Subgroups — #36356423

Order 22144.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011377	C₂	1	2	yes	maximal, complemented
2	#26011377	C₂	1	2	yes	maximal, complemented
3	#26011366	C₂	1	2	yes	maximal
4	#26011359	C₂	1	1	yes	maximal, characteristic, complemented
5	#26011357	C₂	1	1	yes	maximal, characteristic, complemented
6	#26011354	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×Dic₁₇₃	C₂	1	2	yes	maximal
9	#26011337	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19149586	C₂	1	1	yes	maximal, characteristic
11	C₂×#117×C₁₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₇₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₇₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₇₃	C₂²	1	2	yes	—
18	#19149586	C₂²	1	1	yes	characteristic
19	#19149586	C₂²	1	1	yes	characteristic
20	#19149586	C₂²	1	2	yes	—
21	#19149586	C₂²	1	1	yes	characteristic
22	#19149586	C₂²	1	2	yes	—
23	#19149586	C₂²	1	2	yes	—
24	#19149586	C₂²	1	2	yes	—
25	#19149586	C₂²	1	1	yes	characteristic
26	#19149586	C₂²	1	2	yes	—
27	#19149586	C₂²	1	2	yes	—
28	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
29	#117×C₁₇₃	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₇₃	—	2	4	no	—
37	C₂²×Dic₁₇₃	—	2	4	no	—
38	C₂²×Dic₁₇₃	—	4	2	no	—
39	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
41	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
42	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
43	C₄⋊Dic₁₇₃	—	2	4	no	—
44	C₄⋊Dic₁₇₃	—	4	2	no	—
45	C₂²⋊Dic₁₇₃	—	2	4	no	—
46	C₄×Dic₁₇₃	—	2	4	no	—
47	C₄×Dic₁₇₃	—	4	2	no	—
48	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₇₃	—	2	4	no	—
50	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
60	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
61	C₄²×C₁₇₃	C₂×C₄	1	4	yes	complemented
62	C₄²×C₁₇₃	C₂×C₄	1	4	yes	complemented
63	C₂×Dic₁₇₃	—	2	2	no	—
64	C₂×Dic₁₇₃	—	2	2	no	—
65	C₂×Dic₁₇₃	—	2	2	no	—
66	C₂×Dic₁₇₃	—	2	2	no	—
67	C₂×Dic₁₇₃	—	2	2	no	—
68	C₂×Dic₁₇₃	—	2	2	no	—
69	C₂×Dic₁₇₃	—	2	2	no	—
70	C₂×Dic₁₇₃	—	2	2	no	—
71	C₂×Dic₁₇₃	—	2	2	no	—
72	C₂×Dic₁₇₃	—	2	2	no	—
73	C₂×Dic₁₇₃	—	2	2	no	—
74	C₂×Dic₁₇₃	—	4	8	no	—
75	C₂×Dic₁₇₃	#41	1	4	yes	—
76	C₂³×C₁₇₃	—	2	2	no	—
77	C₂³×C₁₇₃	—	2	1	no	—
78	C₂³×C₁₇₃	—	2	1	no	—
79	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₇₃	#41	1	4	yes	—
81	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₁₇₃	—	2	8	no	—
83	C₂×C₄×C₁₇₃	—	2	2	no	—
84	C₂×C₄×C₁₇₃	—	2	2	no	—
85	C₂×C₄×C₁₇₃	—	2	2	no	—
86	C₂×C₄×C₁₇₃	—	2	2	no	—
87	C₂×C₄×C₁₇₃	—	2	1	no	—
88	C₂×C₄×C₁₇₃	—	2	1	no	—
89	C₂×C₄×C₁₇₃	—	2	1	no	—
90	C₂×C₄×C₁₇₃	—	2	1	no	—
91	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₁₇₃	—	2	2	no	—
98	C₂²×C₁₇₃	—	2	2	no	—
99	C₂²×C₁₇₃	—	2	4	no	—
100	C₂²×C₁₇₃	—	2	4	no	—
101	C₂²×C₁₇₃	#117	1	4	yes	—
102	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
104	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
105	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₁₇₃	—	2	2	no	—
108	C₄×C₁₇₃	—	2	2	no	—
109	C₄×C₁₇₃	—	2	2	no	—
110	C₄×C₁₇₃	—	2	2	no	—
111	C₄×C₁₇₃	—	2	2	no	—
112	C₄×C₁₇₃	—	2	2	no	—
113	Dic₁₇₃	—	2	4	no	—
114	Dic₁₇₃	—	4	2	no	—
115	Dic₁₇₃	—	4	2	no	—
116	Dic₁₇₃	—	4	2	no	—
117	C₂×C₁₇₃	—	2	4	no	—
118	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
119	C₂×C₁₇₃	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₇₃	#520	1	1	yes	characteristic
121	C₂×C₁₇₃	#551	1	2	yes	—
122	C₂×C₁₇₃	#552	1	2	yes	—
123	C₁₇₃	#2343	1	1	yes	characteristic, complemented, Sylow
124	#2343	—	173	1	no	maximal, Sylow
125	C₂×#117	D₁₇₃	1	1	yes	characteristic
126	#387	—	173	2	no	—
127	#387	—	173	2	no	—
128	#385	—	173	2	no	—
129	#383	—	173	1	no	—
130	#382	—	173	1	no	—
131	#381	—	173	1	no	—
132	C₄×C₄⋊C₄	—	173	1	no	—
133	C₄×C₂²⋊C₄	—	173	2	no	—
134	#376	—	173	1	no	—
135	C₂×#95	—	173	1	no	—
136	C₂³×C₄	—	173	2	no	—
137	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
138	#117	Dic₁₇₃	1	8	yes	complemented
139	C₂×C₄⋊C₄	—	173	2	no	—
140	C₂×C₄⋊C₄	—	173	2	no	—
141	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	—	173	2	no	—
144	C₂×C₂²⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
146	C₂×C₄²	—	173	2	no	—
147	C₂×C₄²	—	173	2	no	—
148	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
150	#95	—	173	1	no	—
151	#95	—	173	1	no	—
152	#95	—	173	2	no	—
153	#95	—	173	1	no	—
154	#95	—	173	2	no	—
155	#95	—	173	2	no	—
156	#95	—	173	2	no	—
157	#95	—	173	1	no	—
158	#95	—	173	2	no	—
159	#95	—	173	2	no	—
160	C₂⁴	C₂²×D₁₇₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	346	4	no	—
163	C₂²×C₄	—	346	4	no	—
164	C₂²×C₄	—	173	2	no	—
165	C₂²×C₄	—	173	2	no	—
166	C₂²×C₄	—	173	2	no	—
167	C₂²×C₄	—	173	2	no	—
168	C₂²×C₄	—	692	2	no	—
169	C₂²×C₄	C₂×Dic₁₇₃	1	4	yes	—
170	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
176	C₄⋊C₄	—	346	4	no	—
177	C₄⋊C₄	—	692	2	no	—
178	C₄⋊C₄	C₂×Dic₁₇₃	1	4	yes	complemented
179	C₄⋊C₄	C₂×Dic₁₇₃	1	4	yes	complemented
180	C₂²⋊C₄	—	346	4	no	—
181	C₂²⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
182	C₂²⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
183	C₄²	—	346	4	no	—
184	C₄²	—	692	2	no	—
185	C₄²	C₂×Dic₁₇₃	1	4	yes	complemented
186	C₄²	C₂×Dic₁₇₃	1	4	yes	complemented
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	C₂²×Dic₁₇₃	1	2	yes	—
191	C₂³	#14583826	1	4	yes	—
192	C₂³	C₂³×D₁₇₃	1	1	yes	characteristic
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	346	2	no	—
203	C₂×C₄	—	346	2	no	—
204	C₂×C₄	—	346	2	no	—
205	C₂×C₄	—	346	2	no	—
206	C₂×C₄	—	346	2	no	—
207	C₂×C₄	—	346	2	no	—
208	C₂×C₄	—	346	2	no	—
209	C₂×C₄	—	346	2	no	—
210	C₂×C₄	—	346	2	no	—
211	C₂×C₄	—	346	2	no	—
212	C₂×C₄	—	346	2	no	—
213	C₂×C₄	—	173	4	no	—
214	C₂×C₄	—	692	8	no	—
215	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
216	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
217	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
218	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
219	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
220	C₂×C₄	C₂²×Dic₁₇₃	1	2	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#19149638	1	4	yes	—
226	C₂²	C₂³×Dic₁₇₃	1	1	yes	characteristic
227	C₂²	#41×D₁₇₃	1	2	yes	—
228	C₂²	C₂×#14583826	1	2	yes	—
229	C₂²	#19149725	1	1	yes	characteristic
230	C₂²	#19149726	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	346	4	no	—
238	C₄	—	692	2	no	—
239	C₄	—	692	2	no	—
240	C₄	—	692	2	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#19149638	1	1	yes	characteristic
243	C₂	#41×Dic₁₇₃	1	1	yes	characteristic
244	C₂	#26011974	1	1	yes	characteristic
245	C₂	#26012165	1	2	yes	—
246	C₂	#26012174	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇₃
commutator	C₂²×C₁₇₃	C₂³×C₄
Fitting	C₂×#117×C₁₇₃	C₂
socle, cosocle	C₂³×C₁₇₃	C₂⁴
prime radical	C₂⁴×C₁₇₃	C₂³
hypercenter	C₂×#117	D₁₇₃
hypocenter	C₁₇₃	#2343
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #26011377	C₂	semidirect
2. #26011377	C₂	semidirect
4. #26011359	C₂	semidirect
5. #26011357	C₂	semidirect
7. C₄×C₄⋊Dic₁₇₃	C₂	semidirect
9. #26011337	C₂	semidirect
29. #117×C₁₇₃	C₄	semidirect
57. C₄⋊C₄×C₁₇₃	C₂×C₄	semidirect
58. C₄⋊C₄×C₁₇₃	C₂×C₄	semidirect
61. C₄²×C₁₇₃	C₂×C₄	semidirect
62. C₄²×C₁₇₃	C₂×C₄	semidirect
123. C₁₇₃	#2343	semidirect
138. #117	Dic₁₇₃	semidirect
178. C₄⋊C₄	C₂×Dic₁₇₃	semidirect
179. C₄⋊C₄	C₂×Dic₁₇₃	semidirect
185. C₄²	C₂×Dic₁₇₃	semidirect
186. C₄²	C₂×Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011377	2
#26011366	2
#26011359	2
#26011357	2
#26011354	2
C₄×C₄⋊Dic₁₇₃	2
C₂²⋊C₄×Dic₁₇₃	2
#26011337	2
C₂×#19149586	2
C₂×#117×C₁₇₃	2
#2343	173