Subgroups — #36363113

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149709	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19149699	C₂	1	1	yes	maximal, characteristic
3	#26012067	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011713	C₂	1	2	yes	maximal, complemented
5	#26011706	C₂	1	2	yes	maximal, complemented
6	#26011701	C₂	1	2	yes	maximal, complemented
7	#26011696	C₂	1	2	yes	maximal, complemented
8	#26011518	C₂	1	1	yes	maximal, characteristic, complemented
9	#26011490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19149602	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583826	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₄₆	C₂²	1	1	yes	characteristic
14	#19149709	—	2	4	no	—
15	#19149699	—	2	4	no	—
16	C₂×#14583816	C₂²	1	1	yes	characteristic
17	C₂×#14583816	C₂²	1	2	yes	—
18	C₂×#14583814	C₂²	1	1	yes	characteristic
19	C₂×#14583814	C₂²	1	2	yes	—
20	C₂×#14583813	C₂²	1	1	yes	characteristic
21	C₂×#14583811	C₂²	1	1	yes	characteristic
22	C₂×#14583810	C₂²	1	1	yes	characteristic
23	C₂²×#13063436	C₂²	1	1	yes	characteristic
24	#19149679	C₂²	1	2	yes	—
25	#19149667	C₂²	1	2	yes	—
26	#19149663	—	2	2	no	—
27	#19149659	—	2	1	no	—
28	#19149659	—	2	1	no	—
29	C₄⋊#13063436	C₂²	1	2	yes	—
30	C₄⋊#13063436	C₂²	1	2	yes	—
31	#19149602	C₂²	1	2	yes	—
32	#19149602	C₂²	1	2	yes	—
33	#19149602	C₂²	1	2	yes	—
34	#19149599	C₂²	1	2	yes	—
35	C₂×#41×C₁₇₃	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₇₃	C₂²	1	2	yes	—
37	C₄×D₄×C₁₇₃	—	2	2	no	—
38	C₄×D₄×C₁₇₃	C₂²	1	2	yes	—
39	#117×C₁₇₃	—	2	1	no	—
40	#117×C₁₇₃	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₇₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
43	#14583826	—	2	2	no	—
44	#14583826	—	2	2	no	—
45	C₂×Dic₃₄₆	—	2	1	no	—
46	C₂×Dic₃₄₆	—	2	2	no	—
47	C₂×Dic₃₄₆	—	2	2	no	—
48	C₂×Dic₃₄₆	C₂³	1	2	yes	—
49	C₂×Dic₃₄₆	C₂³	1	1	yes	characteristic
50	C₂×D₆₉₂	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
52	C₂²×Dic₁₇₃	—	2	1	no	—
53	C₂×C₄×D₁₇₃	—	2	1	no	—
54	#14583816	—	2	2	no	—
55	#14583816	—	2	4	no	—
56	#14583814	—	2	2	no	—
57	#14583814	—	2	4	no	—
58	#14583813	—	2	2	no	—
59	#14583811	—	2	2	no	—
60	#14583810	—	2	2	no	—
61	C₂×#13063436	—	2	2	no	—
62	C₂×#13063436	C₂³	1	2	yes	—
63	C₂×#13063436	C₂³	1	1	yes	characteristic
64	C₂×#13063436	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₇₃	—	2	2	no	—
66	C₄⋊Dic₁₇₃	—	2	2	no	—
67	#14583803	—	2	2	no	—
68	#14583803	—	2	2	no	—
69	#41×C₁₇₃	—	2	2	no	—
70	#41×C₁₇₃	D₄	1	4	yes	—
71	C₂×Q₈×C₁₇₃	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₇₃	—	2	1	no	—
73	C₂×D₄×C₁₇₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₇₃	—	2	1	no	—
75	C₂²×C₄×C₁₇₃	—	2	1	no	—
76	C₂²×C₄×C₁₇₃	—	2	2	no	—
77	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇₃	—	2	1	no	—
79	C₄⋊C₄×C₁₇₃	—	2	2	no	—
80	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
84	C₂²⋊C₄×C₁₇₃	D₄	1	2	yes	—
85	C₄²×C₁₇₃	—	2	2	no	—
86	C₄²×C₁₇₃	C₂³	1	2	yes	—
87	C₂²×D₁₇₃	—	2	1	no	—
88	Dic₃₄₆	—	2	2	no	—
89	Dic₃₄₆	—	2	1	no	—
90	Dic₃₄₆	—	2	2	no	—
91	Dic₃₄₆	—	2	1	no	—
92	Dic₃₄₆	—	4	2	no	—
93	D₆₉₂	—	2	1	no	—
94	D₆₉₂	—	2	1	no	—
95	C₂²⋊D₁₇₃	—	4	2	no	—
96	C₂×Dic₁₇₃	—	2	1	no	—
97	C₂×Dic₁₇₃	—	2	2	no	—
98	C₂×Dic₁₇₃	—	4	2	no	—
99	C₄×D₁₇₃	—	4	2	no	—
100	#13063436	—	2	1	no	—
101	#13063436	—	2	1	no	—
102	#13063436	—	2	2	no	—
103	C₂³×C₁₇₃	—	2	1	no	—
104	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₁₇₃	—	2	1	no	—
106	Q₈×C₁₇₃	C₂×D₄	1	2	yes	—
107	D₄×C₁₇₃	—	2	4	no	—
108	D₄×C₁₇₃	—	2	1	no	—
109	D₄×C₁₇₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₇₃	—	2	1	no	—
111	C₂×C₄×C₁₇₃	—	2	4	no	—
112	C₂×C₄×C₁₇₃	—	2	2	no	—
113	C₂×C₄×C₁₇₃	—	8	2	no	—
114	C₂×C₄×C₁₇₃	—	8	1	no	—
115	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₇₃	#41	1	2	yes	—
120	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
121	C₂²×C₁₇₃	—	2	1	no	—
122	C₂²×C₁₇₃	—	2	1	no	—
123	C₂²×C₁₇₃	—	2	2	no	—
124	C₂²×C₁₇₃	—	4	1	no	—
125	C₂²×C₁₇₃	—	4	1	no	—
126	C₂²×C₁₇₃	#121	1	2	yes	—
127	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₇₃	—	4	1	no	—
129	C₂×D₁₇₃	—	4	1	no	—
130	C₂×D₁₇₃	—	4	1	no	—
131	C₄×C₁₇₃	—	2	2	no	—
132	C₄×C₁₇₃	—	2	2	no	—
133	C₄×C₁₇₃	—	2	2	no	—
134	C₄×C₁₇₃	—	4	2	no	—
135	C₄×C₁₇₃	#121	1	2	yes	—
136	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
138	Dic₁₇₃	—	4	1	no	—
139	Dic₁₇₃	—	4	2	no	—
140	C₂×C₁₇₃	—	2	2	no	—
141	C₂×C₁₇₃	—	4	2	no	—
142	C₂×C₁₇₃	C₂×#121	1	1	yes	characteristic
143	C₂×C₁₇₃	#577	1	1	yes	characteristic
144	C₂×C₁₇₃	#578	1	1	yes	characteristic
145	D₁₇₃	—	8	1	no	—
146	C₁₇₃	#3046	1	1	yes	characteristic, complemented, Sylow
147	#3046	—	173	1	no	maximal, Sylow
148	C₂×#137	—	173	1	no	—
149	C₂×#135	—	173	1	no	—
150	#536	—	173	1	no	—
151	#520	D₁₇₃	1	1	yes	characteristic, complemented
152	#464	—	173	2	no	—
153	#463	—	173	2	no	—
154	#462	—	173	2	no	—
155	#461	—	173	2	no	—
156	#424	—	173	1	no	—
157	#417	—	173	1	no	—
158	C₂×#103	—	173	1	no	—
159	C₂×#41	—	173	1	no	—
160	C₂×#41	C₂×D₁₇₃	1	1	yes	characteristic
161	C₂²×Q₈	—	173	1	no	—
162	#137	—	346	4	no	—
163	#135	—	346	4	no	—
164	C₂×Q₁₆	—	173	1	no	—
165	C₂×Q₁₆	—	173	2	no	—
166	C₂×SD₁₆	—	173	1	no	—
167	C₂×SD₁₆	—	173	1	no	—
168	C₂×SD₁₆	—	173	2	no	—
169	C₂×D₈	—	173	1	no	—
170	C₂×M₁₆	—	173	1	no	—
171	C₂²×C₈	—	173	1	no	—
172	C₄⋊Q₈	—	173	2	no	—
173	#124	—	173	2	no	—
174	#123	—	346	2	no	—
175	C₂²⋊Q₈	—	346	1	no	—
176	C₂²⋊Q₈	—	346	1	no	—
177	C₄×Q₈	C₂×D₁₇₃	1	2	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂×D₁₇₃	1	2	yes	—
180	#117	—	2	1	no	—
181	#117	C₂×D₁₇₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
184	C₄⋊C₈	—	173	2	no	—
185	C₄⋊C₈	—	173	2	no	—
186	#103	—	173	2	no	—
187	#103	—	173	2	no	—
188	#103	—	173	2	no	—
189	#102	—	173	2	no	—
190	#41	—	2	2	no	—
191	#41	—	346	2	no	—
192	#41	—	346	2	no	—
193	#41	D₆₉₂	1	4	yes	—
194	C₂×Q₈	—	346	1	no	—
195	C₂×Q₈	—	346	2	no	—
196	C₂×Q₈	—	346	2	no	—
197	C₂×Q₈	—	173	2	no	—
198	C₂×Q₈	—	173	1	no	—
199	C₂×Q₈	C₂²×D₁₇₃	1	1	yes	characteristic
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	346	1	no	—
202	C₂×D₄	—	173	1	no	—
203	C₂×D₄	C₂²×D₁₇₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	346	1	no	—
208	C₂²×C₄	—	346	1	no	—
209	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
210	Q₁₆	—	346	2	no	—
211	Q₁₆	—	346	4	no	—
212	SD₁₆	—	346	2	no	—
213	SD₁₆	—	346	2	no	—
214	SD₁₆	—	346	4	no	—
215	D₈	—	346	2	no	—
216	M₁₆	—	346	2	no	—
217	C₂×C₈	—	346	2	no	—
218	C₂×C₈	—	173	2	no	—
219	C₂×C₈	—	173	1	no	—
220	C₂×C₈	—	173	1	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	346	2	no	—
224	C₄⋊C₄	—	346	2	no	—
225	C₄⋊C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
226	C₄⋊C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₇₃	1	2	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	346	2	no	—
230	C₂²⋊C₄	—	346	2	no	—
231	C₂²⋊C₄	C₂²⋊D₁₇₃	1	2	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂²×D₁₇₃	1	2	yes	—
234	C₂³	—	2	1	no	—
235	C₂³	—	346	1	no	—
236	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
237	Q₈	—	2	1	no	—
238	Q₈	—	346	2	no	—
239	Q₈	—	346	1	no	—
240	Q₈	—	346	2	no	—
241	Q₈	—	346	1	no	—
242	Q₈	—	692	2	no	—
243	Q₈	C₂×D₆₉₂	1	2	yes	—
244	D₄	—	2	4	no	—
245	D₄	—	2	1	no	—
246	D₄	—	346	1	no	—
247	D₄	—	346	1	no	—
248	D₄	—	692	2	no	—
249	D₄	C₂×D₆₉₂	1	2	yes	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	346	2	no	—
256	C₂×C₄	—	346	1	no	—
257	C₂×C₄	—	692	2	no	—
258	C₂×C₄	—	692	2	no	—
259	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₆₉₂	1	2	yes	—
263	C₂×C₄	#14583826	1	2	yes	—
264	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
265	C₈	—	346	1	no	—
266	C₈	—	346	1	no	—
267	C₈	—	346	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	692	1	no	—
274	C₂²	—	692	1	no	—
275	C₂²	—	692	1	no	—
276	C₂²	#19149653	1	2	yes	—
277	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	692	1	no	—
283	C₄	—	692	2	no	—
284	C₄	#19149653	1	2	yes	—
285	C₄	C₂²×D₆₉₂	1	1	yes	characteristic
286	C₄	C₂×#14583826	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	1384	1	no	—
290	C₂	C₂×#19149653	1	1	yes	characteristic
291	C₂	#26012302	1	1	yes	characteristic
292	C₂	#26012306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	#520×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
prime radical	C₂×#19149699	C₂
hypercenter	#520	D₁₇₃
hypocenter	C₁₇₃	#3046
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#19149709	C₂	semidirect
3. #26012067	C₂	semidirect
4. #26011713	C₂	semidirect
5. #26011706	C₂	semidirect
6. #26011701	C₂	semidirect
7. #26011696	C₂	semidirect
8. #26011518	C₂	semidirect
9. #26011490	C₂	semidirect
10. C₂×#19149602	C₂	semidirect
11. #520×C₁₇₃	C₂	semidirect
146. C₁₇₃	#3046	semidirect
151. #520	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149709	2
C₂×#19149699	2
#26012067	2
#26011713	2
#26011706	2
#26011701	2
#26011696	2
#26011518	2
#26011490	2
C₂×#19149602	2
#520×C₁₇₃	2
#3046	173