Subgroups — #36363189

Order 22144.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149710	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14583810	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012002	C₂	1	1	yes	maximal, characteristic
4	#26011747	C₂	1	2	yes	maximal, complemented
5	#26011739	C₂	1	2	yes	maximal, complemented
6	#26011730	C₂	1	2	yes	maximal, complemented
7	#26011722	C₂	1	2	yes	maximal, complemented
8	#26011531	C₂	1	1	yes	maximal, characteristic, complemented
9	#26011500	C₂	1	1	yes	maximal, characteristic, complemented
10	#26011490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583826	C₂²	1	1	yes	characteristic
13	#19149710	—	2	4	no	—
14	C₂×#14583816	C₂²	1	2	yes	complemented
15	C₂×#14583813	C₂²	1	2	yes	complemented
16	C₂×#14583810	—	2	4	no	—
17	C₂×#14583810	C₂²	1	2	yes	—
18	C₂×#14583810	C₂²	1	1	yes	characteristic
19	C₂×#14583810	C₂²	1	1	yes	characteristic
20	C₂²×#13063436	C₂²	1	2	yes	—
21	#19149662	—	2	1	no	—
22	#19149658	C₂²	1	1	yes	characteristic
23	#19149658	C₂²	1	1	yes	characteristic
24	#19149653	C₂²	1	1	yes	characteristic
25	#19149653	C₂²	1	1	yes	characteristic
26	#19149648	—	2	1	no	—
27	C₄×C₂²⋊D₁₇₃	—	2	1	no	—
28	C₄×C₂²⋊D₁₇₃	—	2	1	no	—
29	#19149638	C₂²	1	1	yes	characteristic
30	#19149612	C₂²	1	2	yes	complemented
31	#19149610	C₂²	1	2	yes	complemented
32	#19149603	C₂²	1	2	yes	complemented
33	#19149603	C₂²	1	2	yes	complemented
34	#19149602	C₂²	1	2	yes	complemented
35	#19149599	C₂²	1	2	yes	complemented
36	C₂²×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
37	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
38	C₂²⋊Q₈×C₁₇₃	—	2	1	no	—
39	C₂²⋊Q₈×C₁₇₃	—	2	1	no	—
40	C₂²⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
41	C₂²⋊Q₈×C₁₇₃	C₂²	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₇₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₁₇₃	—	2	1	no	—
45	#14583826	—	2	2	no	—
46	#14583826	—	2	2	no	—
47	C₂×Dic₃₄₆	C₂³	1	1	yes	characteristic
48	C₂×D₆₉₂	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
50	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
51	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
52	C₂×C₄×D₁₇₃	—	2	1	no	—
53	#14583816	—	2	4	no	—
54	#14583813	—	2	4	no	—
55	#14583810	—	2	4	no	—
56	#14583810	—	2	2	no	—
57	#14583810	D₄	1	4	yes	complemented
58	C₂×#13063436	—	2	4	no	—
59	C₂×#13063436	C₂³	1	2	yes	—
60	C₂×#13063436	C₂³	1	2	yes	—
61	C₄⋊Dic₁₇₃	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₇₃	C₂³	1	1	yes	characteristic
63	#14583805	—	2	1	no	—
64	#14583805	—	2	1	no	—
65	C₂²⋊Dic₁₇₃	—	2	1	no	—
66	C₂²⋊Dic₁₇₃	—	2	1	no	—
67	C₂²⋊Dic₁₇₃	—	2	1	no	—
68	#14583803	—	2	1	no	—
69	#14583803	—	2	1	no	—
70	C₄×Dic₁₇₃	—	2	1	no	—
71	C₂⁴×C₁₇₃	D₄	1	1	yes	characteristic
72	C₂×Q₈×C₁₇₃	—	2	1	no	—
73	C₂×Q₈×C₁₇₃	—	2	2	no	—
74	C₂×Q₈×C₁₇₃	—	2	2	no	—
75	C₂×Q₈×C₁₇₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₇₃	—	2	2	no	—
77	C₂²×C₄×C₁₇₃	—	2	2	no	—
78	C₂²×C₄×C₁₇₃	—	2	1	no	—
79	C₂²×C₄×C₁₇₃	—	2	1	no	—
80	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₇₃	—	2	2	no	—
87	C₄⋊C₄×C₁₇₃	—	2	2	no	—
88	C₄⋊C₄×C₁₇₃	—	2	1	no	—
89	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
90	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
91	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
92	C₂²×D₁₇₃	—	2	1	no	—
93	Dic₃₄₆	—	2	1	no	—
94	Dic₃₄₆	—	2	1	no	—
95	D₆₉₂	—	2	1	no	—
96	D₆₉₂	—	2	1	no	—
97	C₂²⋊D₁₇₃	—	4	2	no	—
98	C₂²⋊D₁₇₃	—	4	2	no	—
99	C₂²⋊D₁₇₃	—	4	2	no	—
100	C₂×Dic₁₇₃	—	2	1	no	—
101	C₂×Dic₁₇₃	—	2	1	no	—
102	C₂×Dic₁₇₃	—	2	1	no	—
103	C₄×D₁₇₃	—	4	2	no	—
104	#13063436	—	2	2	no	—
105	#13063436	C₂×D₄	1	4	yes	—
106	C₂³×C₁₇₃	—	2	2	no	—
107	C₂³×C₁₇₃	—	2	2	no	—
108	C₂³×C₁₇₃	—	4	1	no	—
109	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₇₃	—	2	2	no	—
113	Q₈×C₁₇₃	—	2	2	no	—
114	Q₈×C₁₇₃	—	4	2	no	—
115	C₂×C₄×C₁₇₃	—	2	2	no	—
116	C₂×C₄×C₁₇₃	—	2	2	no	—
117	C₂×C₄×C₁₇₃	—	2	2	no	—
118	C₂×C₄×C₁₇₃	—	2	1	no	—
119	C₂×C₄×C₁₇₃	—	2	1	no	—
120	C₂×C₄×C₁₇₃	—	2	1	no	—
121	C₂×C₄×C₁₇₃	—	2	1	no	—
122	C₂×C₄×C₁₇₃	—	2	2	no	—
123	C₂×C₄×C₁₇₃	—	2	2	no	—
124	C₂×C₄×C₁₇₃	—	4	2	no	—
125	C₂×C₄×C₁₇₃	—	4	2	no	—
126	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇₃	#41	1	2	yes	—
131	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
132	C₂²×C₁₇₃	—	2	1	no	—
133	C₂²×C₁₇₃	—	2	1	no	—
134	C₂²×C₁₇₃	—	2	1	no	—
135	C₂²×C₁₇₃	—	2	1	no	—
136	C₂²×C₁₇₃	—	2	1	no	—
137	C₂²×C₁₇₃	—	2	1	no	—
138	C₂²×C₁₇₃	—	2	1	no	—
139	C₂²×C₁₇₃	—	4	4	no	—
140	C₂²×C₁₇₃	#121	1	2	yes	—
141	C₂²×C₁₇₃	#137	1	2	yes	complemented
142	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₇₃	—	4	1	no	—
144	C₂×D₁₇₃	—	4	1	no	—
145	C₂×D₁₇₃	—	4	1	no	—
146	C₄×C₁₇₃	—	2	1	no	—
147	C₄×C₁₇₃	—	2	1	no	—
148	C₄×C₁₇₃	—	4	2	no	—
149	C₄×C₁₇₃	—	4	2	no	—
150	C₄×C₁₇₃	#121	1	2	yes	—
151	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
153	Dic₁₇₃	—	4	1	no	—
154	Dic₁₇₃	—	4	1	no	—
155	Dic₁₇₃	—	4	1	no	—
156	C₂×C₁₇₃	—	2	2	no	—
157	C₂×C₁₇₃	—	2	2	no	—
158	C₂×C₁₇₃	—	4	1	no	—
159	C₂×C₁₇₃	C₂×#121	1	1	yes	characteristic
160	C₂×C₁₇₃	C₂×#137	1	1	yes	characteristic
161	C₂×C₁₇₃	#575	1	1	yes	characteristic
162	D₁₇₃	—	8	1	no	—
163	C₁₇₃	#3056	1	1	yes	characteristic, complemented, Sylow
164	#3056	—	173	1	no	maximal, Sylow
165	C₂×#137	—	173	1	no	—
166	C₂²×M₁₆	—	173	1	no	—
167	#525	—	173	1	no	—
168	C₂×C₂²⋊Q₈	D₁₇₃	1	1	yes	characteristic, complemented
169	#470	—	173	2	no	—
170	#468	—	173	2	no	—
171	#467	—	173	2	no	—
172	#465	—	173	2	no	—
173	#428	—	173	1	no	—
174	#419	—	173	1	no	—
175	#417	—	173	1	no	—
176	C₂×#41	—	173	1	no	—
177	C₂²×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
178	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
179	#137	—	346	4	no	—
180	C₂×Q₁₆	—	173	2	no	—
181	C₂×SD₁₆	—	173	2	no	—
182	C₂×M₁₆	—	346	4	no	—
183	C₂×M₁₆	—	173	1	no	—
184	C₂×M₁₆	—	173	2	no	—
185	C₂×M₁₆	—	173	1	no	—
186	C₂²×C₈	—	173	2	no	—
187	#123	—	346	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₂²⋊Q₈	—	2	1	no	—
190	C₂²⋊Q₈	—	173	1	no	—
191	C₂²⋊Q₈	—	173	1	no	—
192	C₂²⋊Q₈	C₂×D₁₇₃	1	2	yes	—
193	C₂²⋊Q₈	C₂×D₁₇₃	1	2	yes	—
194	#121	—	173	1	no	—
195	#121	—	173	1	no	—
196	#120	—	346	1	no	—
197	C₄×D₄	—	346	1	no	—
198	C₄×D₄	—	346	1	no	—
199	#117	—	173	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	#107	—	173	2	no	—
204	#106	—	173	2	no	—
205	#103	—	173	2	no	—
206	#103	—	173	2	no	—
207	#103	—	173	2	no	—
208	#102	—	173	2	no	—
209	C₂⁴	C₂²⋊D₁₇₃	1	1	yes	characteristic
210	#41	—	346	2	no	—
211	#41	—	346	2	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	173	1	no	—
216	C₂×Q₈	C₂²×D₁₇₃	1	2	yes	—
217	C₂×D₄	—	346	1	no	—
218	C₂×D₄	—	346	1	no	—
219	C₂×D₄	—	346	1	no	—
220	C₂×D₄	—	173	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	346	1	no	—
226	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
232	Q₁₆	—	346	4	no	—
233	SD₁₆	—	346	4	no	—
234	M₁₆	—	346	2	no	—
235	M₁₆	—	346	4	no	—
236	M₁₆	—	173	4	no	—
237	C₂×C₈	—	346	4	no	—
238	C₂×C₈	—	173	2	no	—
239	C₂×C₈	—	173	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	346	1	no	—
244	C₄⋊C₄	—	346	1	no	—
245	C₄⋊C₄	—	173	1	no	—
246	C₄⋊C₄	—	173	1	no	—
247	C₄⋊C₄	C₂²×D₁₇₃	1	2	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	346	1	no	—
251	C₂²⋊C₄	—	346	1	no	—
252	C₂²⋊C₄	—	346	1	no	—
253	C₂²⋊C₄	—	346	1	no	—
254	C₂²⋊C₄	—	346	1	no	—
255	C₄²	—	346	1	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	346	1	no	—
260	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	Q₈	—	2	2	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	346	1	no	—
267	Q₈	—	346	1	no	—
268	D₄	—	346	1	no	—
269	D₄	—	346	1	no	—
270	D₄	—	692	2	no	—
271	D₄	—	692	2	no	—
272	D₄	—	692	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	346	1	no	—
285	C₂×C₄	—	346	1	no	—
286	C₂×C₄	—	346	1	no	—
287	C₂×C₄	—	692	2	no	—
288	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
290	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
292	C₂×C₄	#14583825	1	2	yes	—
293	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
294	C₈	—	346	2	no	—
295	C₈	—	173	4	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	692	1	no	—
305	C₂²	—	692	1	no	—
306	C₂²	—	692	1	no	—
307	C₂²	#19149651	1	2	yes	—
308	C₂²	#19149710	1	2	yes	complemented
309	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	692	1	no	—
315	C₄	—	692	1	no	—
316	C₄	—	692	1	no	—
317	C₄	#19149651	1	2	yes	—
318	C₄	C₂×D₄×D₁₇₃	1	1	yes	characteristic
319	C₄	C₂×#14583825	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	1384	1	no	—
324	C₂	C₂×#19149651	1	1	yes	characteristic
325	C₂	C₂×#19149710	1	1	yes	characteristic
326	C₂	#26012289	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	C₂×C₂²⋊Q₈×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
prime radical	#26012002	C₂
hypercenter	C₂×C₂²⋊Q₈	D₁₇₃
hypocenter	C₁₇₃	#3056
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19149710	C₂	semidirect
2. C₂²×#14583810	C₂	semidirect
4. #26011747	C₂	semidirect
5. #26011739	C₂	semidirect
6. #26011730	C₂	semidirect
7. #26011722	C₂	semidirect
8. #26011531	C₂	semidirect
9. #26011500	C₂	semidirect
10. #26011490	C₂	semidirect
11. C₂×C₂²⋊Q₈×C₁₇₃	C₂	semidirect
14. C₂×#14583816	C₂²	semidirect
15. C₂×#14583813	C₂²	semidirect
30. #19149612	C₂²	semidirect
31. #19149610	C₂²	semidirect
32. #19149603	C₂²	semidirect
33. #19149603	C₂²	semidirect
34. #19149602	C₂²	semidirect
35. #19149599	C₂²	semidirect
57. #14583810	D₄	semidirect
141. C₂²×C₁₇₃	#137	semidirect
163. C₁₇₃	#3056	semidirect
168. C₂×C₂²⋊Q₈	D₁₇₃	semidirect
308. C₂²	#19149710	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149710	2
C₂²×#14583810	2
#26012002	2
#26011747	2
#26011739	2
#26011730	2
#26011722	2
#26011531	2
#26011500	2
#26011490	2
C₂×C₂²⋊Q₈×C₁₇₃	2
#3056	173