Subgroups — C₂×#14545199 #17978653

Order 4352.

542 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	#14545199	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#63216	C₂	1	2	yes	maximal
4	C₂×#11763697	C₂	1	2	yes	maximal
5	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
7	C₂²×#63216	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	2	yes	—
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	C₄⋊#63216	C₂²	1	4	yes	—
11	C₄⋊#63216	C₂²	1	8	yes	—
12	#11763697	C₂²	1	8	yes	—
13	C₄×#63216	C₂²	1	4	yes	—
14	Q₈×#3622	C₄	1	4	yes	—
15	Q₈×#3622	C₂²	1	4	yes	—
16	C₂×C₄⋊#3622	C₄	1	2	yes	—
17	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	C₂×C₄×#3622	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
21	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₈	1	4	yes	—
23	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
29	C₂²×#3622	C₂×C₄	1	2	yes	—
30	C₂²×#3622	C₂³	1	1	yes	characteristic
31	C₂×#63216	—	2	2	no	—
32	C₂×#63216	—	2	4	no	—
33	C₂×#63216	Q₈	1	4	yes	complemented
34	C₂×#63216	C₂³	1	2	yes	—
35	C₂×#63216	C₂³	1	4	yes	—
36	C₂×#63216	C₂³	1	2	yes	—
37	C₄⋊#3622	C₂×C₄	1	8	yes	—
38	C₄⋊#3622	C₂×C₄	1	4	yes	—
39	C₄×#3622	C₂×C₄	1	2	yes	—
40	C₄×#3622	C₂×C₄	1	4	yes	—
41	C₄×#3622	C₂³	1	2	yes	—
42	C₄×#3622	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₇	—	2	4	no	—
44	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
51	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
52	C₄²×C₁₇	C₂×C₈	1	4	yes	—
53	C₄²×C₁₇	C₂×C₈	1	2	yes	—
54	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
55	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	—	2	4	no	—
58	C₂×#3622	C₂²×C₄	1	2	yes	—
59	C₂×#3622	C₂²×C₄	1	4	yes	—
60	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
61	C₂×#3622	C₂×Q₈	1	2	yes	—
62	C₂×#3622	#41	1	2	yes	—
63	C₂×#3622	C₂⁴	1	1	yes	characteristic
64	#63216	—	2	2	no	—
65	#63216	—	2	4	no	—
66	#63216	C₂×Q₈	1	4	yes	complemented
67	C₂×#2183	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	Q₈×C₁₇	—	2	8	no	—
70	C₂×C₄×C₁₇	—	2	4	no	—
71	C₂×C₄×C₁₇	—	2	4	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
74	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₁₇	#131	1	2	yes	—
81	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
82	#3622	—	2	2	no	—
83	#3622	—	2	4	no	—
84	#3622	C₄×Q₈	1	2	yes	—
85	#3622	C₂²×Q₈	1	1	yes	characteristic
86	#3622	C₂×#41	1	1	yes	characteristic
87	C₂×C₈×Q₈	—	17	1	no	—
88	#2183	—	17	8	no	—
89	C₂×C₄⋊C₁₆	—	17	1	no	—
90	C₂×C₄⋊C₁₆	—	17	2	no	—
91	C₂×#346	—	17	2	no	—
92	C₂×C₄×C₁₆	—	17	1	no	—
93	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
94	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
95	C₂²×C₁₇	#504	1	2	yes	—
96	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₁₇	—	2	4	no	—
98	C₄×C₁₇	—	2	4	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	C₈×Q₈	1	2	yes	—
101	C₄×C₁₇	C₂×M₃₂	1	4	yes	—
102	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
104	C₂×C₄×Q₈	#595	1	1	yes	characteristic
105	C₂²×C₁₆	—	17	1	no	—
106	C₂²×C₁₆	—	17	1	no	—
107	C₂²×C₁₆	—	17	2	no	—
108	C₈×Q₈	—	17	4	no	—
109	C₈×Q₈	—	17	4	no	—
110	C₂×C₄⋊C₈	—	17	2	no	—
111	C₂×C₄⋊C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	17	2	no	—
114	C₄⋊C₁₆	—	17	4	no	—
115	C₄⋊C₁₆	—	17	8	no	—
116	#346	—	17	8	no	—
117	C₄×C₁₆	—	17	4	no	—
118	C₂×C₁₇	#2183	1	4	yes	complemented
119	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
120	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
121	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
122	C₂²×Q₈	#3624	1	1	yes	characteristic
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	—	17	2	no	—
126	C₄×Q₈	#3624	1	4	yes	—
127	C₄×Q₈	C₂×#595	1	4	yes	—
128	C₂×C₄⋊C₄	#3624	1	2	yes	—
129	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
130	C₂×C₄²	C₂×#595	1	2	yes	—
131	C₂×C₄²	C₂×#595	1	1	yes	characteristic
132	C₂×C₁₆	—	17	2	no	—
133	C₂×C₁₆	—	17	4	no	—
134	C₂×C₁₆	—	17	2	no	—
135	C₂×C₁₆	—	17	4	no	—
136	C₂×C₁₆	—	34	2	no	—
137	C₂×C₁₆	—	34	4	no	—
138	C₄⋊C₈	—	17	8	no	—
139	C₄⋊C₈	—	17	4	no	—
140	C₄×C₈	—	17	2	no	—
141	C₄×C₈	—	17	4	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	17	4	no	—
144	C₁₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₂×#3624	1	4	yes	—
147	C₂²×C₄	C₂×#3624	1	2	yes	—
148	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
149	C₂²×C₄	C₂×#3624	1	2	yes	—
150	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
151	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
152	C₂×C₈	—	17	2	no	—
153	C₂×C₈	—	17	2	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	17	2	no	—
157	C₂×C₈	—	17	4	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	34	4	no	—
160	C₄⋊C₄	C₂×#3624	1	8	yes	—
161	C₄⋊C₄	C₂×#3624	1	4	yes	—
162	C₄²	C₂×#3624	1	4	yes	—
163	C₄²	C₂×#3624	1	2	yes	—
164	C₄²	C₂²×#595	1	4	yes	—
165	C₄²	C₂²×#595	1	2	yes	—
166	C₁₆	—	17	4	no	—
167	C₁₆	—	34	2	no	—
168	C₁₆	—	34	4	no	—
169	C₂³	C₂²×#3624	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#10587538	1	4	yes	—
175	C₂×C₄	#10587680	1	2	yes	—
176	C₂×C₄	Q₈×#595	1	2	yes	—
177	C₂×C₄	C₂²×#3624	1	4	yes	—
178	C₂×C₄	C₂²×#3624	1	2	yes	—
179	C₂×C₄	C₂²×#3624	1	4	yes	—
180	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
181	C₂×C₄	C₂²×#3624	1	2	yes	—
182	C₂×C₄	C₂³×#595	1	1	yes	characteristic
183	C₈	—	17	1	no	—
184	C₈	—	17	1	no	—
185	C₈	—	17	2	no	—
186	C₈	—	34	2	no	—
187	C₈	—	34	4	no	—
188	C₂²	Q₈×#3624	1	2	yes	—
189	C₂²	C₂×#10587538	1	2	yes	—
190	C₂²	#11763867	1	2	yes	—
191	C₂²	C₂³×#3624	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	Q₈×#3624	1	2	yes	—
196	C₄	C₂×#10587538	1	4	yes	—
197	C₄	C₂×Q₈×#595	1	1	yes	characteristic
198	C₄	C₂×#10587680	1	1	yes	characteristic
199	C₂	#14545199	1	4	yes	direct product factor
200	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
201	C₂	C₂²×#10587538	1	1	yes	characteristic
202	C₂	C₂×#11763867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂×C₄×Q₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×C₄×Q₈	#595
hypocenter	C₁₇	C₂×#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	direct (N × H)
33. C₂×#63216	Q₈	semidirect
66. #63216	C₂×Q₈	semidirect
118. C₂×C₁₇	#2183	semidirect
144. C₁₇	C₂×#2183	semidirect
199. C₂	#14545199	direct (N × H)

Maximal subgroups

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

subgroup	index
#14545199	2
C₂×C₄⋊#63216	2
C₂×#11763697	2
C₂×C₄×#63216	2
C₂×Q₈×#3622	2
C₂×#2183	17

Subgroups — C2×#14545199 #17978653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14545199 #17978653