Subgroups — #2360×C₁₇₃ #36350145

Order 22144.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#391×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#389×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#389×C₁₇₃	C₂	1	2	yes	maximal, complemented
7	#387×C₁₇₃	C₂	1	2	yes	maximal, complemented
8	#385×C₁₇₃	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₄×C₁₇₃	C₂	1	2	yes	maximal
11	C₄×C₂²⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₇₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂²⋊Q₈×C₁₇₃	C₄	1	8	yes	complemented
16	C₄×Q₈×C₁₇₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₇₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₇₃	C₂²	1	2	yes	—
32	#95×C₁₇₃	C₂²	1	1	yes	characteristic
33	#95×C₁₇₃	C₂²	1	2	yes	—
34	#95×C₁₇₃	C₂²	1	1	yes	characteristic
35	#95×C₁₇₃	C₂²	1	2	yes	—
36	#95×C₁₇₃	C₂²	1	2	yes	—
37	#95×C₁₇₃	C₂²	1	2	yes	—
38	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₇₃	—	2	2	no	—
40	C₂×Q₈×C₁₇₃	—	2	2	no	—
41	C₂×Q₈×C₁₇₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₁₇₃	—	2	2	no	—
43	C₂²×C₄×C₁₇₃	—	2	2	no	—
44	C₂²×C₄×C₁₇₃	—	2	4	no	—
45	C₂²×C₄×C₁₇₃	—	2	4	no	—
46	C₂²×C₄×C₁₇₃	—	4	2	no	—
47	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₇₃	—	2	4	no	—
59	C₄⋊C₄×C₁₇₃	—	2	2	no	—
60	C₄⋊C₄×C₁₇₃	—	2	4	no	—
61	C₄⋊C₄×C₁₇₃	—	4	2	no	—
62	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	complemented
63	C₄⋊C₄×C₁₇₃	C₂×C₄	1	8	yes	complemented
64	C₄⋊C₄×C₁₇₃	D₄	1	4	yes	—
65	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
66	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	8	yes	—
67	C₂²⋊C₄×C₁₇₃	Q₈	1	4	yes	complemented
68	C₄²×C₁₇₃	—	2	4	no	—
69	C₄²×C₁₇₃	—	2	2	no	—
70	C₄²×C₁₇₃	—	2	2	no	—
71	C₄²×C₁₇₃	—	4	2	no	—
72	C₂³×C₁₇₃	—	2	1	no	—
73	C₂³×C₁₇₃	—	2	1	no	—
74	C₂³×C₁₇₃	—	2	1	no	—
75	C₂³×C₁₇₃	—	2	1	no	—
76	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₁₇₃	C₂×Q₈	1	2	yes	—
78	C₂³×C₁₇₃	#41	1	2	yes	—
79	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
80	Q₈×C₁₇₃	—	2	8	no	—
81	C₂×C₄×C₁₇₃	—	2	1	no	—
82	C₂×C₄×C₁₇₃	—	2	2	no	—
83	C₂×C₄×C₁₇₃	—	2	1	no	—
84	C₂×C₄×C₁₇₃	—	2	1	no	—
85	C₂×C₄×C₁₇₃	—	2	1	no	—
86	C₂×C₄×C₁₇₃	—	2	2	no	—
87	C₂×C₄×C₁₇₃	—	2	2	no	—
88	C₂×C₄×C₁₇₃	—	2	1	no	—
89	C₂×C₄×C₁₇₃	—	2	2	no	—
90	C₂×C₄×C₁₇₃	—	2	2	no	—
91	C₂×C₄×C₁₇₃	—	2	2	no	—
92	C₂×C₄×C₁₇₃	—	2	1	no	—
93	C₂×C₄×C₁₇₃	—	2	8	no	—
94	C₂×C₄×C₁₇₃	—	2	1	no	—
95	C₂×C₄×C₁₇₃	—	2	2	no	—
96	C₂×C₄×C₁₇₃	—	2	1	no	—
97	C₂×C₄×C₁₇₃	—	2	1	no	—
98	C₂×C₄×C₁₇₃	—	2	1	no	—
99	C₂×C₄×C₁₇₃	—	2	2	no	—
100	C₂×C₄×C₁₇₃	—	2	1	no	—
101	C₂×C₄×C₁₇₃	—	2	2	no	—
102	C₂×C₄×C₁₇₃	—	2	1	no	—
103	C₂×C₄×C₁₇₃	—	4	8	no	—
104	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
109	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₁₇₃	#41	1	2	yes	—
113	C₂²×C₁₇₃	—	2	2	no	—
114	C₂²×C₁₇₃	—	2	2	no	—
115	C₂²×C₁₇₃	—	2	2	no	—
116	C₂²×C₁₇₃	—	2	2	no	—
117	C₂²×C₁₇₃	—	2	2	no	—
118	C₂²×C₁₇₃	—	2	2	no	—
119	C₂²×C₁₇₃	C₄×Q₈	1	4	yes	complemented
120	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₁₇₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₁₇₃	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₄×C₁₇₃	—	2	2	no	—
128	C₄×C₁₇₃	—	2	4	no	—
129	C₄×C₁₇₃	—	2	4	no	—
130	C₄×C₁₇₃	—	2	4	no	—
131	C₄×C₁₇₃	—	2	2	no	—
132	C₄×C₁₇₃	—	2	2	no	—
133	C₄×C₁₇₃	—	4	1	no	—
134	C₄×C₁₇₃	—	4	2	no	—
135	C₄×C₁₇₃	—	4	1	no	—
136	C₄×C₁₇₃	C₄×D₄	1	4	yes	—
137	C₂×C₁₇₃	—	2	4	no	—
138	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₁₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₂×C₁₇₃	#520	1	1	yes	characteristic
141	C₂×C₁₇₃	#546	1	1	yes	characteristic
142	C₂×C₁₇₃	D₄×Q₈	1	1	yes	characteristic
143	C₂×C₁₇₃	#552	1	1	yes	characteristic
144	C₂×C₁₇₃	#554	1	1	yes	characteristic
145	C₁₇₃	#2360	1	1	yes	characteristic, direct product factor, Sylow
146	#2360	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₂²⋊Q₈	C₂×C₁₇₃	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂×C₁₇₃	1	1	yes	characteristic, complemented
149	C₂²×C₄⋊C₄	C₂×C₁₇₃	1	1	yes	characteristic
150	#391	C₂×C₁₇₃	1	1	yes	characteristic, complemented
151	#389	C₂×C₁₇₃	1	1	yes	characteristic, complemented
152	#389	C₂×C₁₇₃	1	2	yes	complemented
153	#387	C₂×C₁₇₃	1	2	yes	complemented
154	#385	C₂×C₁₇₃	1	2	yes	—
155	C₄×C₄⋊C₄	C₂×C₁₇₃	1	1	yes	characteristic, complemented
156	C₄×C₂²⋊C₄	C₂×C₁₇₃	1	1	yes	characteristic
157	C₄×C₂²⋊C₄	C₂×C₁₇₃	1	2	yes	—
158	C₂²×Q₈	C₂²×C₁₇₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₁₇₃	1	2	yes	—
160	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
161	C₂²⋊Q₈	C₄×C₁₇₃	1	8	yes	complemented
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	2	yes	—
170	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₁₇₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₁₇₃	1	2	yes	—
175	C₂×C₄²	C₂²×C₁₇₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×C₁₇₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²×C₁₇₃	1	2	yes	—
178	#95	C₂²×C₁₇₃	1	1	yes	characteristic
179	#95	C₂²×C₁₇₃	1	2	yes	—
180	#95	C₂²×C₁₇₃	1	1	yes	characteristic
181	#95	C₂²×C₁₇₃	1	2	yes	—
182	#95	C₂²×C₁₇₃	1	2	yes	—
183	#95	C₂²×C₁₇₃	1	2	yes	—
184	C₂⁴	C₂³×C₁₇₃	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₂×C₄×C₁₇₃	1	4	yes	complemented
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×C₄×C₁₇₃	1	4	yes	—
194	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
196	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	C₂×C₄×C₁₇₃	1	4	yes	complemented
209	C₄⋊C₄	C₂×C₄×C₁₇₃	1	8	yes	complemented
210	C₄⋊C₄	D₄×C₁₇₃	1	4	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	C₂×C₄×C₁₇₃	1	8	yes	—
213	C₂²⋊C₄	Q₈×C₁₇₃	1	4	yes	complemented
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	C₂²×C₄×C₁₇₃	1	2	yes	—
223	C₂³	C₂×Q₈×C₁₇₃	1	2	yes	—
224	C₂³	#41×C₁₇₃	1	2	yes	—
225	C₂³	C₂⁴×C₁₇₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
251	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
252	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
253	C₂×C₄	C₂²×C₄×C₁₇₃	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₁₇₃	1	4	yes	—
255	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
256	C₂×C₄	C₂×Q₈×C₁₇₃	1	2	yes	—
257	C₂×C₄	C₂×Q₈×C₁₇₃	1	2	yes	—
258	C₂×C₄	#41×C₁₇₃	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	C₄×Q₈×C₁₇₃	1	4	yes	complemented
266	C₂²	C₂³×C₄×C₁₇₃	1	1	yes	characteristic
267	C₂²	C₂²×D₄×C₁₇₃	1	1	yes	characteristic
268	C₂²	C₂²×Q₈×C₁₇₃	1	1	yes	characteristic
269	C₂²	C₂×#41×C₁₇₃	1	1	yes	characteristic
270	C₂²	C₂×#41×C₁₇₃	1	1	yes	characteristic
271	C₂²	H(5,𝔽₂)×C₁₇₃	1	1	yes	characteristic
272	C₂²	2₋¹⁺⁴×C₁₇₃	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₄×D₄×C₁₇₃	1	4	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₂×C₄×D₄×C₁₇₃	1	1	yes	characteristic
285	C₂	C₂×C₄×Q₈×C₁₇₃	1	1	yes	characteristic
286	C₂	#520×C₁₇₃	1	1	yes	characteristic
287	C₂	#546×C₁₇₃	1	1	yes	characteristic
288	C₂	D₄×Q₈×C₁₇₃	1	1	yes	characteristic
289	C₂	#552×C₁₇₃	1	1	yes	characteristic
290	C₂	#554×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇₃
Frattini, cosocle	C₂³	C₂⁴×C₁₇₃
prime radical	C₂⁴×C₁₇₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₁₇₃	C₂	semidirect
4. #391×C₁₇₃	C₂	semidirect
5. #389×C₁₇₃	C₂	semidirect
6. #389×C₁₇₃	C₂	semidirect
7. #387×C₁₇₃	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₇₃	C₂	semidirect
15. C₂²⋊Q₈×C₁₇₃	C₄	semidirect
41. C₂×Q₈×C₁₇₃	C₂×C₄	semidirect
62. C₄⋊C₄×C₁₇₃	C₂×C₄	semidirect
63. C₄⋊C₄×C₁₇₃	C₂×C₄	semidirect
67. C₂²⋊C₄×C₁₇₃	Q₈	semidirect
119. C₂²×C₁₇₃	C₄×Q₈	semidirect
145. C₁₇₃	#2360	direct (N × H)
146. #2360	C₁₇₃	direct (N × H)
148. C₂×C₄×Q₈	C₂×C₁₇₃	semidirect
150. #391	C₂×C₁₇₃	semidirect
151. #389	C₂×C₁₇₃	semidirect
152. #389	C₂×C₁₇₃	semidirect
153. #387	C₂×C₁₇₃	semidirect
155. C₄×C₄⋊C₄	C₂×C₁₇₃	semidirect
161. C₂²⋊Q₈	C₄×C₁₇₃	semidirect
187. C₂×Q₈	C₂×C₄×C₁₇₃	semidirect
208. C₄⋊C₄	C₂×C₄×C₁₇₃	semidirect
209. C₄⋊C₄	C₂×C₄×C₁₇₃	semidirect
213. C₂²⋊C₄	Q₈×C₁₇₃	semidirect
265. C₂²	C₄×Q₈×C₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₁₇₃	2
C₂×C₄×Q₈×C₁₇₃	2
C₂²×C₄⋊C₄×C₁₇₃	2
#391×C₁₇₃	2
#389×C₁₇₃	2
#387×C₁₇₃	2
#385×C₁₇₃	2
C₄×C₄⋊C₄×C₁₇₃	2
C₄×C₂²⋊C₄×C₁₇₃	2
#2360	173

Subgroups — #2360×C173 #36350145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2360×C₁₇₃ #36350145