Subgroups — C₂×#19151727 #26029178

Order 11392.

400 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#19151727	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667941	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667923	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667922	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₄	1	1	yes	characteristic
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₇₈	C₄	1	4	yes	—
13	C₄×Dic₁₇₈	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
15	C₂×#13083553	C₄	1	1	yes	characteristic
16	C₂×#13083553	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
19	#15667941	C₂²	1	4	yes	—
20	#15667941	C₂²	1	4	yes	—
21	#15667923	C₂²	1	4	yes	—
22	#15667922	C₂²	1	4	yes	—
23	C₈×Dic₈₉	C₂²	1	4	yes	—
24	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
28	C₂×Dic₁₇₈	—	2	4	no	—
29	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
30	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	Q₈	1	4	yes	complemented
36	C₂×#10623494	C₂³	1	2	yes	—
37	C₂×#10623494	C₂³	1	2	yes	—
38	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
39	#13083553	C₂×C₄	1	4	yes	—
40	#13083553	C₂×C₄	1	4	yes	—
41	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
42	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
43	C₄×Dic₈₉	C₂³	1	2	yes	—
44	C₄×Dic₈₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	C₂³	1	2	yes	—
51	C₂×C₈×C₈₉	C₂³	1	2	yes	—
52	C₄²×C₈₉	C₂×C₄	1	2	yes	—
53	C₄²×C₈₉	C₂³	1	2	yes	—
54	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₈₉	#41	1	2	yes	—
62	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₈₉	—	2	2	no	—
64	C₂×Dic₈₉	—	2	2	no	—
65	C₂×Dic₈₉	—	2	2	no	—
66	C₂×Dic₈₉	M₁₆	1	4	yes	—
67	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
68	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
69	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
70	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
71	Dic₁₇₈	—	2	8	no	—
72	C₈×C₈₉	—	2	2	no	—
73	C₈×C₈₉	—	2	2	no	—
74	#10623494	—	2	2	no	—
75	#10623494	C₂×Q₈	1	4	yes	complemented
76	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
77	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
78	C₂²×C₈₉	#131	1	2	yes	—
79	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	C₄×Q₈	1	2	yes	—
83	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
85	Dic₈₉	—	2	2	no	—
86	Dic₈₉	—	2	2	no	—
87	Dic₈₉	—	2	2	no	—
88	Dic₈₉	C₂×M₁₆	1	4	yes	—
89	C₂×C₈₉	#446	1	4	yes	complemented
90	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
92	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
93	C₂×#446	—	89	1	no	maximal, Sylow
94	C₈₉	C₂×#446	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄×Q₈	—	89	1	no	—
96	#446	—	89	8	no	—
97	C₂×C₄⋊C₈	—	89	1	no	—
98	C₂×C₄⋊C₈	—	89	1	no	—
99	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
100	C₂×#97	—	89	1	no	—
101	C₂×#97	—	89	1	no	—
102	C₂×C₄×C₈	—	89	1	no	—
103	C₂²×Q₈	—	89	1	no	—
104	C₂²×C₈	—	89	1	no	—
105	C₂²×C₈	—	89	1	no	—
106	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
108	C₄×Q₈	—	89	4	no	—
109	C₄×Q₈	—	89	4	no	—
110	C₂×C₄⋊C₄	—	89	1	no	—
111	C₂×C₄⋊C₄	—	89	1	no	—
112	C₂×C₄⋊C₄	—	89	1	no	—
113	C₂×C₄²	—	89	1	no	—
114	C₂×C₄²	—	89	1	no	—
115	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
116	C₄⋊C₈	—	89	4	no	—
117	C₄⋊C₈	—	89	4	no	—
118	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
119	#97	—	89	4	no	—
120	#97	—	89	4	no	—
121	C₄×C₈	—	89	4	no	—
122	C₂×Q₈	—	89	4	no	—
123	C₂×Q₈	—	178	4	no	—
124	C₂²×C₄	—	89	1	no	—
125	C₂²×C₄	—	89	1	no	—
126	C₂²×C₄	—	89	1	no	—
127	C₂²×C₄	—	89	1	no	—
128	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	89	4	no	—
134	C₂×C₈	—	89	2	no	—
135	C₂×C₈	—	89	2	no	—
136	C₂×C₈	—	178	2	no	—
137	C₂×C₈	C₂²×D₈₉	1	2	yes	—
138	C₂×C₈	C₂²×D₈₉	1	2	yes	—
139	C₄⋊C₄	—	89	4	no	—
140	C₄⋊C₄	—	89	4	no	—
141	C₄⋊C₄	—	89	4	no	—
142	C₄²	—	89	2	no	—
143	C₄²	—	89	2	no	—
144	C₄²	—	89	2	no	—
145	C₄²	—	89	2	no	—
146	C₄²	C₄×D₈₉	1	2	yes	—
147	C₄²	C₂²×D₈₉	1	2	yes	—
148	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
149	Q₈	—	178	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	89	4	no	—
153	C₂×C₄	—	89	2	no	—
154	C₂×C₄	—	89	2	no	—
155	C₂×C₄	—	89	2	no	—
156	C₂×C₄	—	89	2	no	—
157	C₂×C₄	—	178	2	no	—
158	C₂×C₄	—	178	2	no	—
159	C₂×C₄	—	178	2	no	—
160	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
163	C₂×C₄	Q₈×D₈₉	1	2	yes	—
164	C₂×C₄	#13083573	1	2	yes	—
165	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	89	4	no	—
169	C₈	—	178	2	no	—
170	C₂²	#15667977	1	2	yes	—
171	C₂²	M₁₆×D₈₉	1	2	yes	—
172	C₂²	#15668018	1	2	yes	—
173	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	89	4	no	—
177	C₄	—	178	2	no	—
178	C₄	—	178	2	no	—
179	C₄	—	178	2	no	—
180	C₄	#15667977	1	2	yes	—
181	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
182	C₄	C₂×#13083573	1	1	yes	characteristic
183	C₂	#19151727	1	4	yes	direct product factor
184	C₂	C₂×#15667977	1	1	yes	characteristic
185	C₂	C₂×M₁₆×D₈₉	1	1	yes	characteristic
186	C₂	C₂×#15668018	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	C₂×#446
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #19151727	C₂	direct (N × H)
35. C₂×#10623494	Q₈	semidirect
75. #10623494	C₂×Q₈	semidirect
89. C₂×C₈₉	#446	semidirect
94. C₈₉	C₂×#446	semidirect
183. C₂	#19151727	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×C₄×Dic₁₇₈	2
#19151727	2
C₂×#15667941	2
C₂×#15667923	2
C₂×#15667922	2
C₂×C₈×Dic₈₉	2
C₂×C₄⋊C₈×C₈₉	2
C₂×#446	89

Subgroups — C2×#19151727 #26029178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151727 #26029178