Subgroups — #36370394

Order 22144.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012541	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012540	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012526	C₂	1	1	yes	maximal, characteristic, complemented
4	#26012526	C₂	1	2	yes	maximal, complemented
5	#26012522	C₂	1	2	yes	maximal, complemented
6	D₄×#13063443	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012515	C₂	1	2	yes	maximal, complemented
8	#26012507	C₂	1	1	yes	maximal, characteristic, complemented
9	#26012455	C₂	1	2	yes	maximal, complemented
10	C₄×#14583835	C₂	1	1	yes	maximal, characteristic, complemented
11	#26012022	C₂	1	1	yes	maximal, characteristic
12	C₂×#14583835	C₂²	1	1	yes	characteristic
13	C₂×#14583835	C₂²	1	1	yes	characteristic
14	C₂×#14583835	C₂²	1	2	yes	—
15	C₂²×#13063443	C₂²	1	2	yes	—
16	#19149746	C₂²	1	2	yes	—
17	C₄⋊#13063443	C₂²	1	1	yes	characteristic
18	C₄⋊#13063443	C₂²	1	1	yes	characteristic
19	C₄⋊#13063443	C₂²	1	2	yes	—
20	C₂²⋊#13063443	C₂²	1	2	yes	—
21	C₂²⋊#13063443	C₂²	1	2	yes	—
22	C₂²⋊#13063443	C₂²	1	2	yes	—
23	#19149734	C₂²	1	1	yes	characteristic
24	#19149734	C₂²	1	1	yes	characteristic
25	#19149734	C₂²	1	2	yes	—
26	C₄×#13063443	C₂²	1	1	yes	characteristic
27	C₄×#13063443	C₂²	1	1	yes	characteristic
28	C₄×#13063443	C₂²	1	2	yes	—
29	C₄×#13063443	C₂²	1	2	yes	—
30	#19149672	C₄	1	2	yes	—
31	#19149670	C₄	1	1	yes	characteristic
32	#19149666	C₄	1	1	yes	characteristic
33	#19149660	C₄	1	2	yes	—
34	#19149658	C₄	1	1	yes	characteristic
35	#19149644	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₇₃	C₂²	1	2	yes	—
38	#19149635	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
40	#121×C₁₇₃	C₄	1	1	yes	characteristic
41	#14583835	—	2	2	no	—
42	#14583835	—	2	2	no	—
43	#14583835	—	2	4	no	—
44	C₂×#13063443	—	2	4	no	—
45	C₂×#13063443	C₂³	1	2	yes	—
46	C₂×#13063443	C₂³	1	1	yes	characteristic
47	C₂×#13063443	C₂³	1	2	yes	—
48	C₂×#13063443	C₂³	1	1	yes	characteristic
49	C₂×#13063443	C₂³	1	1	yes	characteristic
50	C₂×#13063443	C₂³	1	1	yes	characteristic
51	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₇₃	C₂³	1	2	yes	—
55	C₄⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
57	#14583805	C₂×C₄	1	2	yes	—
58	#14583805	C₂×C₄	1	2	yes	—
59	#14583805	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
63	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
68	C₄×Dic₁₇₃	C₂³	1	2	yes	—
69	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₇₃	C₂×C₄	1	2	yes	—
71	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
74	#13063443	—	2	2	no	—
75	#13063443	—	2	1	no	—
76	#13063443	—	2	2	no	—
77	#13063443	—	2	1	no	—
78	#13063443	#41	1	2	yes	—
79	#13063443	#41	1	2	yes	—
80	Dic₃₄₆	—	2	2	no	—
81	C₂×Dic₁₇₃	—	2	2	no	—
82	C₂×Dic₁₇₃	—	2	1	no	—
83	C₂×Dic₁₇₃	—	2	1	no	—
84	C₂×Dic₁₇₃	—	2	2	no	—
85	C₂×Dic₁₇₃	—	4	1	no	—
86	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
92	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
94	D₄×C₁₇₃	—	2	2	no	—
95	D₄×C₁₇₃	—	4	2	no	—
96	C₂×C₄×C₁₇₃	—	2	2	no	—
97	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₁₇₃	—	2	1	no	—
101	C₂²×C₁₇₃	—	2	2	no	—
102	C₂²×C₁₇₃	—	2	1	no	—
103	C₂²×C₁₇₃	—	2	2	no	—
104	C₂²×C₁₇₃	—	2	1	no	—
105	C₂²×C₁₇₃	—	2	2	no	—
106	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₁₇₃	—	2	1	no	—
108	C₄×C₁₇₃	—	2	2	no	—
109	C₄×C₁₇₃	#131	1	2	yes	—
110	Dic₁₇₃	—	2	2	no	—
111	Dic₁₇₃	—	2	1	no	—
112	Dic₁₇₃	—	2	2	no	—
113	Dic₁₇₃	—	2	1	no	—
114	Dic₁₇₃	—	2	1	no	—
115	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
116	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₇₃	—	4	1	no	—
118	C₂×C₁₇₃	—	4	2	no	—
119	C₂×C₁₇₃	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
121	C₂×C₁₇₃	#568	1	1	yes	characteristic
122	C₁₇₃	#2969	1	1	yes	characteristic, complemented, Sylow
123	#2969	—	173	1	no	maximal, Sylow
124	#529	—	173	1	no	—
125	#446	—	173	1	no	—
126	C₈×Q₈	—	173	1	no	—
127	#436	—	173	1	no	—
128	#436	—	173	2	no	—
129	#435	—	173	2	no	—
130	C₈×D₄	—	173	1	no	—
131	#433	—	173	2	no	—
132	#431	—	173	1	no	—
133	#405	—	173	2	no	—
134	C₄×M₁₆	—	173	1	no	—
135	C₂×M₁₆	—	173	1	no	—
136	C₂×M₁₆	—	173	1	no	—
137	C₂×M₁₆	—	173	2	no	—
138	C₂²×C₈	—	173	2	no	—
139	#126	—	173	2	no	—
140	#125	—	173	1	no	—
141	#124	—	173	1	no	—
142	#123	—	173	2	no	—
143	C₂²⋊Q₈	—	173	1	no	—
144	#121	#10621924	1	1	yes	characteristic
145	C₄×Q₈	—	173	1	no	—
146	C₄×D₄	—	173	1	no	—
147	C₄×D₄	—	173	2	no	—
148	#117	—	173	2	no	—
149	C₂×C₄²	—	173	1	no	—
150	C₄⋊C₈	—	173	1	no	—
151	C₄⋊C₈	—	173	1	no	—
152	C₄⋊C₈	—	173	2	no	—
153	C₄⋊C₈	—	173	2	no	—
154	C₂²⋊C₈	—	173	2	no	—
155	C₂²⋊C₈	—	173	2	no	—
156	C₂²⋊C₈	—	173	2	no	—
157	#97	—	173	1	no	—
158	#97	—	173	1	no	—
159	#97	—	173	2	no	—
160	C₄×C₈	—	173	1	no	—
161	C₄×C₈	—	173	1	no	—
162	C₄×C₈	—	173	2	no	—
163	C₄×C₈	—	173	2	no	—
164	C₂×Q₈	—	173	1	no	—
165	C₂×D₄	C₂×#10621924	1	1	yes	characteristic
166	C₂×D₄	C₂×#10621924	1	2	yes	—
167	C₂²×C₄	—	173	1	no	—
168	C₂²×C₄	—	173	2	no	—
169	C₂²×C₄	—	173	1	no	—
170	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
171	M₁₆	—	346	2	no	—
172	M₁₆	—	346	2	no	—
173	M₁₆	—	346	4	no	—
174	C₂×C₈	—	346	4	no	—
175	C₂×C₈	—	173	2	no	—
176	C₂×C₈	—	173	1	no	—
177	C₂×C₈	—	173	2	no	—
178	C₂×C₈	—	173	1	no	—
179	C₂×C₈	—	173	1	no	—
180	C₂×C₈	—	173	1	no	—
181	C₄⋊C₄	—	173	2	no	—
182	C₄⋊C₄	—	173	2	no	—
183	C₄⋊C₄	—	173	1	no	—
184	C₄⋊C₄	—	173	2	no	—
185	C₄⋊C₄	—	173	2	no	—
186	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
187	C₂²⋊C₄	—	173	2	no	—
188	C₂²⋊C₄	—	173	2	no	—
189	C₂²⋊C₄	—	173	2	no	—
190	C₂²⋊C₄	—	173	2	no	—
191	C₂²⋊C₄	C₂×#10621924	1	2	yes	—
192	C₄²	—	173	1	no	—
193	C₄²	—	173	1	no	—
194	C₄²	—	173	2	no	—
195	C₄²	—	173	1	no	—
196	C₄²	—	173	1	no	—
197	C₂³	C₂²×#10621924	1	2	yes	—
198	C₂³	C₂²×#10621924	1	1	yes	characteristic
199	Q₈	—	346	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	346	2	no	—
204	C₂×C₄	—	346	1	no	—
205	C₂×C₄	—	346	1	no	—
206	C₂×C₄	—	346	2	no	—
207	C₂×C₄	—	173	1	no	—
208	C₂×C₄	—	173	1	no	—
209	C₂×C₄	—	173	1	no	—
210	C₂×C₄	—	173	2	no	—
211	C₂×C₄	—	173	2	no	—
212	C₂×C₄	—	173	1	no	—
213	C₂×C₄	—	692	1	no	—
214	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
215	C₂×C₄	C₂²×#10621924	1	2	yes	—
216	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
217	C₈	—	346	2	no	—
218	C₈	—	346	1	no	—
219	C₈	—	346	1	no	—
220	C₈	—	346	2	no	—
221	C₈	—	173	2	no	—
222	C₈	—	173	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	2	no	—
229	C₂²	C₂³×#10621924	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	346	2	no	—
233	C₄	—	346	1	no	—
234	C₄	—	346	2	no	—
235	C₄	—	346	1	no	—
236	C₄	—	346	1	no	—
237	C₄	—	173	1	no	—
238	C₄	—	173	1	no	—
239	C₄	#19149766	1	2	yes	—
240	C₂	—	4	1	no	—
241	C₂	—	4	2	no	—
242	C₂	#41×#10621924	1	1	yes	characteristic
243	C₂	C₂×#19149766	1	1	yes	characteristic
244	C₂	#26012569	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting, prime radical	#121×C₁₇₃	C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	#121	#10621924
hypocenter	C₁₇₃	#2969
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. #26012541	C₂	semidirect
2. #26012540	C₂	semidirect
3. #26012526	C₂	semidirect
4. #26012526	C₂	semidirect
5. #26012522	C₂	semidirect
6. D₄×#13063443	C₂	semidirect
7. #26012515	C₂	semidirect
8. #26012507	C₂	semidirect
9. #26012455	C₂	semidirect
10. C₄×#14583835	C₂	semidirect
122. C₁₇₃	#2969	semidirect

Maximal subgroups

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

subgroup	index
#26012541	2
#26012540	2
#26012526	2
#26012522	2
D₄×#13063443	2
#26012515	2
#26012507	2
#26012455	2
C₄×#14583835	2
#26012022	2
#2969	173