Subgroups — #36370492

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012543	C₂	1	2	yes	maximal, complemented
2	#26012521	C₂	1	2	yes	maximal, complemented
3	#26012520	C₂	1	2	yes	maximal, complemented
4	#26012518	C₂	1	2	yes	maximal, complemented
5	#26012514	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012511	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012509	C₂	1	1	yes	maximal, characteristic, complemented
8	#26012496	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19149746	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₇₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#26011959	C₂	1	1	yes	maximal, characteristic
12	C₂×#14583836	C₂²	1	2	yes	—
13	C₂×D₁₇₃⋊C₈	C₂²	1	2	yes	—
14	C₂²×#13063443	C₂²	1	2	yes	—
15	#19149746	C₂²	1	1	yes	characteristic
16	#19149746	C₂²	1	1	yes	characteristic
17	#19149746	C₂²	1	1	yes	characteristic
18	#19149746	C₂²	1	2	yes	—
19	#19149746	C₂²	1	2	yes	—
20	#19149746	C₂²	1	1	yes	characteristic
21	C₄⋊#13063443	C₂²	1	2	yes	—
22	#19149736	C₂²	1	2	yes	—
23	#19149736	C₂²	1	2	yes	—
24	C₂²⋊#13063443	C₂²	1	2	yes	—
25	#19149734	C₂²	1	2	yes	—
26	#19149734	C₂²	1	2	yes	—
27	C₄×#13063443	C₂²	1	2	yes	—
28	C₂×#14583826	C₄	1	1	yes	characteristic
29	#19149644	C₄	1	1	yes	characteristic
30	#19149644	C₂²	1	1	yes	characteristic
31	#19149641	C₄	1	1	yes	characteristic
32	#19149641	C₂²	1	1	yes	characteristic
33	#19149639	C₄	1	1	yes	characteristic
34	#19149639	C₄	1	1	yes	characteristic
35	#19149639	C₂²	1	1	yes	characteristic
36	#19149639	C₂²	1	1	yes	characteristic
37	#19149636	C₄	1	1	yes	characteristic
38	#19149636	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
40	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
41	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
42	#117×C₁₇₃	C₄	1	1	yes	characteristic
43	#14583836	—	4	2	no	—
44	D₁₇₃⋊C₈	—	4	2	no	—
45	C₂×#13063443	—	4	2	no	—
46	C₂×#13063443	C₂³	1	2	yes	—
47	C₂×#13063443	C₂³	1	2	yes	—
48	C₂×#13063443	C₂³	1	2	yes	—
49	C₂×#13063443	C₂³	1	2	yes	—
50	#14583826	—	2	4	no	—
51	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
52	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
63	#14583805	C₂×C₄	1	1	yes	characteristic
64	#14583805	C₂×C₄	1	1	yes	characteristic
65	#14583805	C₂×C₄	1	1	yes	characteristic
66	#14583805	C₂×C₄	1	1	yes	characteristic
67	#14583803	C₂×C₄	1	1	yes	characteristic
68	#14583803	C₂×C₄	1	1	yes	characteristic
69	#14583803	C₂×C₄	1	1	yes	characteristic
70	#14583803	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
78	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
86	#13063443	—	2	2	no	—
87	#13063443	—	2	2	no	—
88	#13063443	—	2	2	no	—
89	#13063443	—	2	2	no	—
90	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
92	Dic₃₄₆	—	2	2	no	—
93	D₆₉₂	—	2	2	no	—
94	C₂²⋊D₁₇₃	—	2	2	no	—
95	C₂²⋊D₁₇₃	M₁₆	1	4	yes	complemented
96	C₂×Dic₁₇₃	—	2	2	no	—
97	C₂×Dic₁₇₃	—	2	1	no	—
98	C₂×Dic₁₇₃	—	2	1	no	—
99	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
105	C₄×D₁₇₃	—	2	2	no	—
106	C₄×D₁₇₃	—	2	2	no	—
107	C₄×D₁₇₃	—	2	2	no	—
108	C₄×D₁₇₃	—	2	2	no	—
109	C₄×D₁₇₃	—	2	1	no	—
110	C₄×D₁₇₃	—	2	1	no	—
111	C₄×D₁₇₃	—	2	1	no	—
112	C₄×D₁₇₃	—	2	1	no	—
113	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇₃	—	2	2	no	—
115	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₁₇₃	—	2	1	no	—
122	C₂²×C₁₇₃	—	2	1	no	—
123	C₂²×C₁₇₃	C₂×M₁₆	1	2	yes	—
124	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₁₇₃	—	2	1	no	—
126	C₂×D₁₇₃	—	2	1	no	—
127	C₂×D₁₇₃	—	2	1	no	—
128	C₂×D₁₇₃	—	2	1	no	—
129	C₂×D₁₇₃	—	2	1	no	—
130	C₂×D₁₇₃	C₂×M₁₆	1	2	yes	—
131	C₄×C₁₇₃	—	2	1	no	—
132	C₄×C₁₇₃	—	2	1	no	—
133	C₄×C₁₇₃	—	2	1	no	—
134	C₄×C₁₇₃	—	2	1	no	—
135	C₄×C₁₇₃	—	2	1	no	—
136	C₄×C₁₇₃	—	2	1	no	—
137	Dic₁₇₃	—	2	1	no	—
138	Dic₁₇₃	—	2	1	no	—
139	Dic₁₇₃	—	2	1	no	—
140	Dic₁₇₃	C₂×M₁₆	1	2	yes	—
141	Dic₁₇₃	#131	1	2	yes	—
142	Dic₁₇₃	H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×C₁₇₃	—	2	2	no	—
145	C₂×C₁₇₃	#520	1	1	yes	characteristic
146	C₂×C₁₇₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
148	D₁₇₃	—	2	2	no	—
149	D₁₇₃	—	4	1	no	—
150	C₁₇₃	#2990	1	1	yes	characteristic, complemented, Sylow
151	#2990	—	173	1	no	maximal, Sylow
152	C₄×#41	—	173	1	no	—
153	#446	—	173	2	no	—
154	#435	—	173	2	no	—
155	#435	—	173	2	no	—
156	C₈×D₄	—	173	2	no	—
157	#432	—	173	1	no	—
158	#432	—	173	1	no	—
159	#431	—	173	1	no	—
160	#423	—	173	1	no	—
161	C₂×C₄⋊C₈	—	173	1	no	—
162	C₂×C₄⋊C₈	—	173	1	no	—
163	C₂×#41	—	173	1	no	—
164	C₂×M₁₆	—	173	2	no	—
165	C₂²×C₈	—	173	2	no	—
166	C₂²×C₈	—	173	2	no	—
167	C₄×Q₈	—	173	1	no	—
168	C₄×Q₈	—	173	1	no	—
169	C₄×D₄	—	173	1	no	—
170	C₄×D₄	—	173	1	no	—
171	C₄×D₄	—	173	1	no	—
172	C₄×D₄	—	173	1	no	—
173	C₄×D₄	—	173	1	no	—
174	C₄×D₄	—	173	1	no	—
175	#117	—	173	1	no	—
176	#117	—	173	1	no	—
177	#117	#10621924	1	1	yes	characteristic
178	C₂×C₄²	—	173	1	no	—
179	C₂×C₄²	—	173	1	no	—
180	C₂×C₄²	—	173	1	no	—
181	C₄⋊C₈	—	173	1	no	—
182	C₄⋊C₈	—	173	1	no	—
183	C₄⋊C₈	—	173	1	no	—
184	C₄⋊C₈	—	173	2	no	—
185	C₄⋊C₈	—	173	2	no	—
186	C₄⋊C₈	—	173	2	no	—
187	C₄⋊C₈	—	173	1	no	—
188	C₂²⋊C₈	—	173	2	no	—
189	C₂²⋊C₈	—	173	2	no	—
190	C₂²⋊C₈	—	173	2	no	—
191	#97	—	173	2	no	—
192	#97	—	173	2	no	—
193	C₄×C₈	—	173	2	no	—
194	#41	—	346	4	no	—
195	C₂×Q₈	—	173	1	no	—
196	C₂×D₄	—	173	1	no	—
197	C₂×D₄	—	173	1	no	—
198	C₂×D₄	—	173	1	no	—
199	C₂²×C₄	—	173	1	no	—
200	C₂²×C₄	—	173	1	no	—
201	C₂²×C₄	—	173	1	no	—
202	C₂²×C₄	—	173	1	no	—
203	C₂²×C₄	—	173	1	no	—
204	C₂²×C₄	—	173	1	no	—
205	C₂²×C₄	—	173	1	no	—
206	C₂²×C₄	—	173	1	no	—
207	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
208	M₁₆	—	692	2	no	—
209	C₂×C₈	—	173	2	no	—
210	C₂×C₈	—	173	2	no	—
211	C₂×C₈	—	173	2	no	—
212	C₂×C₈	—	173	2	no	—
213	C₂×C₈	—	692	2	no	—
214	C₂×C₈	—	692	2	no	—
215	C₄⋊C₄	—	173	1	no	—
216	C₄⋊C₄	—	173	1	no	—
217	C₄⋊C₄	—	173	1	no	—
218	C₄⋊C₄	—	173	1	no	—
219	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
221	C₂²⋊C₄	—	173	1	no	—
222	C₂²⋊C₄	—	173	1	no	—
223	C₂²⋊C₄	—	173	1	no	—
224	C₂²⋊C₄	—	173	1	no	—
225	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
227	C₄²	—	173	1	no	—
228	C₄²	—	173	1	no	—
229	C₄²	—	173	1	no	—
230	C₄²	—	173	1	no	—
231	C₄²	—	173	1	no	—
232	C₄²	—	173	1	no	—
233	C₄²	—	173	1	no	—
234	C₄²	—	173	1	no	—
235	C₄²	C₂×#10621924	1	1	yes	characteristic
236	C₄²	C₂×#10621924	1	1	yes	characteristic
237	C₂³	—	173	1	no	—
238	C₂³	—	173	1	no	—
239	C₂³	C₂²×#10621924	1	1	yes	characteristic
240	Q₈	—	346	2	no	—
241	D₄	—	346	2	no	—
242	D₄	—	346	2	no	—
243	D₄	—	173	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	346	2	no	—
246	C₂×C₄	—	346	1	no	—
247	C₂×C₄	—	346	1	no	—
248	C₂×C₄	—	346	1	no	—
249	C₂×C₄	—	346	1	no	—
250	C₂×C₄	—	346	2	no	—
251	C₂×C₄	—	346	2	no	—
252	C₂×C₄	—	346	2	no	—
253	C₂×C₄	—	346	1	no	—
254	C₂×C₄	—	346	1	no	—
255	C₂×C₄	—	346	2	no	—
256	C₂×C₄	—	173	1	no	—
257	C₂×C₄	—	173	1	no	—
258	C₂×C₄	—	173	1	no	—
259	C₂×C₄	—	173	1	no	—
260	C₂×C₄	—	173	1	no	—
261	C₂×C₄	—	173	1	no	—
262	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
263	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
264	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
265	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
266	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
267	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
268	C₈	—	346	2	no	—
269	C₈	—	346	2	no	—
270	C₈	—	346	2	no	—
271	C₈	—	346	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	346	1	no	—
275	C₂²	—	346	1	no	—
276	C₂²	—	346	1	no	—
277	C₂²	—	346	1	no	—
278	C₂²	—	346	1	no	—
279	C₂²	—	173	2	no	—
280	C₂²	D₁₇₃⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#10621924	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	346	1	no	—
289	C₄	—	346	1	no	—
290	C₄	—	346	1	no	—
291	C₄	—	173	1	no	—
292	C₄	—	173	1	no	—
293	C₄	—	173	2	no	—
294	C₄	—	173	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	346	2	no	—
297	C₂	—	692	1	no	—
298	C₂	#26012560	1	1	yes	characteristic
299	C₂	C₂×D₁₇₃⋊M₁₆	1	1	yes	characteristic
300	C₂	D₁₇₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#117×C₁₇₃	C₄
prime radical	C₂×#14583826	C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	#117	#10621924
hypocenter	C₁₇₃	#2990
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #26012543	C₂	semidirect
2. #26012521	C₂	semidirect
3. #26012520	C₂	semidirect
4. #26012518	C₂	semidirect
5. #26012514	C₂	semidirect
6. #26012511	C₂	semidirect
7. #26012509	C₂	semidirect
8. #26012496	C₂	semidirect
9. C₂×#19149746	C₂	semidirect
10. D₁₇₃⋊(C₄⋊C₈)	C₂	semidirect
95. C₂²⋊D₁₇₃	M₁₆	semidirect
150. C₁₇₃	#2990	semidirect

Maximal subgroups

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

subgroup	index
#26012543	2
#26012521	2
#26012520	2
#26012518	2
#26012514	2
#26012511	2
#26012509	2
#26012496	2
C₂×#19149746	2
D₁₇₃⋊(C₄⋊C₈)	2
#26011959	2
#2990	173