Subgroups — C₂×#26012144 #36367022

Order 22144.

1632 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14583827	C₂	1	2	yes	maximal, complemented
2	#26012144	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19149664	C₂	1	4	yes	maximal, complemented
4	C₂×#19149655	C₂	1	2	yes	maximal, complemented
5	C₂×#19149654	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₆₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₆₉₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄×D₁₇₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14583827	—	2	16	no	—
11	C₂×#14583827	C₂²	1	16	yes	complemented
12	C₂²×D₆₉₂	C₂²	1	1	yes	characteristic
13	C₂²×D₆₉₂	C₂²	1	4	yes	—
14	C₂²×D₆₉₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₇₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₇₃	C₂²	1	4	yes	—
17	C₂²×C₄×D₁₇₃	C₂²	1	2	yes	—
18	#19149664	C₂²	1	32	yes	complemented
19	#19149655	C₂²	1	16	yes	complemented
20	#19149654	C₂²	1	16	yes	complemented
21	C₄×D₆₉₂	C₂²	1	8	yes	complemented
22	C₄×D₆₉₂	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇₃	C₂²	1	1	yes	characteristic
25	C₂×#14583805	C₂²	1	4	yes	—
26	C₄⋊C₄×D₁₇₃	C₂²	1	16	yes	complemented
27	C₂×#14583803	C₂²	1	2	yes	—
28	C₂×#14583803	C₂²	1	2	yes	—
29	C₂×#14583803	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₇₃	C₂²	1	2	yes	—
31	C₄⋊Q₈×C₁₇₃	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
35	C₂³×D₁₇₃	C₂³	1	2	yes	—
36	C₂³×D₁₇₃	C₂³	1	2	yes	—
37	#14583827	—	2	64	no	—
38	C₂×D₆₉₂	—	2	16	no	—
39	C₂×D₆₉₂	—	2	4	no	—
40	C₂×D₆₉₂	D₄	1	8	yes	complemented
41	C₂×D₆₉₂	C₂³	1	4	yes	—
42	C₂×D₆₉₂	C₂³	1	16	yes	—
43	C₂×D₆₉₂	C₂³	1	4	yes	—
44	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
46	C₂×C₄×D₁₇₃	—	2	16	no	—
47	C₂×C₄×D₁₇₃	—	2	8	no	—
48	C₂×C₄×D₁₇₃	—	2	8	no	—
49	C₂×C₄×D₁₇₃	C₂³	1	8	yes	—
50	C₂×C₄×D₁₇₃	C₂³	1	16	yes	—
51	C₂×C₄×D₁₇₃	C₂³	1	8	yes	—
52	C₄⋊Dic₁₇₃	C₂³	1	4	yes	—
53	C₄⋊Dic₁₇₃	C₂³	1	4	yes	—
54	#14583805	C₂³	1	16	yes	—
55	#14583803	C₂³	1	8	yes	—
56	#14583803	C₂³	1	8	yes	—
57	#14583803	C₂³	1	16	yes	—
58	C₂×Q₈×C₁₇₃	—	2	8	no	—
59	C₂×Q₈×C₁₇₃	C₂³	1	8	yes	—
60	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇₃	C₂³	1	8	yes	—
65	C₄⋊C₄×C₁₇₃	C₂³	1	8	yes	—
66	C₄²×C₁₇₃	C₂³	1	4	yes	—
67	C₂²×D₁₇₃	—	2	4	no	—
68	C₂²×D₁₇₃	—	2	4	no	—
69	C₂²×D₁₇₃	—	2	4	no	—
70	C₂²×D₁₇₃	—	2	4	no	—
71	C₂²×D₁₇₃	—	2	4	no	—
72	C₂²×D₁₇₃	C₂×D₄	1	8	yes	—
73	C₂²×D₁₇₃	C₂⁴	1	4	yes	—
74	C₂²×D₁₇₃	C₂⁴	1	4	yes	—
75	D₆₉₂	—	2	32	no	—
76	D₆₉₂	—	2	8	no	—
77	D₆₉₂	C₂×D₄	1	16	yes	complemented
78	C₂×Dic₁₇₃	—	2	4	no	—
79	C₂×Dic₁₇₃	—	2	4	no	—
80	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
81	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
82	C₄×D₁₇₃	—	2	32	no	—
83	C₄×D₁₇₃	—	2	16	no	—
84	C₄×D₁₇₃	—	2	16	no	—
85	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
86	Q₈×C₁₇₃	—	2	16	no	—
87	C₂×C₄×C₁₇₃	—	2	8	no	—
88	C₂×C₄×C₁₇₃	—	2	2	no	—
89	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₁₇₃	#41	1	4	yes	complemented
91	C₂×C₄×C₁₇₃	C₂⁴	1	8	yes	—
92	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
93	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
94	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
95	C₂²×C₁₇₃	C₂²×D₄	1	2	yes	—
96	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
97	C₂²×C₁₇₃	2₋¹⁺⁴	1	2	yes	—
98	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₇₃	—	2	16	no	—
100	C₂×D₁₇₃	—	2	4	no	—
101	C₂×D₁₇₃	—	2	4	no	—
102	C₂×D₁₇₃	—	2	4	no	—
103	C₂×D₁₇₃	—	2	4	no	—
104	C₂×D₁₇₃	—	2	4	no	—
105	C₂×D₁₇₃	—	4	8	no	—
106	C₂×D₁₇₃	C₂²×D₄	1	8	yes	—
107	C₄×C₁₇₃	—	2	8	no	—
108	C₄×C₁₇₃	—	2	2	no	—
109	C₄×C₁₇₃	C₂²×D₄	1	4	yes	—
110	C₄×C₁₇₃	C₂×#41	1	4	yes	complemented
111	Dic₁₇₃	—	2	4	no	—
112	Dic₁₇₃	—	2	4	no	—
113	C₂×C₁₇₃	#547	1	4	yes	complemented
114	C₂×C₁₇₃	C₂³×D₄	1	1	yes	characteristic
115	C₂×C₁₇₃	C₂²×#41	1	1	yes	characteristic
116	C₂×C₁₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
117	D₁₇₃	—	2	8	no	—
118	D₁₇₃	—	4	4	no	—
119	C₁₇₃	C₂×#547	1	1	yes	characteristic, complemented, Sylow
120	C₂×#547	—	173	1	no	maximal, Sylow
121	C₂²×#41	—	173	2	no	—
122	#547	—	173	16	no	—
123	C₂×C₄⋊Q₈	D₁₇₃	1	1	yes	characteristic, complemented
124	C₂×#123	—	173	4	no	—
125	C₂×C₂²⋊Q₈	—	173	2	no	—
126	C₂×#121	—	173	2	no	—
127	C₂×C₄×D₄	—	173	1	no	—
128	C₂×C₄×D₄	—	173	1	no	—
129	C₂²×C₄⋊C₄	—	173	2	no	—
130	C₂×#41	—	346	16	no	—
131	C₂×#41	—	173	16	no	—
132	C₂²×Q₈	C₂×D₁₇₃	1	2	yes	—
133	C₂²×D₄	—	173	1	no	—
134	C₂²×D₄	—	173	4	no	—
135	C₂²×D₄	—	173	1	no	—
136	C₂³×C₄	—	173	2	no	—
137	C₂³×C₄	—	173	4	no	—
138	C₂³×C₄	—	173	2	no	—
139	C₄⋊Q₈	C₂×D₁₇₃	1	8	yes	complemented
140	#123	—	173	32	no	—
141	C₂²⋊Q₈	—	173	16	no	—
142	#121	—	173	16	no	—
143	C₄×D₄	—	173	8	no	—
144	C₄×D₄	—	173	8	no	—
145	C₂×C₄⋊C₄	—	173	1	no	—
146	C₂×C₄⋊C₄	—	173	4	no	—
147	C₂×C₄⋊C₄	—	173	1	no	—
148	C₂×C₄⋊C₄	—	173	16	no	—
149	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
150	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
151	C₂×C₂²⋊C₄	—	173	2	no	—
152	C₂×C₂²⋊C₄	—	173	2	no	—
153	C₂×C₂²⋊C₄	—	173	4	no	—
154	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
155	C₂⁴	—	173	2	no	—
156	C₂⁴	—	173	2	no	—
157	#41	—	346	64	no	—
158	C₂×Q₈	—	2	8	no	—
159	C₂×Q₈	C₂²×D₁₇₃	1	8	yes	—
160	C₂×D₄	—	346	16	no	—
161	C₂×D₄	—	346	4	no	—
162	C₂×D₄	—	173	8	no	—
163	C₂×D₄	—	173	4	no	—
164	C₂×D₄	—	173	16	no	—
165	C₂×D₄	—	173	4	no	—
166	C₂²×C₄	—	346	16	no	—
167	C₂²×C₄	—	346	8	no	—
168	C₂²×C₄	—	346	8	no	—
169	C₂²×C₄	—	173	2	no	—
170	C₂²×C₄	—	173	2	no	—
171	C₂²×C₄	—	173	8	no	—
172	C₂²×C₄	—	173	16	no	—
173	C₂²×C₄	—	173	8	no	—
174	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
176	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
178	C₄⋊C₄	—	173	4	no	—
179	C₄⋊C₄	—	173	16	no	—
180	C₄⋊C₄	—	173	4	no	—
181	C₄⋊C₄	C₂²×D₁₇₃	1	8	yes	—
182	C₄⋊C₄	C₂²×D₁₇₃	1	8	yes	—
183	C₂²⋊C₄	—	173	8	no	—
184	C₂²⋊C₄	—	173	8	no	—
185	C₂²⋊C₄	—	173	16	no	—
186	C₄²	C₂²×D₁₇₃	1	4	yes	—
187	C₂³	—	346	4	no	—
188	C₂³	—	346	4	no	—
189	C₂³	—	346	4	no	—
190	C₂³	—	346	4	no	—
191	C₂³	—	346	4	no	—
192	C₂³	—	173	4	no	—
193	C₂³	—	173	8	no	—
194	C₂³	—	173	4	no	—
195	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
196	Q₈	—	2	16	no	—
197	D₄	—	346	32	no	—
198	D₄	—	346	8	no	—
199	D₄	—	173	16	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	346	32	no	—
203	C₂×C₄	—	346	4	no	—
204	C₂×C₄	—	346	4	no	—
205	C₂×C₄	—	346	16	no	—
206	C₂×C₄	—	346	16	no	—
207	C₂×C₄	—	173	4	no	—
208	C₂×C₄	—	173	4	no	—
209	C₂×C₄	D₄×D₁₇₃	1	4	yes	—
210	C₂×C₄	#14583827	1	4	yes	complemented
211	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
212	C₂×C₄	C₂³×D₁₇₃	1	8	yes	—
213	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
214	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
215	C₂²	—	346	4	no	—
216	C₂²	—	346	4	no	—
217	C₂²	—	346	16	no	—
218	C₂²	—	346	4	no	—
219	C₂²	—	346	4	no	—
220	C₂²	—	346	4	no	—
221	C₂²	—	173	8	no	—
222	C₂²	—	692	8	no	—
223	C₂²	C₂×D₄×D₁₇₃	1	2	yes	—
224	C₂²	C₂×#14583827	1	2	yes	—
225	C₂²	#19149727	1	2	yes	—
226	C₂²	C₂⁴×D₁₇₃	1	1	yes	characteristic
227	C₄	—	2	8	no	—
228	C₄	—	2	2	no	—
229	C₄	—	346	4	no	—
230	C₄	—	346	4	no	—
231	C₄	C₂×D₄×D₁₇₃	1	4	yes	—
232	C₄	C₂×#14583827	1	4	yes	complemented
233	C₂	—	346	8	no	—
234	C₂	—	692	4	no	—
235	C₂	#26012144	1	4	yes	direct product factor
236	C₂	C₂²×D₄×D₁₇₃	1	1	yes	characteristic
237	C₂	C₂²×#14583827	1	1	yes	characteristic
238	C₂	C₂×#19149727	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₇₃
commutator, cosocle	C₂²×C₁₇₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₇₃
Fitting	C₂×C₄⋊Q₈×C₁₇₃	C₂
socle	C₂³×C₁₇₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₇₃
hypocenter	C₁₇₃	C₂×#547
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14583827	C₂	semidirect
2. #26012144	C₂	direct (N × H)
3. C₂×#19149664	C₂	semidirect
4. C₂×#19149655	C₂	semidirect
5. C₂×#19149654	C₂	semidirect
6. C₂×C₄×D₆₉₂	C₂	semidirect
7. C₂×C₄×D₆₉₂	C₂	semidirect
8. C₂×C₄⋊C₄×D₁₇₃	C₂	semidirect
9. C₂×C₄⋊Q₈×C₁₇₃	C₂	semidirect
11. C₂×#14583827	C₂²	semidirect
18. #19149664	C₂²	semidirect
19. #19149655	C₂²	semidirect
20. #19149654	C₂²	semidirect
21. C₄×D₆₉₂	C₂²	semidirect
22. C₄×D₆₉₂	C₂²	semidirect
26. C₄⋊C₄×D₁₇₃	C₂²	semidirect
31. C₄⋊Q₈×C₁₇₃	C₂²	semidirect
40. C₂×D₆₉₂	D₄	semidirect
77. D₆₉₂	C₂×D₄	semidirect
90. C₂×C₄×C₁₇₃	#41	semidirect
110. C₄×C₁₇₃	C₂×#41	semidirect
113. C₂×C₁₇₃	#547	semidirect
119. C₁₇₃	C₂×#547	semidirect
123. C₂×C₄⋊Q₈	D₁₇₃	semidirect
139. C₄⋊Q₈	C₂×D₁₇₃	semidirect
210. C₂×C₄	#14583827	semidirect
232. C₄	C₂×#14583827	semidirect
235. C₂	#26012144	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×#14583827	2
#26012144	2
C₂×#19149664	2
C₂×#19149655	2
C₂×#19149654	2
C₂×C₄×D₆₉₂	2
C₂×C₄⋊C₄×D₁₇₃	2
C₂×C₄⋊Q₈×C₁₇₃	2
C₂×#547	173

Subgroups — C2×#26012144 #36367022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26012144 #36367022