Subgroups — #36370422

Order 22144.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012524	C₂	1	2	yes	maximal, complemented
2	#26012523	C₂	1	2	yes	maximal, complemented
3	#26012520	C₂	1	2	yes	maximal, complemented
4	#26012518	C₂	1	2	yes	maximal, complemented
5	#26012511	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012507	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012462	C₂	1	1	yes	maximal, characteristic, complemented
8	#26012461	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₇₃⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂²⋊#13063443	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#19149639	C₂	1	1	yes	maximal, characteristic
12	C₂×#14583836	C₂²	1	2	yes	complemented
13	C₂×#14583835	C₂²	1	2	yes	complemented
14	C₂×D₁₇₃⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×#13063443	C₂²	1	2	yes	complemented
16	#19149746	C₂²	1	2	yes	complemented
17	C₄⋊#13063443	C₂²	1	2	yes	complemented
18	#19149736	C₂²	1	2	yes	complemented
19	#19149736	C₂²	1	2	yes	complemented
20	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
21	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
22	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
23	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
24	C₂²⋊#13063443	C₂²	1	2	yes	complemented
25	C₂²⋊#13063443	C₂²	1	2	yes	complemented
26	#19149734	C₂²	1	2	yes	complemented
27	C₄×#13063443	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊D₁₇₃	C₄	1	1	yes	characteristic
29	C₂³×Dic₁₇₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
31	#19149639	C₄	1	1	yes	characteristic
32	#19149639	C₄	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	#19149639	C₂²	1	1	yes	characteristic
38	#19149639	C₂²	1	1	yes	characteristic
39	C₂×#14583805	C₄	1	1	yes	characteristic
40	C₂×#14583803	C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₁₇₃	C₄	1	1	yes	characteristic
43	#14583836	—	4	2	no	—
44	#14583835	—	4	2	no	—
45	D₁₇₃⋊C₈	—	4	2	no	—
46	C₂×#13063443	—	4	2	no	—
47	C₂×#13063443	C₂³	1	2	yes	—
48	C₂×#13063443	C₂³	1	2	yes	—
49	C₂×#13063443	C₂³	1	2	yes	—
50	C₂×#13063443	C₂³	1	2	yes	—
51	C₂³×D₁₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₇₃	—	2	4	no	—
53	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₇₃	—	4	2	no	—
58	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇₃	—	4	2	no	—
66	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
70	#14583805	C₂×C₄	1	1	yes	characteristic
71	#14583805	C₂×C₄	1	1	yes	characteristic
72	#14583805	C₂×C₄	1	1	yes	characteristic
73	#14583805	C₂×C₄	1	1	yes	characteristic
74	#14583803	C₂×C₄	1	1	yes	characteristic
75	#14583803	C₂×C₄	1	1	yes	characteristic
76	#14583803	C₂×C₄	1	1	yes	characteristic
77	#14583803	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
82	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
89	#13063443	—	2	2	no	—
90	#13063443	—	2	2	no	—
91	#13063443	—	2	2	no	—
92	#13063443	—	2	2	no	—
93	C₂²×D₁₇₃	—	2	2	no	—
94	C₂²×D₁₇₃	—	2	2	no	—
95	C₂²×D₁₇₃	—	4	1	no	—
96	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²⋊D₁₇₃	—	2	2	no	—
99	C₂²⋊D₁₇₃	—	2	2	no	—
100	C₂²⋊D₁₇₃	—	2	2	no	—
101	C₂²⋊D₁₇₃	M₁₆	1	4	yes	complemented
102	C₂×Dic₁₇₃	—	2	2	no	—
103	C₂×Dic₁₇₃	—	2	2	no	—
104	C₂×Dic₁₇₃	—	2	2	no	—
105	C₂×Dic₁₇₃	—	2	2	no	—
106	C₂×Dic₁₇₃	—	4	1	no	—
107	C₂×Dic₁₇₃	—	4	1	no	—
108	C₂×Dic₁₇₃	—	4	1	no	—
109	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
115	C₄×D₁₇₃	—	2	2	no	—
116	C₄×D₁₇₃	—	2	2	no	—
117	C₄×D₁₇₃	—	4	1	no	—
118	C₄×D₁₇₃	—	4	1	no	—
119	C₂³×C₁₇₃	—	2	2	no	—
120	C₂³×C₁₇₃	—	2	2	no	—
121	C₂³×C₁₇₃	—	4	1	no	—
122	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇₃	—	2	2	no	—
126	C₂×C₄×C₁₇₃	—	4	1	no	—
127	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₁₇₃	—	2	1	no	—
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	—	4	4	no	—
139	C₂²×C₁₇₃	C₂×M₁₆	1	2	yes	—
140	C₂²×C₁₇₃	#131	1	2	yes	—
141	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×D₁₇₃	—	2	1	no	—
143	C₂×D₁₇₃	—	2	1	no	—
144	C₂×D₁₇₃	—	2	1	no	—
145	C₂×D₁₇₃	—	2	1	no	—
146	C₂×D₁₇₃	—	2	1	no	—
147	C₂×D₁₇₃	—	4	4	no	—
148	C₂×D₁₇₃	C₂×M₁₆	1	2	yes	—
149	C₄×C₁₇₃	—	2	1	no	—
150	C₄×C₁₇₃	—	2	1	no	—
151	C₄×C₁₇₃	—	2	1	no	—
152	C₄×C₁₇₃	—	2	1	no	—
153	Dic₁₇₃	—	2	1	no	—
154	Dic₁₇₃	—	2	1	no	—
155	Dic₁₇₃	—	2	1	no	—
156	Dic₁₇₃	—	2	1	no	—
157	Dic₁₇₃	C₂×M₁₆	1	2	yes	—
158	Dic₁₇₃	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₁₇₃	—	2	2	no	—
160	C₂×C₁₇₃	—	2	2	no	—
161	C₂×C₁₇₃	—	4	1	no	—
162	C₂×C₁₇₃	#518	1	1	yes	characteristic
163	C₂×C₁₇₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
165	D₁₇₃	—	2	2	no	—
166	D₁₇₃	—	4	1	no	—
167	C₁₇₃	#2974	1	1	yes	characteristic, complemented, Sylow
168	#2974	—	173	1	no	maximal, Sylow
169	C₂×C₄×D₄	—	173	1	no	—
170	#436	—	173	2	no	—
171	#435	—	173	2	no	—
172	#435	—	173	2	no	—
173	C₈×D₄	—	173	2	no	—
174	#432	—	173	1	no	—
175	#431	—	173	1	no	—
176	#407	—	173	1	no	—
177	#407	—	173	1	no	—
178	C₂×C₂²⋊C₈	—	173	1	no	—
179	C₂×C₂²⋊C₈	—	173	1	no	—
180	C₂²×D₄	—	173	1	no	—
181	C₂³×C₄	—	173	1	no	—
182	C₂³×C₄	—	173	1	no	—
183	C₂×M₁₆	—	173	2	no	—
184	C₂×M₁₆	—	173	2	no	—
185	C₂²×C₈	—	173	2	no	—
186	C₂²×C₈	—	173	2	no	—
187	C₄×D₄	—	173	1	no	—
188	C₄×D₄	—	173	1	no	—
189	C₄×D₄	—	173	1	no	—
190	C₄×D₄	—	173	1	no	—
191	C₄×D₄	—	173	1	no	—
192	C₄×D₄	—	173	1	no	—
193	C₄×D₄	—	173	1	no	—
194	C₄×D₄	—	173	1	no	—
195	C₂×C₄⋊C₄	—	173	1	no	—
196	C₂×C₂²⋊C₄	—	173	1	no	—
197	C₂×C₂²⋊C₄	#10621924	1	1	yes	characteristic
198	C₂×C₄²	—	173	1	no	—
199	C₄⋊C₈	—	173	2	no	—
200	C₄⋊C₈	—	173	2	no	—
201	C₂²⋊C₈	—	173	1	no	—
202	C₂²⋊C₈	—	173	1	no	—
203	C₂²⋊C₈	—	173	1	no	—
204	C₂²⋊C₈	—	173	2	no	—
205	C₂²⋊C₈	—	173	2	no	—
206	C₂²⋊C₈	—	173	1	no	—
207	C₂²⋊C₈	—	173	2	no	—
208	C₂²⋊C₈	—	173	2	no	—
209	#97	—	173	2	no	—
210	C₄×C₈	—	173	2	no	—
211	C₂⁴	—	173	1	no	—
212	C₂⁴	C₂×#10621924	1	1	yes	characteristic
213	C₂×D₄	—	346	4	no	—
214	C₂×D₄	—	173	1	no	—
215	C₂×D₄	—	173	1	no	—
216	C₂×D₄	—	173	1	no	—
217	C₂×D₄	—	173	1	no	—
218	C₂²×C₄	—	173	1	no	—
219	C₂²×C₄	—	173	1	no	—
220	C₂²×C₄	—	173	1	no	—
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	—	173	1	no	—
226	C₂²×C₄	—	173	1	no	—
227	C₂²×C₄	—	173	1	no	—
228	C₂²×C₄	—	173	1	no	—
229	C₂²×C₄	—	692	2	no	—
230	C₂²×C₄	—	692	2	no	—
231	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
232	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
233	M₁₆	—	692	2	no	—
234	M₁₆	—	692	2	no	—
235	C₂×C₈	—	173	2	no	—
236	C₂×C₈	—	173	2	no	—
237	C₂×C₈	—	173	2	no	—
238	C₂×C₈	—	173	2	no	—
239	C₂×C₈	—	692	2	no	—
240	C₂×C₈	—	692	2	no	—
241	C₄⋊C₄	—	173	1	no	—
242	C₄⋊C₄	—	173	1	no	—
243	C₄⋊C₄	—	173	1	no	—
244	C₄⋊C₄	—	173	1	no	—
245	C₂²⋊C₄	—	173	1	no	—
246	C₂²⋊C₄	—	173	1	no	—
247	C₂²⋊C₄	—	173	1	no	—
248	C₂²⋊C₄	—	173	1	no	—
249	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
250	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
251	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
252	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
253	C₄²	—	173	1	no	—
254	C₄²	—	173	1	no	—
255	C₄²	—	173	1	no	—
256	C₄²	—	173	1	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	346	2	no	—
261	C₂³	—	346	2	no	—
262	C₂³	—	173	1	no	—
263	C₂³	—	173	1	no	—
264	C₂³	—	692	1	no	—
265	C₂³	C₂²×#10621924	1	1	yes	characteristic
266	C₂³	C₂²×#10621924	1	1	yes	characteristic
267	C₂³	C₂²×#10621924	1	1	yes	characteristic
268	D₄	—	346	2	no	—
269	D₄	—	346	2	no	—
270	D₄	—	346	2	no	—
271	D₄	—	173	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	346	2	no	—
275	C₂×C₄	—	346	2	no	—
276	C₂×C₄	—	346	2	no	—
277	C₂×C₄	—	346	2	no	—
278	C₂×C₄	—	346	2	no	—
279	C₂×C₄	—	346	2	no	—
280	C₂×C₄	—	173	1	no	—
281	C₂×C₄	—	173	1	no	—
282	C₂×C₄	—	173	1	no	—
283	C₂×C₄	—	173	1	no	—
284	C₂×C₄	—	173	1	no	—
285	C₂×C₄	—	173	1	no	—
286	C₂×C₄	—	692	1	no	—
287	C₂×C₄	—	692	1	no	—
288	C₂×C₄	—	692	1	no	—
289	C₂×C₄	—	692	1	no	—
290	C₂×C₄	—	692	1	no	—
291	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
292	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
293	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
294	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
295	C₈	—	346	2	no	—
296	C₈	—	346	2	no	—
297	C₈	—	346	2	no	—
298	C₈	—	346	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	346	1	no	—
308	C₂²	—	346	1	no	—
309	C₂²	—	346	1	no	—
310	C₂²	—	346	1	no	—
311	C₂²	—	346	1	no	—
312	C₂²	—	173	2	no	—
313	C₂²	—	692	4	no	—
314	C₂²	D₁₇₃⋊M₁₆	1	2	yes	—
315	C₂²	#19149766	1	2	yes	—
316	C₂²	C₂³×#10621924	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	346	1	no	—
322	C₄	—	346	1	no	—
323	C₄	—	346	1	no	—
324	C₄	—	346	1	no	—
325	C₄	—	173	2	no	—
326	C₄	—	173	2	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	346	2	no	—
331	C₂	—	692	1	no	—
332	C₂	#26012558	1	1	yes	characteristic
333	C₂	C₂×D₁₇₃⋊M₁₆	1	1	yes	characteristic
334	C₂	C₂×#19149766	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	C₂×C₂²⋊C₄×C₁₇₃	C₄
prime radical	C₂²×C₂²⋊D₁₇₃	C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#10621924
hypocenter	C₁₇₃	#2974
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #26012524	C₂	semidirect
2. #26012523	C₂	semidirect
3. #26012520	C₂	semidirect
4. #26012518	C₂	semidirect
5. #26012511	C₂	semidirect
6. #26012507	C₂	semidirect
7. #26012462	C₂	semidirect
8. #26012461	C₂	semidirect
9. D₁₇₃⋊(C₂²⋊C₈)	C₂	semidirect
10. C₂×C₂²⋊#13063443	C₂	semidirect
12. C₂×#14583836	C₂²	semidirect
13. C₂×#14583835	C₂²	semidirect
14. C₂×D₁₇₃⋊C₈	C₂²	semidirect
15. C₂²×#13063443	C₂²	semidirect
16. #19149746	C₂²	semidirect
17. C₄⋊#13063443	C₂²	semidirect
18. #19149736	C₂²	semidirect
19. #19149736	C₂²	semidirect
24. C₂²⋊#13063443	C₂²	semidirect
25. C₂²⋊#13063443	C₂²	semidirect
26. #19149734	C₂²	semidirect
27. C₄×#13063443	C₂²	semidirect
101. C₂²⋊D₁₇₃	M₁₆	semidirect
167. C₁₇₃	#2974	semidirect

Maximal subgroups

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

subgroup	index
#26012524	2
#26012523	2
#26012520	2
#26012518	2
#26012511	2
#26012507	2
#26012462	2
#26012461	2
D₁₇₃⋊(C₂²⋊C₈)	2
C₂×C₂²⋊#13063443	2
C₂×#19149639	2
#2974	173