Subgroups — #135×#10621924 #36370313

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10621924	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#10621924	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#10621924	C₂	1	4	yes	maximal, complemented
4	D₈×#10621924	C₂	1	2	yes	maximal, complemented
5	#26012501	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012481	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#10621924	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#10621924	C₂²	1	4	yes	—
11	D₄×#10621924	—	2	4	no	—
12	D₄×#10621924	C₂²	1	4	yes	—
13	#19149758	—	2	2	no	—
14	#19149758	C₂²	1	2	yes	—
15	C₂×C₄×#10621924	—	2	2	no	—
16	C₂×C₄×#10621924	C₂²	1	1	yes	characteristic
17	#19149748	C₂²	1	2	yes	—
18	#19149747	C₂²	1	2	yes	—
19	#19149742	C₂²	1	4	yes	—
20	#19149741	C₂²	1	4	yes	—
21	C₈×#10621924	C₂²	1	2	yes	—
22	C₈×#10621924	C₂²	1	2	yes	—
23	#41×D₁₇₃	C₂²	1	2	yes	—
24	#19149702	C₄	1	1	yes	characteristic, complemented
25	#19149701	C₄	1	1	yes	characteristic, complemented
26	#19149700	C₄	1	2	yes	complemented
27	#19149699	C₄	1	2	yes	complemented
28	#19149698	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
30	SD₁₆×D₁₇₃	C₂²	1	2	yes	—
31	D₈×D₁₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
33	#135×C₁₇₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×#10621924	—	2	4	no	—
35	C₂²×#10621924	D₄	1	2	yes	—
36	D₁₇₃⋊C₈	—	2	1	no	—
37	D₁₇₃⋊C₈	C₂³	1	2	yes	—
38	C₂×#13063443	—	2	1	no	—
39	C₄⋊#10621924	—	2	4	no	—
40	C₄⋊#10621924	C₂³	1	4	yes	—
41	C₂²⋊#10621924	—	2	4	no	—
42	C₂²⋊#10621924	—	2	2	no	—
43	C₄×#10621924	—	2	4	no	—
44	C₄×#10621924	—	2	2	no	—
45	C₄×#10621924	D₄	1	1	yes	characteristic
46	C₄×#10621924	D₄	1	1	yes	characteristic
47	C₄×#10621924	C₂³	1	2	yes	—
48	#14583827	C₂×C₄	1	2	yes	complemented
49	#14583826	—	2	2	no	—
50	#14583826	C₂×C₄	1	2	yes	complemented
51	#14583825	—	2	2	no	—
52	#14583825	C₂×C₄	1	2	yes	complemented
53	Q₈×D₁₇₃	C₂³	1	2	yes	—
54	D₄×D₁₇₃	—	2	2	no	—
55	D₄×D₁₇₃	C₂³	1	2	yes	—
56	C₂×C₄×D₁₇₃	—	2	2	no	—
57	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
58	Dic₆₉₂	C₂×C₄	1	1	yes	characteristic, complemented
59	#14583816	C₂×C₄	1	2	yes	complemented
60	#14583815	C₂×C₄	1	2	yes	complemented
61	#14583814	C₂×C₄	1	2	yes	complemented
62	#14583813	C₂×C₄	1	2	yes	complemented
63	D₁₃₈₄	C₂×C₄	1	1	yes	characteristic, complemented
64	#14583811	C₂×C₄	1	2	yes	complemented
65	C₂×#13063436	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
68	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
69	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
70	#41×C₁₇₃	C₂×C₄	1	2	yes	complemented
71	Q₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
72	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
73	D₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
75	C₂×#10621924	—	2	4	no	—
76	C₂×#10621924	—	2	4	no	—
77	C₂×#10621924	—	4	8	no	—
78	C₂×#10621924	C₂×D₄	1	2	yes	—
79	C₂×#10621924	C₂×D₄	1	2	yes	—
80	#13063443	—	2	2	no	—
81	C₂²×D₁₇₃	—	2	2	no	—
82	C₂²×D₁₇₃	C₂×D₄	1	1	yes	characteristic
83	Dic₃₄₆	—	2	2	no	—
84	Dic₃₄₆	C₂²×C₄	1	2	yes	—
85	D₆₉₂	—	2	2	no	—
86	D₆₉₂	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₇₃	—	2	2	no	—
88	C₂²⋊D₁₇₃	—	2	2	no	—
89	C₂²⋊D₁₇₃	—	2	2	no	—
90	C₂×Dic₁₇₃	—	2	2	no	—
91	C₂×Dic₁₇₃	#41	1	1	yes	characteristic
92	C₄×D₁₇₃	—	2	2	no	—
93	C₄×D₁₇₃	—	2	2	no	—
94	C₄×D₁₇₃	—	2	2	no	—
95	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₇₃	#41	1	1	yes	characteristic
98	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
99	#13063436	C₂²×C₄	1	1	yes	characteristic
100	#13063436	C₂²×C₄	1	1	yes	characteristic
101	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
102	D₄×C₁₇₃	—	2	2	no	—
103	D₄×C₁₇₃	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₇₃	—	2	2	no	—
105	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₁₇₃	—	2	2	no	—
109	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
110	C₂×D₁₇₃	—	2	2	no	—
111	C₂×D₁₇₃	—	2	2	no	—
112	C₂×D₁₇₃	—	4	4	no	—
113	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
114	C₂×D₁₇₃	C₂²×D₄	1	1	yes	characteristic
115	#10621924	—	2	2	no	—
116	#10621924	—	4	4	no	—
117	#10621924	#135	1	4	yes	direct product factor
118	C₄×C₁₇₃	—	2	2	no	—
119	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
120	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
121	Dic₁₇₃	—	2	2	no	—
122	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
123	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₁₇₃	—	2	1	no	—
125	C₂×C₁₇₃	—	4	2	no	—
126	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
127	D₁₇₃	—	2	1	no	—
128	D₁₇₃	—	4	2	no	—
129	D₁₇₃	C₂×#135	1	2	yes	—
130	C₁₇₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
131	C₄×#135	—	173	1	no	maximal, Sylow
132	C₂×#135	—	173	1	no	—
133	C₄×#41	—	173	2	no	—
134	C₄×Q₁₆	—	173	2	no	—
135	C₄×SD₁₆	—	173	4	no	—
136	C₄×D₈	—	173	2	no	—
137	#427	—	173	1	no	—
138	#416	—	173	2	no	—
139	C₂×C₄×C₈	—	173	1	no	—
140	C₂×#41	—	173	2	no	—
141	#135	—	173	2	no	—
142	#135	—	173	1	no	—
143	#135	—	173	2	no	—
144	#135	—	173	1	no	—
145	#135	—	173	1	no	—
146	#135	#10621924	1	1	yes	characteristic, direct product factor
147	C₂×Q₁₆	—	173	1	no	—
148	C₂×SD₁₆	—	173	2	no	—
149	C₂×D₈	—	173	1	no	—
150	C₂²×C₈	—	173	1	no	—
151	C₂²×C₈	—	173	1	no	—
152	C₄×Q₈	—	173	4	no	—
153	C₄×D₄	—	346	4	no	—
154	C₄×D₄	—	173	4	no	—
155	#117	—	346	2	no	—
156	#117	—	173	2	no	—
157	C₂×C₄²	—	346	2	no	—
158	C₂×C₄²	—	173	1	no	—
159	#107	—	173	2	no	—
160	#106	—	173	2	no	—
161	#103	—	173	4	no	—
162	#102	—	173	4	no	—
163	C₄×C₈	—	173	2	no	—
164	C₄×C₈	—	173	2	no	—
165	#41	—	346	2	no	—
166	#41	—	346	2	no	—
167	#41	—	173	2	no	—
168	#41	—	173	2	no	—
169	#41	—	173	2	no	—
170	#41	C₂×#10621924	1	2	yes	complemented
171	C₂×Q₈	—	173	2	no	—
172	C₂×D₄	—	346	2	no	—
173	C₂×D₄	—	173	2	no	—
174	C₂²×C₄	—	346	2	no	—
175	C₂²×C₄	—	346	4	no	—
176	C₂²×C₄	—	173	2	no	—
177	C₂²×C₄	—	173	1	no	—
178	Q₁₆	—	173	2	no	—
179	Q₁₆	—	173	1	no	—
180	Q₁₆	C₂×#10621924	1	1	yes	characteristic, complemented
181	SD₁₆	—	173	2	no	—
182	SD₁₆	—	173	2	no	—
183	SD₁₆	—	173	2	no	—
184	SD₁₆	C₂×#10621924	1	2	yes	complemented
185	D₈	—	173	2	no	—
186	D₈	—	173	1	no	—
187	D₈	C₂×#10621924	1	1	yes	characteristic, complemented
188	C₂×C₈	—	346	1	no	—
189	C₂×C₈	—	346	1	no	—
190	C₂×C₈	—	173	2	no	—
191	C₂×C₈	—	173	1	no	—
192	C₂×C₈	—	173	1	no	—
193	C₂×C₈	—	173	1	no	—
194	C₂×C₈	—	173	1	no	—
195	C₂×C₈	—	173	1	no	—
196	C₂×C₈	C₂×#10621924	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	346	4	no	—
198	C₄⋊C₄	—	173	4	no	—
199	C₂²⋊C₄	—	346	2	no	—
200	C₂²⋊C₄	—	346	4	no	—
201	C₄²	—	346	2	no	—
202	C₄²	—	346	4	no	—
203	C₄²	—	173	1	no	—
204	C₄²	—	173	1	no	—
205	C₄²	—	173	2	no	—
206	C₂³	—	346	2	no	—
207	C₂³	—	173	1	no	—
208	Q₈	—	346	2	no	—
209	Q₈	—	173	2	no	—
210	Q₈	C₂²×#10621924	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	346	2	no	—
213	D₄	—	346	2	no	—
214	D₄	—	346	2	no	—
215	D₄	—	346	2	no	—
216	D₄	—	173	2	no	—
217	D₄	C₂²×#10621924	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	346	2	no	—
220	C₂×C₄	—	346	2	no	—
221	C₂×C₄	—	346	2	no	—
222	C₂×C₄	—	346	4	no	—
223	C₂×C₄	—	346	2	no	—
224	C₂×C₄	—	346	4	no	—
225	C₂×C₄	—	173	2	no	—
226	C₂×C₄	—	173	2	no	—
227	C₂×C₄	—	173	1	no	—
228	C₂×C₄	—	173	1	no	—
229	C₂×C₄	—	173	1	no	—
230	C₂×C₄	—	173	1	no	—
231	C₂×C₄	—	173	1	no	—
232	C₂×C₄	—	692	8	no	—
233	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
234	C₈	—	346	2	no	—
235	C₈	—	173	1	no	—
236	C₈	—	173	1	no	—
237	C₈	C₂²×#10621924	1	1	yes	characteristic
238	C₈	C₂²×#10621924	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	346	2	no	—
241	C₂²	—	346	2	no	—
242	C₂²	—	173	1	no	—
243	C₂²	—	173	1	no	—
244	C₂²	—	692	4	no	—
245	C₂²	D₄×#10621924	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	346	2	no	—
248	C₄	—	346	2	no	—
249	C₄	—	173	4	no	—
250	C₄	—	173	1	no	—
251	C₄	—	173	1	no	—
252	C₄	—	692	4	no	—
253	C₄	D₄×#10621924	1	1	yes	characteristic
254	C₄	C₂³×#10621924	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	346	1	no	—
258	C₂	—	173	2	no	—
259	C₂	—	692	2	no	—
260	C₂	C₂×D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10621924
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₄	C₂³×#10621924
Fitting	#135×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	#135×D₁₇₃	C₂
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	#135	#10621924
hypocenter	C₁₇₃	C₄×#135
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#10621924	C₂	semidirect
2. Q₁₆×#10621924	C₂	semidirect
3. SD₁₆×#10621924	C₂	semidirect
4. D₈×#10621924	C₂	semidirect
5. #26012501	C₂	semidirect
6. #26012481	C₂	semidirect
7. C₂×C₈×#10621924	C₂	semidirect
24. #19149702	C₄	semidirect
25. #19149701	C₄	semidirect
26. #19149700	C₄	semidirect
27. #19149699	C₄	semidirect
28. #19149698	C₄	semidirect
33. #135×C₁₇₃	C₄	semidirect
48. #14583827	C₂×C₄	semidirect
50. #14583826	C₂×C₄	semidirect
52. #14583825	C₂×C₄	semidirect
58. Dic₆₉₂	C₂×C₄	semidirect
59. #14583816	C₂×C₄	semidirect
60. #14583815	C₂×C₄	semidirect
61. #14583814	C₂×C₄	semidirect
62. #14583813	C₂×C₄	semidirect
63. D₁₃₈₄	C₂×C₄	semidirect
64. #14583811	C₂×C₄	semidirect
65. C₂×#13063436	C₂×C₄	semidirect
66. C₈×D₁₇₃	C₂×C₄	semidirect
67. C₈×D₁₇₃	C₂×C₄	semidirect
70. #41×C₁₇₃	C₂×C₄	semidirect
71. Q₁₆×C₁₇₃	C₂×C₄	semidirect
72. SD₁₆×C₁₇₃	C₂×C₄	semidirect
73. D₈×C₁₇₃	C₂×C₄	semidirect
74. C₂×C₈×C₁₇₃	C₂×C₄	semidirect
117. #10621924	#135	direct (N × H)
130. C₁₇₃	C₄×#135	semidirect
146. #135	#10621924	direct (N × H)
170. #41	C₂×#10621924	semidirect
180. Q₁₆	C₂×#10621924	semidirect
184. SD₁₆	C₂×#10621924	semidirect
187. D₈	C₂×#10621924	semidirect
196. C₂×C₈	C₂×#10621924	semidirect

Maximal subgroups

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

subgroup	index
#41×#10621924	2
Q₁₆×#10621924	2
SD₁₆×#10621924	2
D₈×#10621924	2
#26012501	2
#26012481	2
C₂×C₈×#10621924	2
#135×D₁₇₃	2
C₄×#135	173