Subgroups — C₂×#26018323 #36466005

Order 22784.

262 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018323	C₂	1	4	yes	maximal, direct product factor
4	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₄	1	1	yes	characteristic
6	C₂²×#13083549	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#10623494	—	2	1	no	—
8	C₄×#13083549	C₄	1	2	yes	—
9	C₄×#13083549	C₂²	1	2	yes	—
10	#19151351	—	2	1	no	—
11	#19151351	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#335×C₈₉	—	2	1	no	—
15	#335×C₈₉	C₂²	1	2	yes	—
16	C₂²×#10623494	—	4	1	no	—
17	C₂×#13083549	—	2	2	no	—
18	C₂×#13083549	C₈	1	4	yes	complemented
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₂×#13083549	C₂×C₄	1	2	yes	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	4	1	no	—
23	C₂²×C₈×C₈₉	—	4	1	no	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₄⋊C₈×C₈₉	—	4	1	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	8	2	no	—
35	#13083549	—	2	2	no	—
36	#13083549	C₂×C₈	1	4	yes	complemented
37	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	4	2	no	—
42	C₂×C₈×C₈₉	—	8	2	no	—
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₈₉	D₈	1	1	yes	characteristic
51	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₈₉	#107	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#108	1	2	yes	—
60	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
61	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₈×C₈₉	—	2	1	no	—
66	C₈×C₈₉	—	2	1	no	—
67	C₈×C₈₉	—	8	2	no	—
68	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
69	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
71	#10623494	—	8	2	no	—
72	C₂²×C₈₉	#335	1	2	yes	—
73	C₂²×C₈₉	#366	1	2	yes	—
74	C₂²×C₈₉	#368	1	2	yes	—
75	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	#335	1	2	yes	—
79	C₄×C₈₉	#366	1	2	yes	—
80	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
82	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
83	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
84	C₂×#1371	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	#1371	1	4	yes	complemented
86	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
87	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
88	C₂×C₈₉	C₂×#368	1	1	yes	characteristic
89	C₂×C₄×C₁₆	—	89	1	no	—
90	C₂×#335	—	89	1	no	—
91	C₂×#335	D₈₉	1	1	yes	characteristic
92	#1371	—	89	4	no	—
93	C₈₉	C₂×#1371	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₁₆	—	89	1	no	—
95	C₂²×C₁₆	—	89	1	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₄⋊C₈	—	178	1	no	—
98	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄×C₁₆	—	89	2	no	—
100	C₄×C₁₆	—	89	2	no	—
101	#335	—	2	1	no	—
102	#335	—	89	2	no	—
103	#335	—	178	1	no	—
104	#335	C₂×D₈₉	1	2	yes	—
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	—	356	1	no	—
107	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂×C₁₆	—	89	4	no	—
111	C₂×C₁₆	—	89	2	no	—
112	C₂×C₁₆	—	89	2	no	—
113	C₂×C₁₆	—	178	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	356	1	no	—
117	C₄⋊C₈	—	178	2	no	—
118	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	C₂²×C₄	#13083553	1	1	yes	characteristic
123	C₂²×C₄	#13083559	1	1	yes	characteristic
124	C₂²×C₄	#13083564	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	356	2	no	—
129	C₂×C₈	—	712	2	no	—
130	C₂×C₈	#13083553	1	1	yes	characteristic
131	C₂×C₈	#13083553	1	1	yes	characteristic
132	C₂×C₈	C₈×D₈₉	1	2	yes	—
133	C₂×C₈	#13083557	1	2	yes	—
134	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
136	C₄²	#13083553	1	1	yes	characteristic
137	C₄²	#13083559	1	1	yes	characteristic
138	C₄²	#13083564	1	1	yes	characteristic
139	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₁₆	—	89	4	no	—
141	C₁₆	—	178	2	no	—
142	C₂³	#15667945	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	#15667941	1	2	yes	—
145	C₂×C₄	#15667945	1	1	yes	characteristic
146	C₂×C₄	#15667945	1	1	yes	characteristic
147	C₂×C₄	#15667945	1	1	yes	characteristic
148	C₂×C₄	#15667947	1	2	yes	—
149	C₂×C₄	#15667953	1	2	yes	—
150	C₂×C₄	#15667958	1	2	yes	—
151	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
152	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
153	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	8	2	no	—
157	C₈	—	712	2	no	—
158	C₈	#15667941	1	2	yes	—
159	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
160	C₈	C₂×#13083557	1	1	yes	characteristic
161	C₂²	#19151350	1	2	yes	—
162	C₂²	#19151423	1	2	yes	—
163	C₂²	#19151427	1	2	yes	—
164	C₂²	C₂×#15667945	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	#19151350	1	2	yes	—
168	C₄	#19151423	1	2	yes	—
169	C₄	C₂×#15667941	1	1	yes	characteristic
170	C₄	C₂×#15667947	1	1	yes	characteristic
171	C₄	C₂×#15667953	1	1	yes	characteristic
172	C₄	C₂×#15667958	1	1	yes	characteristic
173	C₂	#26018323	1	4	yes	direct product factor
174	C₂	C₂×#19151350	1	1	yes	characteristic
175	C₂	C₂×#19151423	1	1	yes	characteristic
176	C₂	C₂×#19151427	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#13083559
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#107
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	C₂×#1371
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #26018323	C₂	direct (N × H)
18. C₂×#13083549	C₈	semidirect
36. #13083549	C₂×C₈	semidirect
85. C₂×C₈₉	#1371	semidirect
93. C₈₉	C₂×#1371	semidirect
173. C₂	#26018323	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×#13083549	2
C₂×#19151351	2
#26018323	2
C₂×#335×C₈₉	2
C₂×#1371	89

Subgroups — C2×#26018323 #36466005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018323 #36466005