	C₂²×C₄	—	34	2	no	—
139	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	17	2	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	34	2	no	—
147	C₂×C₈	—	34	2	no	—
148	C₂×C₈	C₂²×D₁₇	1	2	yes	—
149	C₂×C₈	C₂²×D₁₇	1	2	yes	—
150	C₄⋊C₄	—	17	4	no	—
151	C₄⋊C₄	—	17	4	no	—
152	C₂²⋊C₄	—	17	4	no	—
153	C₂²⋊C₄	—	17	4	no	—
154	C₄²	—	17	2	no	—
155	C₄²	—	17	2	no	—
156	C₄²	C₄×D₁₇	1	2	yes	—
157	C₄²	C₂²×D₁₇	1	2	yes	—
158	C₂³	—	17	2	no	—
159	C₂³	—	34	2	no	—
160	C₂³	—	34	2	no	—
161	C₂³	—	34	2	no	—
162	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	17	2	no	—
166	C₂×C₄	—	17	2	no	—
167	C₂×C₄	—	17	2	no	—
168	C₂×C₄	—	34	2	no	—
169	C₂×C₄	—	34	2	no	—
170	C₂×C₄	—	34	2	no	—
171	C₂×C₄	—	34	2	no	—
172	C₂×C₄	—	34	2	no	—
173	C₂×C₄	—	34	4	no	—
174	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
175	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
176	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
177	C₂×C₄	#63254	1	2	yes	—
178	C₂×C₄	#63256	1	2	yes	—
179	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	34	2	no	—
183	C₈	—	34	2	no	—
184	C₂²	—	34	2	no	—
185	C₂²	—	34	2	no	—
186	C₂²	—	34	2	no	—
187	C₂²	—	68	4	no	—
188	C₂²	#10587578	1	2	yes	—
189	C₂²	#10587587	1	2	yes	—
190	C₂²	#10587593	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	34	2	no	—
195	C₄	—	34	2	no	—
196	C₄	—	34	2	no	—
197	C₄	#10587578	1	2	yes	—
198	C₄	C₂×#63254	1	1	yes	characteristic
199	C₄	C₂×#63256	1	1	yes	characteristic
200	C₂	—	68	2	no	—
201	C₂	#11762869	1	4	yes	direct product factor
202	C₂	C₂×#10587578	1	1	yes	characteristic
203	C₂	C₂×#10587587	1	1	yes	characteristic
204	C₂	C₂×#10587593	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₈	C₂
socle	C₂³×C₁₇	C₂²×C₄
prime radical	C₂³×D₁₇	C₂×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₇
hypocenter	C₁₇	C₂×#433
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #11762869	C₂	direct (N × H)
3. C₂×C₄⋊#3622	C₂	semidirect
6. C₂×#10587486	C₂	semidirect
7. C₂×C₈×Dic₁₇	C₂	semidirect
8. C₂×C₁₇×C₄⋊C₈	C₂	semidirect
22. C₁₇×C₄⋊C₈	C₂²	semidirect
25. #10587486	C₂²	semidirect
26. C₈×Dic₁₇	C₂²	semidirect
27. C₄⋊#3622	C₂²	semidirect
101. C₂×C₄⋊C₈	D₁₇	semidirect
106. C₂×C₁₇	#433	semidirect
125. C₄⋊C₈	C₂×D₁₇	semidirect
130. C₁₇	C₂×#433	semidirect
201. C₂	#11762869	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×#10587558	2
#11762869	2
C₂×C₄⋊#3622	2
C₂×#10587491	2
C₂×#10587486	2
C₂×C₈×Dic₁₇	2
C₂×C₁₇×C₄⋊C₈	2
C₂×#433	17

Subgroups — C2×#11762869 #14537409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11762869 #14537409