Subgroups — #36367172

Order 22144.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012201	C₂	1	2	yes	maximal, complemented
2	#26012191	C₂	1	4	yes	maximal, complemented
3	#26012169	C₂	1	4	yes	maximal, complemented
4	#26012134	C₂	1	4	yes	maximal, complemented
5	#26012126	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012039	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012020	C₂	1	4	yes	maximal, complemented
8	#26011999	C₂	1	4	yes	maximal, complemented
9	C₄×#14583827	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₁₇₃	C₂	1	4	yes	maximal, complemented
11	#559×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583827	C₂²	1	2	yes	—
13	C₂×#14583825	C₂²	1	2	yes	complemented
14	C₂×D₄×D₁₇₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₇₃	C₂²	1	4	yes	complemented
16	#19149679	C₂²	1	1	yes	characteristic, complemented
17	#19149672	C₂²	1	8	yes	complemented
18	#19149671	C₂²	1	4	yes	complemented
19	#19149666	C₂²	1	2	yes	complemented
20	#19149662	C₂²	1	4	yes	complemented
21	#19149661	C₂²	1	8	yes	complemented
22	#19149657	C₂²	1	8	yes	complemented
23	#19149655	C₂²	1	4	yes	complemented
24	#19149652	C₂²	1	8	yes	complemented
25	#19149650	C₂²	1	4	yes	—
26	#19149650	C₂²	1	4	yes	complemented
27	#19149649	C₂²	1	4	yes	complemented
28	#19149647	C₂²	1	4	yes	—
29	Q₈×Dic₁₇₃	C₂²	1	2	yes	complemented
30	C₄×D₆₉₂	C₂²	1	4	yes	—
31	C₄×D₆₉₂	C₂²	1	2	yes	—
32	C₄×D₆₉₂	C₂²	1	4	yes	complemented
33	D₄×Dic₁₇₃	C₂²	1	4	yes	complemented
34	#19149641	C₂²	1	2	yes	—
35	#19149641	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₁₇₃	C₂²	1	8	yes	complemented
37	#19149639	C₂²	1	8	yes	complemented
38	#19149637	C₂²	1	4	yes	complemented
39	#19149637	C₂²	1	2	yes	complemented
40	#19149635	C₂²	1	4	yes	complemented
41	C₄⋊C₄×D₁₇₃	C₂²	1	4	yes	complemented
42	C₂²⋊C₄×D₁₇₃	C₂²	1	8	yes	complemented
43	C₄²×D₁₇₃	C₂²	1	4	yes	complemented
44	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic, complemented
45	#127×C₁₇₃	C₂²	1	1	yes	characteristic
46	#124×C₁₇₃	C₂²	1	4	yes	complemented
47	#123×C₁₇₃	C₂²	1	4	yes	complemented
48	C₄×Q₈×C₁₇₃	C₂²	1	2	yes	complemented
49	C₄×D₄×C₁₇₃	C₂²	1	4	yes	complemented
50	C₂³×D₁₇₃	C₂³	1	4	yes	—
51	#14583827	—	2	8	no	—
52	#14583825	—	4	4	no	—
53	C₂×Dic₃₄₆	C₂³	1	2	yes	—
54	C₂×D₆₉₂	C₂³	1	4	yes	—
55	C₂×D₆₉₂	C₂³	1	2	yes	—
56	C₂×C₂²⋊D₁₇₃	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₇₃	C₂³	1	4	yes	—
58	D₄×D₁₇₃	—	2	16	no	—
59	C₂²×Dic₁₇₃	C₂³	1	4	yes	—
60	C₂×C₄×D₁₇₃	—	4	8	no	—
61	C₂×C₄×D₁₇₃	C₂³	1	4	yes	—
62	C₂×C₄×D₁₇₃	C₂³	1	4	yes	—
63	C₂×C₄×D₁₇₃	C₂³	1	2	yes	—
64	C₂×C₄×D₁₇₃	C₂³	1	4	yes	—
65	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₇₃	C₂³	1	8	yes	—
67	C₄⋊Dic₁₇₃	C₂³	1	4	yes	—
68	C₄⋊Dic₁₇₃	C₂³	1	2	yes	—
69	C₄⋊Dic₁₇₃	C₂³	1	4	yes	—
70	#14583805	C₂³	1	8	yes	—
71	#14583805	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₇₃	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₇₃	C₂³	1	4	yes	—
74	#14583803	C₂³	1	4	yes	—
75	#14583803	C₂³	1	8	yes	—
76	#14583803	C₂³	1	4	yes	—
77	#14583803	C₂³	1	4	yes	—
78	C₄×Dic₁₇₃	C₂³	1	2	yes	—
79	C₄×Dic₁₇₃	C₂³	1	4	yes	—
80	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₇₃	C₂³	1	4	yes	—
82	C₂×Q₈×C₁₇₃	C₂³	1	2	yes	—
83	C₂×D₄×C₁₇₃	C₂³	1	4	yes	—
84	C₂×D₄×C₁₇₃	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₇₃	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
87	C₄⋊C₄×C₁₇₃	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₁₇₃	C₂³	1	8	yes	—
89	C₂²⋊C₄×C₁₇₃	C₂³	1	4	yes	—
90	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
91	C₄²×C₁₇₃	C₂³	1	4	yes	—
92	C₂²×D₁₇₃	—	2	16	no	—
93	C₂²×D₁₇₃	—	4	4	no	—
94	C₂²×D₁₇₃	C₂⁴	1	1	yes	characteristic
95	C₂²×D₁₇₃	C₂⁴	1	4	yes	—
96	Dic₃₄₆	—	4	2	no	—
97	D₆₉₂	—	2	8	no	—
98	D₆₉₂	#41	1	8	yes	complemented
99	C₂²⋊D₁₇₃	—	2	16	no	—
100	C₂²⋊D₁₇₃	—	4	4	no	—
101	C₂×Dic₁₇₃	—	2	8	no	—
102	C₂×Dic₁₇₃	C₂⁴	1	2	yes	—
103	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
104	C₂×Dic₁₇₃	C₂⁴	1	4	yes	—
105	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
106	C₄×D₁₇₃	—	2	8	no	—
107	C₄×D₁₇₃	—	2	8	no	—
108	C₄×D₁₇₃	—	2	4	no	—
109	C₄×D₁₇₃	—	2	8	no	—
110	C₄×D₁₇₃	—	2	2	no	—
111	C₄×D₁₇₃	—	4	8	no	—
112	C₂³×C₁₇₃	C₂⁴	1	4	yes	—
113	Q₈×C₁₇₃	—	2	4	no	—
114	D₄×C₁₇₃	—	2	8	no	—
115	D₄×C₁₇₃	—	4	2	no	—
116	C₂×C₄×C₁₇₃	—	4	4	no	—
117	C₂×C₄×C₁₇₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₇₃	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₇₃	C₂⁴	1	2	yes	—
121	C₂²×C₁₇₃	—	2	8	no	—
122	C₂²×C₁₇₃	—	2	4	no	—
123	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
124	C₂×D₁₇₃	—	2	4	no	—
125	C₂×D₁₇₃	—	2	4	no	—
126	C₂×D₁₇₃	—	2	2	no	—
127	C₂×D₁₇₃	—	2	1	no	—
128	C₂×D₁₇₃	—	4	16	no	—
129	C₂×D₁₇₃	C₂×#41	1	8	yes	—
130	C₄×C₁₇₃	—	2	4	no	—
131	C₄×C₁₇₃	—	2	4	no	—
132	C₄×C₁₇₃	—	2	1	no	—
133	C₄×C₁₇₃	C₂×#41	1	4	yes	—
134	Dic₁₇₃	—	2	2	no	—
135	Dic₁₇₃	—	2	4	no	—
136	Dic₁₇₃	—	2	4	no	—
137	Dic₁₇₃	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₁₇₃	—	4	4	no	—
139	C₂×C₁₇₃	C₂²×#41	1	2	yes	—
140	C₂×C₁₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	D₁₇₃	—	2	8	no	—
142	D₁₇₃	—	4	1	no	—
143	C₁₇₃	#3481	1	1	yes	characteristic, complemented, Sylow
144	#3481	—	173	1	no	maximal, Sylow
145	#559	D₁₇₃	1	1	yes	characteristic, complemented
146	#554	—	173	2	no	—
147	#553	—	173	4	no	—
148	#551	—	173	4	no	—
149	#546	—	173	4	no	—
150	D₄²	—	173	1	no	—
151	#532	—	173	1	no	—
152	#529	—	173	4	no	—
153	#525	—	173	4	no	—
154	C₄×#41	—	173	2	no	—
155	C₂×C₄×D₄	—	173	4	no	—
156	C₂×#41	—	173	2	no	—
157	C₂×#41	—	173	2	no	—
158	C₂²×D₄	—	173	4	no	—
159	C₂³×C₄	—	173	4	no	—
160	C₄⋊Q₈	—	173	1	no	—
161	#127	C₂×D₁₇₃	1	1	yes	characteristic
162	#126	—	173	8	no	—
163	#125	—	173	4	no	—
164	#124	—	173	2	no	—
165	#124	C₂×D₁₇₃	1	4	yes	complemented
166	#123	—	173	4	no	—
167	#123	—	173	8	no	—
168	#123	C₂×D₁₇₃	1	4	yes	complemented
169	C₂²⋊Q₈	—	173	4	no	—
170	C₂²⋊Q₈	—	173	8	no	—
171	#121	—	173	4	no	—
172	#121	—	173	4	no	—
173	#121	—	173	8	no	—
174	#120	—	173	4	no	—
175	#120	—	173	4	no	—
176	C₄×Q₈	—	173	2	no	—
177	C₄×Q₈	C₂×D₁₇₃	1	2	yes	complemented
178	C₄×D₄	—	173	4	no	—
179	C₄×D₄	—	173	8	no	—
180	C₄×D₄	—	173	2	no	—
181	C₄×D₄	—	173	4	no	—
182	C₄×D₄	—	173	8	no	—
183	C₄×D₄	—	173	4	no	—
184	C₄×D₄	—	173	2	no	—
185	C₄×D₄	—	173	4	no	—
186	C₄×D₄	C₂×D₁₇₃	1	4	yes	complemented
187	#117	—	173	4	no	—
188	#117	—	173	2	no	—
189	#117	—	173	4	no	—
190	C₂×C₄⋊C₄	—	173	4	no	—
191	C₂×C₂²⋊C₄	—	173	8	no	—
192	C₂×C₄²	—	173	4	no	—
193	C₂×C₄²	—	173	1	no	—
194	C₂⁴	—	173	4	no	—
195	#41	—	346	8	no	—
196	#41	—	692	4	no	—
197	C₂×Q₈	—	173	2	no	—
198	C₂×Q₈	C₂²×D₁₇₃	1	2	yes	—
199	C₂×D₄	—	346	16	no	—
200	C₂×D₄	—	173	8	no	—
201	C₂×D₄	—	173	4	no	—
202	C₂×D₄	—	173	4	no	—
203	C₂×D₄	—	173	2	no	—
204	C₂×D₄	C₂²×D₁₇₃	1	4	yes	—
205	C₂×D₄	C₂²×D₁₇₃	1	2	yes	—
206	C₂²×C₄	—	173	4	no	—
207	C₂²×C₄	—	173	4	no	—
208	C₂²×C₄	—	173	4	no	—
209	C₂²×C₄	—	173	2	no	—
210	C₂²×C₄	—	173	4	no	—
211	C₂²×C₄	—	173	1	no	—
212	C₂²×C₄	—	173	8	no	—
213	C₂²×C₄	—	692	8	no	—
214	C₂²×C₄	C₂²×D₁₇₃	1	4	yes	—
215	C₄⋊C₄	—	173	4	no	—
216	C₄⋊C₄	—	173	8	no	—
217	C₄⋊C₄	—	173	2	no	—
218	C₄⋊C₄	—	173	4	no	—
219	C₄⋊C₄	—	173	8	no	—
220	C₄⋊C₄	C₂²×D₁₇₃	1	2	yes	—
221	C₄⋊C₄	C₂²×D₁₇₃	1	4	yes	—
222	C₂²⋊C₄	—	173	8	no	—
223	C₂²⋊C₄	—	173	4	no	—
224	C₂²⋊C₄	—	173	4	no	—
225	C₂²⋊C₄	—	173	8	no	—
226	C₂²⋊C₄	—	173	4	no	—
227	C₂²⋊C₄	—	173	4	no	—
228	C₂²⋊C₄	C₂²×D₁₇₃	1	8	yes	—
229	C₂²⋊C₄	C₂²×D₁₇₃	1	4	yes	—
230	C₄²	—	173	2	no	—
231	C₄²	—	173	1	no	—
232	C₄²	—	173	4	no	—
233	C₄²	—	173	4	no	—
234	C₄²	C₂²×D₁₇₃	1	4	yes	—
235	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
236	C₂³	—	346	16	no	—
237	C₂³	—	173	1	no	—
238	C₂³	—	173	4	no	—
239	C₂³	—	692	4	no	—
240	C₂³	C₂³×D₁₇₃	1	4	yes	—
241	Q₈	—	2	4	no	—
242	Q₈	—	692	2	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	346	8	no	—
246	D₄	—	346	16	no	—
247	D₄	—	173	8	no	—
248	D₄	—	692	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	346	8	no	—
251	C₂×C₄	—	346	8	no	—
252	C₂×C₄	—	346	2	no	—
253	C₂×C₄	—	346	8	no	—
254	C₂×C₄	—	346	8	no	—
255	C₂×C₄	—	346	4	no	—
256	C₂×C₄	—	173	2	no	—
257	C₂×C₄	—	173	4	no	—
258	C₂×C₄	—	173	4	no	—
259	C₂×C₄	—	173	1	no	—
260	C₂×C₄	—	692	8	no	—
261	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₁₇₃	1	4	yes	—
263	C₂×C₄	C₂³×D₁₇₃	1	4	yes	—
264	C₂×C₄	C₂³×D₁₇₃	1	2	yes	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	346	4	no	—
268	C₂²	—	346	4	no	—
269	C₂²	—	346	2	no	—
270	C₂²	—	346	1	no	—
271	C₂²	—	173	8	no	—
272	C₂²	—	692	16	no	—
273	C₂²	C₂⁴×D₁₇₃	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	346	2	no	—
278	C₄	—	346	4	no	—
279	C₄	—	346	4	no	—
280	C₄	—	173	2	no	—
281	C₄	#41×D₁₇₃	1	4	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	346	8	no	—
284	C₂	—	692	1	no	—
285	C₂	C₂×#41×D₁₇₃	1	2	yes	—
286	C₂	H(5,𝔽₂)×D₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₇₃
commutator, socle, cosocle	C₂²×C₁₇₃	C₂⁵
Fitting	#559×C₁₇₃	C₂
hypercenter	#559	D₁₇₃
hypocenter	C₁₇₃	#3481
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #26012201	C₂	semidirect
2. #26012191	C₂	semidirect
3. #26012169	C₂	semidirect
4. #26012134	C₂	semidirect
5. #26012126	C₂	semidirect
6. #26012039	C₂	semidirect
7. #26012020	C₂	semidirect
8. #26011999	C₂	semidirect
9. C₄×#14583827	C₂	semidirect
10. C₄×D₄×D₁₇₃	C₂	semidirect
11. #559×C₁₇₃	C₂	semidirect
13. C₂×#14583825	C₂²	semidirect
15. C₂²×C₄×D₁₇₃	C₂²	semidirect
16. #19149679	C₂²	semidirect
17. #19149672	C₂²	semidirect
18. #19149671	C₂²	semidirect
19. #19149666	C₂²	semidirect
20. #19149662	C₂²	semidirect
21. #19149661	C₂²	semidirect
22. #19149657	C₂²	semidirect
23. #19149655	C₂²	semidirect
24. #19149652	C₂²	semidirect
26. #19149650	C₂²	semidirect
27. #19149649	C₂²	semidirect
29. Q₈×Dic₁₇₃	C₂²	semidirect
32. C₄×D₆₉₂	C₂²	semidirect
33. D₄×Dic₁₇₃	C₂²	semidirect
35. #19149641	C₂²	semidirect
36. C₄×C₂²⋊D₁₇₃	C₂²	semidirect
37. #19149639	C₂²	semidirect
38. #19149637	C₂²	semidirect
39. #19149637	C₂²	semidirect
40. #19149635	C₂²	semidirect
41. C₄⋊C₄×D₁₇₃	C₂²	semidirect
42. C₂²⋊C₄×D₁₇₃	C₂²	semidirect
43. C₄²×D₁₇₃	C₂²	semidirect
44. C₄²×D₁₇₃	C₂²	semidirect
46. #124×C₁₇₃	C₂²	semidirect
47. #123×C₁₇₃	C₂²	semidirect
48. C₄×Q₈×C₁₇₃	C₂²	semidirect
49. C₄×D₄×C₁₇₃	C₂²	semidirect
98. D₆₉₂	#41	semidirect
143. C₁₇₃	#3481	semidirect
145. #559	D₁₇₃	semidirect
165. #124	C₂×D₁₇₃	semidirect
168. #123	C₂×D₁₇₃	semidirect
177. C₄×Q₈	C₂×D₁₇₃	semidirect
186. C₄×D₄	C₂×D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26012201	2
#26012191	2
#26012169	2
#26012134	2
#26012126	2
#26012039	2
#26012020	2
#26011999	2
C₄×#14583827	2
C₄×D₄×D₁₇₃	2
#559×C₁₇₃	2
#3481	173