Subgroups — C₂×#26011389 #36356014

Order 22144.

800 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	C₂²×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14583805	C₂	1	2	yes	maximal
4	#26011389	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19149586	C₂	1	2	yes	maximal
6	C₂²×C₄²×C₁₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₇₃	C₂²	1	2	yes	—
8	C₂³×Dic₁₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₇₃	—	2	4	no	—
10	C₂×C₄⋊Dic₁₇₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₇₃	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₇₃	C₂²	1	4	yes	—
13	C₂×#14583805	—	2	8	no	—
14	C₂×#14583805	C₂²	1	8	yes	—
15	#19149586	C₂²	1	8	yes	—
16	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₇₃	—	2	4	no	—
20	C₂×C₄²×C₁₇₃	C₂²	1	4	yes	—
21	C₂²×Dic₁₇₃	—	2	4	no	—
22	C₂²×Dic₁₇₃	—	2	4	no	—
23	C₂²×Dic₁₇₃	—	2	4	no	—
24	C₂²×Dic₁₇₃	—	2	4	no	—
25	C₂²×Dic₁₇₃	—	2	4	no	—
26	C₂²×Dic₁₇₃	C₂×C₄	1	8	yes	—
27	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
28	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
29	C₄⋊Dic₁₇₃	—	2	8	no	—
30	C₄⋊Dic₁₇₃	C₂×C₄	1	16	yes	complemented
31	#14583805	—	2	16	no	—
32	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇₃	—	2	2	no	—
34	C₂²×C₄×C₁₇₃	—	2	2	no	—
35	C₂²×C₄×C₁₇₃	—	2	2	no	—
36	C₂²×C₄×C₁₇₃	—	2	2	no	—
37	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₇₃	D₄	1	4	yes	—
39	C₂²×C₄×C₁₇₃	D₄	1	4	yes	—
40	C₂²×C₄×C₁₇₃	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₇₃	Q₈	1	4	yes	—
42	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
45	C₄²×C₁₇₃	—	2	8	no	—
46	C₂×Dic₁₇₃	—	2	16	no	—
47	C₂×Dic₁₇₃	—	2	4	no	—
48	C₂×Dic₁₇₃	—	2	4	no	—
49	C₂×Dic₁₇₃	—	2	4	no	—
50	C₂×Dic₁₇₃	—	2	4	no	—
51	C₂×Dic₁₇₃	—	2	4	no	—
52	C₂×Dic₁₇₃	—	4	8	no	—
53	C₂×Dic₁₇₃	C₂²×C₄	1	8	yes	—
54	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
55	C₂³×C₁₇₃	C₂×D₄	1	2	yes	—
56	C₂³×C₁₇₃	C₂×D₄	1	2	yes	—
57	C₂³×C₁₇₃	C₂×Q₈	1	2	yes	—
58	C₂³×C₁₇₃	C₂×Q₈	1	2	yes	—
59	C₂³×C₁₇₃	#41	1	2	yes	—
60	C₂³×C₁₇₃	#41	1	2	yes	—
61	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₁₇₃	—	2	8	no	—
63	C₂×C₄×C₁₇₃	—	2	8	no	—
64	C₂×C₄×C₁₇₃	—	2	2	no	—
65	C₂×C₄×C₁₇₃	—	2	2	no	—
66	C₂×C₄×C₁₇₃	—	2	2	no	—
67	C₂×C₄×C₁₇₃	—	2	2	no	—
68	C₂×C₄×C₁₇₃	C₄⋊C₄	1	16	yes	complemented
69	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₁₇₃	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₁₇₃	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₁₇₃	C₄×D₄	1	4	yes	—
76	C₂²×C₁₇₃	C₄×Q₈	1	4	yes	—
77	C₂²×C₁₇₃	#121	1	4	yes	—
78	C₂²×C₁₇₃	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₁₇₃	#125	1	4	yes	—
80	C₂²×C₁₇₃	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₇₃	—	2	4	no	—
89	C₄×C₁₇₃	—	2	4	no	—
90	C₄×C₁₇₃	C₂×C₄⋊C₄	1	8	yes	complemented
91	Dic₁₇₃	—	2	8	no	—
92	Dic₁₇₃	—	4	4	no	—
93	C₂×C₁₇₃	#389	1	8	yes	complemented
94	C₂×C₁₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂×C₁₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₇₃	C₂×#121	1	1	yes	characteristic
98	C₂×C₁₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₇₃	C₂×#125	1	1	yes	characteristic
100	C₂×C₁₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₁₇₃	C₂×#389	1	1	yes	characteristic, complemented, Sylow
102	C₂×#389	—	173	1	no	maximal, Sylow
103	C₂²×C₄⋊C₄	—	173	2	no	—
104	C₂²×C₄⋊C₄	—	173	1	no	—
105	C₂²×C₄⋊C₄	—	173	1	no	—
106	C₂²×C₄²	D₁₇₃	1	1	yes	characteristic
107	#389	—	173	8	no	—
108	C₂×#95	—	173	2	no	—
109	C₂³×C₄	—	173	2	no	—
110	C₂³×C₄	—	173	2	no	—
111	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
112	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
113	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	346	4	no	—
115	C₂×C₄⋊C₄	—	346	8	no	—
116	C₂×C₄⋊C₄	—	173	8	no	—
117	C₂×C₄⋊C₄	—	173	4	no	—
118	C₂×C₄⋊C₄	—	173	8	no	—
119	C₂×C₄⋊C₄	—	173	4	no	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂×D₁₇₃	1	4	yes	—
122	#95	—	173	8	no	—
123	C₂⁴	C₂²×D₁₇₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	346	4	no	—
129	C₂²×C₄	—	346	4	no	—
130	C₂²×C₄	—	346	4	no	—
131	C₂²×C₄	—	346	4	no	—
132	C₂²×C₄	—	346	4	no	—
133	C₂²×C₄	—	173	4	no	—
134	C₂²×C₄	—	173	8	no	—
135	C₂²×C₄	—	173	4	no	—
136	C₂²×C₄	C₄×D₁₇₃	1	4	yes	—
137	C₂²×C₄	C₂²⋊D₁₇₃	1	4	yes	—
138	C₂²×C₄	D₆₉₂	1	4	yes	—
139	C₂²×C₄	Dic₃₄₆	1	4	yes	—
140	C₂²×C₄	Dic₃₄₆	1	4	yes	—
141	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
142	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
143	C₂²×C₄	C₂²×D₁₇₃	1	2	yes	—
144	C₄⋊C₄	—	346	8	no	—
145	C₄⋊C₄	—	346	16	no	—
146	C₄⋊C₄	—	173	16	no	—
147	C₄²	—	2	8	no	—
148	C₂³	C₂×C₄×D₁₇₃	1	2	yes	—
149	C₂³	C₂×C₂²⋊D₁₇₃	1	2	yes	—
150	C₂³	C₂×D₆₉₂	1	2	yes	—
151	C₂³	C₂×Dic₃₄₆	1	2	yes	—
152	C₂³	C₂×Dic₃₄₆	1	2	yes	—
153	C₂³	#14583826	1	2	yes	—
154	C₂³	#14583826	1	2	yes	—
155	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	346	16	no	—
163	C₂×C₄	—	346	4	no	—
164	C₂×C₄	—	346	4	no	—
165	C₂×C₄	—	346	4	no	—
166	C₂×C₄	—	346	4	no	—
167	C₂×C₄	—	346	4	no	—
168	C₂×C₄	—	173	8	no	—
169	C₂×C₄	—	692	8	no	—
170	C₂×C₄	#14583805	1	16	yes	complemented
171	C₂×C₄	C₂×C₄×D₁₇₃	1	4	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	4	yes	—
173	C₂×C₄	C₂×D₆₉₂	1	4	yes	—
174	C₂×C₄	C₂×Dic₃₄₆	1	4	yes	—
175	C₂×C₄	C₂×Dic₃₄₆	1	4	yes	—
176	C₂²	C₂×#14583805	1	4	yes	—
177	C₂²	C₄×D₆₉₂	1	4	yes	—
178	C₂²	C₄×Dic₃₄₆	1	4	yes	—
179	C₂²	#19149653	1	4	yes	—
180	C₂²	#19149658	1	4	yes	—
181	C₂²	#19149671	1	4	yes	—
182	C₂²	#19149679	1	4	yes	—
183	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
184	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
185	C₂²	C₂²×D₆₉₂	1	1	yes	characteristic
186	C₂²	C₂²×Dic₃₄₆	1	1	yes	characteristic
187	C₂²	C₂²×Dic₃₄₆	1	1	yes	characteristic
188	C₂²	C₂×#14583826	1	1	yes	characteristic
189	C₂²	C₂×#14583826	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	346	8	no	—
193	C₄	—	692	4	no	—
194	C₄	C₂×#14583805	1	8	yes	complemented
195	C₂	#26011389	1	8	yes	direct product factor
196	C₂	C₂²×#14583805	1	1	yes	characteristic
197	C₂	C₂×C₄×D₆₉₂	1	1	yes	characteristic
198	C₂	C₂×C₄×Dic₃₄₆	1	1	yes	characteristic
199	C₂	C₂×#19149653	1	1	yes	characteristic
200	C₂	C₂×#19149658	1	1	yes	characteristic
201	C₂	C₂×#19149671	1	1	yes	characteristic
202	C₂	C₂×#19149679	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₇₃
commutator	C₂²×C₁₇₃	C₂³×C₄
Frattini	C₂³	C₂³×D₁₇₃
Fitting	C₂²×C₄²×C₁₇₃	C₂
socle, prime radical	C₂⁴×C₁₇₃	C₂³
cosocle	C₂³×C₁₇₃	C₂⁴
hypercenter	C₂²×C₄²	D₁₇₃
hypocenter	C₁₇₃	C₂×#389
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #26011389	C₂	direct (N × H)
10. C₂×C₄⋊Dic₁₇₃	C₄	semidirect
30. C₄⋊Dic₁₇₃	C₂×C₄	semidirect
68. C₂×C₄×C₁₇₃	C₄⋊C₄	semidirect
90. C₄×C₁₇₃	C₂×C₄⋊C₄	semidirect
93. C₂×C₁₇₃	#389	semidirect
101. C₁₇₃	C₂×#389	semidirect
170. C₂×C₄	#14583805	semidirect
194. C₄	C₂×#14583805	semidirect
195. C₂	#26011389	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₇₃	2
C₂²×#14583805	2
#26011389	2
C₂×#19149586	2
C₂²×C₄²×C₁₇₃	2
C₂×#389	173

Subgroups — C2×#26011389 #36356014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26011389 #36356014