Subgroups — #36363178

Order 22144.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19149705	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, complemented
4	#26011742	C₂	1	2	yes	maximal, complemented
5	#26011735	C₂	1	2	yes	maximal, complemented
6	#26011725	C₂	1	2	yes	maximal, complemented
7	#26011718	C₂	1	2	yes	maximal, complemented
8	#26011531	C₂	1	1	yes	maximal, characteristic, complemented
9	#26011496	C₂	1	1	yes	maximal, characteristic, complemented
10	#26011490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#121×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583826	C₂²	1	1	yes	characteristic
13	#19149705	—	2	4	no	—
14	C₂×#14583814	C₂²	1	2	yes	complemented
15	C₂×#14583811	C₂²	1	2	yes	complemented
16	C₂×#14583810	—	2	4	no	—
17	C₂×#14583810	C₂²	1	1	yes	characteristic
18	C₂×#14583810	C₂²	1	1	yes	characteristic
19	C₂×#14583810	C₂²	1	2	yes	complemented
20	C₂²×#13063436	C₂²	1	2	yes	complemented
21	#19149662	—	2	1	no	—
22	#19149658	C₂²	1	1	yes	characteristic, complemented
23	#19149658	C₂²	1	1	yes	characteristic, complemented
24	#19149653	C₂²	1	1	yes	characteristic, complemented
25	#19149653	C₂²	1	1	yes	characteristic, complemented
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	#19149602	C₂²	1	2	yes	complemented
33	#19149600	C₂²	1	2	yes	complemented
34	#19149600	C₂²	1	2	yes	complemented
35	#19149599	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₁₇₃	—	2	1	no	—
37	C₂²×D₄×C₁₇₃	C₂²	1	1	yes	characteristic
38	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
39	#121×C₁₇₃	—	2	1	no	—
40	#121×C₁₇₃	—	2	1	no	—
41	#121×C₁₇₃	C₂²	1	2	yes	—
42	#121×C₁₇₃	C₂²	1	2	yes	—
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	#14583814	—	2	4	no	—
54	#14583811	—	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	—	2	1	no	—
72	C₂⁴×C₁₇₃	D₄	1	1	yes	characteristic
73	C₂×D₄×C₁₇₃	—	2	2	no	—
74	C₂×D₄×C₁₇₃	—	2	2	no	—
75	C₂×D₄×C₁₇₃	—	2	2	no	—
76	C₂×D₄×C₁₇₃	—	2	1	no	—
77	C₂×D₄×C₁₇₃	—	2	2	no	—
78	C₂×D₄×C₁₇₃	—	4	4	no	—
79	C₂×D₄×C₁₇₃	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₇₃	—	2	2	no	—
81	C₂²×C₄×C₁₇₃	—	2	2	no	—
82	C₂²×C₄×C₁₇₃	—	2	1	no	—
83	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₇₃	—	2	1	no	—
90	C₄⋊C₄×C₁₇₃	C₂³	1	2	yes	—
91	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
92	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
93	C₂²×D₁₇₃	—	2	1	no	—
94	Dic₃₄₆	—	2	1	no	—
95	Dic₃₄₆	—	2	1	no	—
96	D₆₉₂	—	2	1	no	—
97	D₆₉₂	—	2	1	no	—
98	C₂²⋊D₁₇₃	—	4	2	no	—
99	C₂²⋊D₁₇₃	—	4	2	no	—
100	C₂²⋊D₁₇₃	—	4	2	no	—
101	C₂×Dic₁₇₃	—	2	1	no	—
102	C₂×Dic₁₇₃	—	2	1	no	—
103	C₂×Dic₁₇₃	—	2	1	no	—
104	C₄×D₁₇₃	—	4	2	no	—
105	#13063436	—	2	2	no	—
106	#13063436	C₂×D₄	1	4	yes	complemented
107	C₂³×C₁₇₃	—	2	2	no	—
108	C₂³×C₁₇₃	—	2	2	no	—
109	C₂³×C₁₇₃	—	2	2	no	—
110	C₂³×C₁₇₃	—	4	1	no	—
111	C₂³×C₁₇₃	—	4	2	no	—
112	C₂³×C₁₇₃	—	4	2	no	—
113	C₂³×C₁₇₃	—	4	2	no	—
114	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
117	D₄×C₁₇₃	—	2	2	no	—
118	D₄×C₁₇₃	—	2	2	no	—
119	D₄×C₁₇₃	—	4	4	no	—
120	D₄×C₁₇₃	—	4	4	no	—
121	D₄×C₁₇₃	—	4	2	no	—
122	C₂×C₄×C₁₇₃	—	2	2	no	—
123	C₂×C₄×C₁₇₃	—	2	2	no	—
124	C₂×C₄×C₁₇₃	—	2	2	no	—
125	C₂×C₄×C₁₇₃	—	2	1	no	—
126	C₂×C₄×C₁₇₃	—	2	1	no	—
127	C₂×C₄×C₁₇₃	—	2	1	no	—
128	C₂×C₄×C₁₇₃	—	2	1	no	—
129	C₂×C₄×C₁₇₃	—	2	2	no	—
130	C₂×C₄×C₁₇₃	—	4	2	no	—
131	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
132	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₇₃	#41	1	2	yes	—
136	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
137	C₂²×C₁₇₃	—	2	1	no	—
138	C₂²×C₁₇₃	—	2	1	no	—
139	C₂²×C₁₇₃	—	2	1	no	—
140	C₂²×C₁₇₃	—	2	1	no	—
141	C₂²×C₁₇₃	—	2	1	no	—
142	C₂²×C₁₇₃	—	2	1	no	—
143	C₂²×C₁₇₃	—	2	1	no	—
144	C₂²×C₁₇₃	—	4	4	no	—
145	C₂²×C₁₇₃	—	4	2	no	—
146	C₂²×C₁₇₃	—	4	2	no	—
147	C₂²×C₁₇₃	—	4	2	no	—
148	C₂²×C₁₇₃	—	8	4	no	—
149	C₂²×C₁₇₃	#121	1	2	yes	—
150	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	complemented
151	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₇₃	—	4	1	no	—
153	C₂×D₁₇₃	—	4	1	no	—
154	C₂×D₁₇₃	—	4	1	no	—
155	C₄×C₁₇₃	—	2	1	no	—
156	C₄×C₁₇₃	—	2	1	no	—
157	C₄×C₁₇₃	—	4	2	no	—
158	C₄×C₁₇₃	#121	1	2	yes	—
159	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
161	Dic₁₇₃	—	4	1	no	—
162	Dic₁₇₃	—	4	1	no	—
163	Dic₁₇₃	—	4	1	no	—
164	C₂×C₁₇₃	—	2	2	no	—
165	C₂×C₁₇₃	—	2	2	no	—
166	C₂×C₁₇₃	—	4	1	no	—
167	C₂×C₁₇₃	—	8	2	no	—
168	C₂×C₁₇₃	C₂×#121	1	1	yes	characteristic
169	C₂×C₁₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₁₇₃	#575	1	1	yes	characteristic
171	D₁₇₃	—	8	1	no	—
172	C₁₇₃	#3055	1	1	yes	characteristic, complemented, Sylow
173	#3055	—	173	1	no	maximal, Sylow
174	C₂×AGL(1,ℤ/8)	—	173	1	no	—
175	C₂²×M₁₆	—	173	1	no	—
176	#525	—	173	1	no	—
177	C₂×#121	D₁₇₃	1	1	yes	characteristic, complemented
178	#469	—	173	2	no	—
179	#468	—	173	2	no	—
180	#466	—	173	2	no	—
181	#465	—	173	2	no	—
182	#428	—	173	1	no	—
183	#418	—	173	1	no	—
184	#417	—	173	1	no	—
185	C₂×#41	—	173	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	C₂×D₁₇₃	1	1	yes	characteristic
188	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
189	AGL(1,ℤ/8)	—	346	4	no	—
190	C₂×SD₁₆	—	173	2	no	—
191	C₂×D₈	—	173	2	no	—
192	C₂×M₁₆	—	346	4	no	—
193	C₂×M₁₆	—	173	1	no	—
194	C₂×M₁₆	—	173	2	no	—
195	C₂×M₁₆	—	173	1	no	—
196	C₂²×C₈	—	173	2	no	—
197	#123	—	346	1	no	—
198	C₂²⋊Q₈	—	173	1	no	—
199	C₂²⋊Q₈	—	173	1	no	—
200	#121	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	—	173	1	no	—
203	#121	—	173	1	no	—
204	#121	C₂×D₁₇₃	1	2	yes	—
205	#121	C₂×D₁₇₃	1	2	yes	—
206	#120	—	346	1	no	—
207	C₄×D₄	—	346	1	no	—
208	C₄×D₄	—	346	1	no	—
209	#117	—	173	1	no	—
210	C₂×C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	#107	—	173	2	no	—
213	#106	—	173	2	no	—
214	#103	—	173	2	no	—
215	#102	—	173	2	no	—
216	#102	—	173	2	no	—
217	#102	—	173	2	no	—
218	C₂⁴	—	2	1	no	—
219	C₂⁴	C₂²⋊D₁₇₃	1	1	yes	characteristic
220	#41	—	346	2	no	—
221	#41	—	346	2	no	—
222	C₂×Q₈	—	173	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	346	1	no	—
230	C₂×D₄	—	346	1	no	—
231	C₂×D₄	—	346	1	no	—
232	C₂×D₄	—	173	1	no	—
233	C₂×D₄	C₂²×D₁₇₃	1	2	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	346	1	no	—
238	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
244	SD₁₆	—	346	4	no	—
245	D₈	—	346	4	no	—
246	M₁₆	—	346	2	no	—
247	M₁₆	—	346	4	no	—
248	M₁₆	—	173	4	no	—
249	C₂×C₈	—	346	4	no	—
250	C₂×C₈	—	173	2	no	—
251	C₂×C₈	—	173	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	346	1	no	—
254	C₄⋊C₄	—	346	1	no	—
255	C₄⋊C₄	—	173	1	no	—
256	C₄⋊C₄	—	173	1	no	—
257	C₄⋊C₄	C₂²×D₁₇₃	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	346	1	no	—
261	C₂²⋊C₄	—	346	1	no	—
262	C₂²⋊C₄	—	346	1	no	—
263	C₂²⋊C₄	—	346	1	no	—
264	C₂²⋊C₄	—	346	1	no	—
265	C₄²	—	346	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	346	1	no	—
274	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
276	C₂³	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
277	Q₈	—	346	1	no	—
278	Q₈	—	346	1	no	—
279	D₄	—	2	2	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	346	1	no	—
285	D₄	—	346	1	no	—
286	D₄	—	692	2	no	—
287	D₄	—	692	2	no	—
288	D₄	—	692	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	346	1	no	—
299	C₂×C₄	—	346	1	no	—
300	C₂×C₄	—	346	1	no	—
301	C₂×C₄	—	692	2	no	—
302	C₂×C₄	D₄×D₁₇₃	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
306	C₂×C₄	#14583825	1	2	yes	—
307	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
308	C₈	—	346	2	no	—
309	C₈	—	173	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	4	no	—
322	C₂²	—	692	1	no	—
323	C₂²	—	692	1	no	—
324	C₂²	—	692	1	no	—
325	C₂²	#19149651	1	2	yes	—
326	C₂²	#19149705	1	2	yes	complemented
327	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	692	1	no	—
332	C₄	—	692	1	no	—
333	C₄	—	692	1	no	—
334	C₄	#19149651	1	2	yes	—
335	C₄	C₂×D₄×D₁₇₃	1	1	yes	characteristic
336	C₄	C₂×#14583825	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	8	2	no	—
341	C₂	—	1384	1	no	—
342	C₂	C₂×#19149651	1	1	yes	characteristic
343	C₂	C₂×#19149705	1	1	yes	characteristic
344	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×#121×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
hypercenter	C₂×#121	D₁₇₃
hypocenter	C₁₇₃	#3055
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. C₂×#19149705	C₂	semidirect
2. C₂²×#14583810	C₂	semidirect
3. #26012002	C₂	semidirect
4. #26011742	C₂	semidirect
5. #26011735	C₂	semidirect
6. #26011725	C₂	semidirect
7. #26011718	C₂	semidirect
8. #26011531	C₂	semidirect
9. #26011496	C₂	semidirect
10. #26011490	C₂	semidirect
11. C₂×#121×C₁₇₃	C₂	semidirect
14. C₂×#14583814	C₂²	semidirect
15. C₂×#14583811	C₂²	semidirect
19. C₂×#14583810	C₂²	semidirect
20. C₂²×#13063436	C₂²	semidirect
22. #19149658	C₂²	semidirect
23. #19149658	C₂²	semidirect
24. #19149653	C₂²	semidirect
25. #19149653	C₂²	semidirect
30. #19149612	C₂²	semidirect
31. #19149610	C₂²	semidirect
32. #19149602	C₂²	semidirect
33. #19149600	C₂²	semidirect
34. #19149600	C₂²	semidirect
35. #19149599	C₂²	semidirect
57. #14583810	D₄	semidirect
106. #13063436	C₂×D₄	semidirect
150. C₂²×C₁₇₃	AGL(1,ℤ/8)	semidirect
172. C₁₇₃	#3055	semidirect
177. C₂×#121	D₁₇₃	semidirect
326. C₂²	#19149705	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19149705	2
C₂²×#14583810	2
#26012002	2
#26011742	2
#26011735	2
#26011725	2
#26011718	2
#26011531	2
#26011496	2
#26011490	2
C₂×#121×C₁₇₃	2
#3055	173