Subgroups — C₂²×#19098370 #34441192

Order 20224.

1174 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14579097	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098370	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#135×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929608	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
7	C₂²×C₇₉⋊C₁₆	—	2	3	no	—
8	C₂²×C₇₉⋊C₁₆	C₂²	1	6	yes	complemented
9	C₂×#14579097	—	2	3	no	—
10	C₂×#14579097	C₂²	1	6	yes	complemented
11	#19098370	C₂²	1	16	yes	direct product factor
12	C₂²×#41×C₇₉	—	2	1	no	—
13	C₂×#135×C₇₉	—	2	3	no	—
14	C₂×#135×C₇₉	—	2	3	no	—
15	C₂×#135×C₇₉	C₄	1	6	yes	—
16	C₂×#135×C₇₉	C₂²	1	6	yes	—
17	C₂²×Q₁₆×C₇₉	C₄	1	1	yes	characteristic
18	C₂²×SD₁₆×C₇₉	—	2	1	no	—
19	C₂²×D₈×C₇₉	C₄	1	1	yes	characteristic
20	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#11929608	—	2	6	no	—
22	C₂×#11929608	—	4	3	no	—
23	C₂²×C₇₉⋊C₈	—	4	1	no	—
24	C₂×C₇₉⋊C₁₆	—	2	6	no	—
25	C₂×C₇₉⋊C₁₆	—	2	6	no	—
26	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	complemented
27	#14579097	—	2	6	no	—
28	#14579097	C₂³	1	4	yes	complemented
29	C₂×#41×C₇₉	—	2	6	no	—
30	C₂×#41×C₇₉	—	2	6	no	—
31	C₂×#41×C₇₉	—	4	3	no	—
32	C₂×#41×C₇₉	—	4	3	no	—
33	C₂²×Q₈×C₇₉	—	2	1	no	—
34	C₂²×D₄×C₇₉	—	2	1	no	—
35	C₂²×D₄×C₇₉	—	4	1	no	—
36	C₂³×C₄×C₇₉	—	4	1	no	—
37	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
38	#135×C₇₉	—	2	12	no	—
39	#135×C₇₉	—	2	6	no	—
40	#135×C₇₉	—	2	6	no	—
41	#135×C₇₉	C₂×C₄	1	12	yes	—
42	#135×C₇₉	C₂³	1	4	yes	—
43	C₂×Q₁₆×C₇₉	—	2	3	no	—
44	C₂×Q₁₆×C₇₉	C₂×C₄	1	6	yes	—
45	C₂×SD₁₆×C₇₉	—	2	6	no	—
46	C₂×SD₁₆×C₇₉	—	2	6	no	—
47	C₂×D₈×C₇₉	—	2	3	no	—
48	C₂×D₈×C₇₉	C₂×C₄	1	6	yes	—
49	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
51	C₂²×C₈×C₇₉	D₄	1	3	yes	—
52	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	D₄	1	3	yes	—
54	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
55	#11929608	—	2	4	no	—
56	#11929608	—	4	6	no	—
57	C₂×C₇₉⋊C₈	—	4	6	no	—
58	C₇₉⋊C₁₆	—	2	4	no	—
59	C₂⁴×C₇₉	—	4	1	no	—
60	C₂⁴×C₇₉	SD₁₆	1	1	yes	characteristic
61	#41×C₇₉	—	2	12	no	—
62	#41×C₇₉	—	2	4	no	—
63	#41×C₇₉	—	4	12	no	—
64	#41×C₇₉	—	4	6	no	—
65	#41×C₇₉	—	4	6	no	—
66	C₂×Q₈×C₇₉	—	2	6	no	—
67	C₂×Q₈×C₇₉	—	4	3	no	—
68	C₂×D₄×C₇₉	—	2	6	no	—
69	C₂×D₄×C₇₉	—	4	6	no	—
70	C₂×D₄×C₇₉	—	4	6	no	—
71	C₂×D₄×C₇₉	—	4	3	no	—
72	C₂²×C₄×C₇₉	—	4	1	no	—
73	C₂²×C₄×C₇₉	—	4	6	no	—
74	C₂²×C₄×C₇₉	—	4	6	no	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	3	yes	—
77	C₂²×C₄×C₇₉	D₈	1	3	yes	—
78	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	SD₁₆	1	3	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
81	Q₁₆×C₇₉	—	2	6	no	—
82	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	—
83	SD₁₆×C₇₉	—	2	12	no	—
84	SD₁₆×C₇₉	—	2	4	no	—
85	D₈×C₇₉	—	2	6	no	—
86	D₈×C₇₉	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	3	yes	—
88	C₂×C₈×C₇₉	C₂²⋊C₄	1	3	yes	—
89	C₂×C₈×C₇₉	C₂²⋊C₄	1	6	yes	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
92	C₂×C₈×C₇₉	C₂×D₄	1	3	yes	—
93	C₂×C₈×C₇₉	C₂×D₄	1	3	yes	—
94	C₂×C₈×C₇₉	C₂×D₄	1	3	yes	—
95	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₇₉	—	2	4	no	—
97	C₂³×C₇₉	—	4	6	no	—
98	C₂³×C₇₉	—	8	4	no	—
99	C₂³×C₇₉	#102	1	3	yes	—
100	C₂³×C₇₉	#102	1	1	yes	characteristic
101	C₂³×C₇₉	C₂×SD₁₆	1	3	yes	—
102	Q₈×C₇₉	—	2	4	no	—
103	Q₈×C₇₉	—	4	6	no	—
104	D₄×C₇₉	—	2	4	no	—
105	D₄×C₇₉	—	4	4	no	—
106	D₄×C₇₉	—	4	12	no	—
107	D₄×C₇₉	—	4	6	no	—
108	C₂×C₄×C₇₉	—	4	6	no	—
109	C₂×C₄×C₇₉	—	4	4	no	—
110	C₂×C₄×C₇₉	—	4	12	no	—
111	C₂×C₄×C₇₉	#102	1	3	yes	—
112	C₂×C₄×C₇₉	#102	1	3	yes	—
113	C₂×C₄×C₇₉	#102	1	6	yes	—
114	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₇₉	C₂×D₈	1	3	yes	—
117	C₂×C₄×C₇₉	C₂×D₈	1	3	yes	—
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	3	yes	—
119	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
121	C₈×C₇₉	C₂×C₂²⋊C₄	1	3	yes	—
122	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
124	C₇₉⋊C₈	—	4	4	no	—
125	C₂²×C₇₉	—	2	6	no	—
126	C₂²×C₇₉	—	2	6	no	—
127	C₂²×C₇₉	—	4	4	no	—
128	C₂²×C₇₉	—	8	12	no	—
129	C₂²×C₇₉	#359	1	4	yes	complemented
130	C₂²×C₇₉	C₂×#102	1	3	yes	—
131	C₂²×C₇₉	C₂×#102	1	3	yes	—
132	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
133	C₄×C₇₉	—	4	4	no	—
134	C₄×C₇₉	C₂×#102	1	3	yes	—
135	C₄×C₇₉	C₂×#102	1	3	yes	—
136	C₄×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
138	C₂²×#359	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	1	no	—
140	C₂×C₇₉	—	2	3	no	—
141	C₂×C₇₉	—	8	4	no	—
142	C₂×C₇₉	C₂×#359	1	6	yes	complemented
143	C₂×C₇₉	C₂²×#102	1	1	yes	characteristic
144	C₂²×#135	D₇₉	1	1	yes	characteristic
145	C₂³×C₁₆	—	79	1	no	—
146	C₂²×#108	—	79	1	no	—
147	C₂×#359	—	79	12	no	—
148	C₇₉	C₂²×#359	1	1	yes	characteristic, complemented, Sylow
149	C₂²×#41	—	2	1	no	—
150	C₂×#135	—	2	3	no	—
151	C₂×#135	—	2	3	no	—
152	C₂×#135	Dic₇₉	1	6	yes	—
153	C₂×#135	C₂×D₇₉	1	6	yes	—
154	C₂²×Q₁₆	Dic₇₉	1	1	yes	characteristic
155	C₂²×SD₁₆	—	2	1	no	—
156	C₂²×D₈	Dic₇₉	1	1	yes	characteristic
157	C₂²×M₁₆	—	158	1	no	—
158	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
159	C₂²×C₁₆	—	79	6	no	—
160	C₂²×C₁₆	—	158	1	no	—
161	C₂²×C₁₆	—	158	3	no	—
162	C₂×#108	—	79	6	no	—
163	C₂×#108	—	158	3	no	—
164	#359	—	79	16	no	—
165	C₂×#41	—	2	6	no	—
166	C₂×#41	—	2	6	no	—
167	C₂×#41	—	4	3	no	—
168	C₂×#41	—	4	3	no	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	4	1	no	—
172	C₂³×C₄	—	4	1	no	—
173	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
174	#135	—	2	12	no	—
175	#135	—	2	6	no	—
176	#135	—	2	6	no	—
177	#135	C₂×Dic₇₉	1	12	yes	—
178	#135	C₂²×D₇₉	1	4	yes	—
179	C₂×Q₁₆	—	2	3	no	—
180	C₂×Q₁₆	C₂×Dic₇₉	1	6	yes	—
181	C₂×SD₁₆	—	2	6	no	—
182	C₂×SD₁₆	—	2	6	no	—
183	C₂×D₈	—	2	3	no	—
184	C₂×D₈	C₂×Dic₇₉	1	6	yes	—
185	C₂×M₁₆	—	316	3	no	—
186	C₂×M₁₆	—	158	6	no	—
187	C₂²×C₈	—	316	1	no	—
188	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
189	C₂²×C₈	C₂×Dic₇₉	1	3	yes	—
190	C₂²×C₈	C₂²⋊D₇₉	1	3	yes	—
191	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₇₉	1	3	yes	—
193	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
194	C₂×C₁₆	—	79	4	no	—
195	C₂×C₁₆	—	158	6	no	—
196	C₂×C₁₆	—	158	6	no	—
197	#108	—	79	4	no	—
198	#108	—	158	6	no	—
199	C₂⁴	—	4	1	no	—
200	C₂⁴	#11929612	1	1	yes	characteristic
201	#41	—	2	12	no	—
202	#41	—	2	4	no	—
203	#41	—	4	12	no	—
204	#41	—	4	6	no	—
205	#41	—	4	6	no	—
206	C₂×Q₈	—	2	6	no	—
207	C₂×Q₈	—	4	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	4	6	no	—
210	C₂×D₄	—	4	6	no	—
211	C₂×D₄	—	4	3	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	6	no	—
214	C₂²×C₄	—	4	6	no	—
215	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₇₉	1	3	yes	—
217	C₂²×C₄	#11929609	1	3	yes	—
218	C₂²×C₄	#11929609	1	1	yes	characteristic
219	C₂²×C₄	#11929612	1	3	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
221	Q₁₆	—	2	6	no	—
222	Q₁₆	C₂²×Dic₇₉	1	4	yes	—
223	SD₁₆	—	2	12	no	—
224	SD₁₆	—	2	4	no	—
225	D₈	—	2	6	no	—
226	D₈	C₂²×Dic₇₉	1	4	yes	—
227	M₁₆	—	316	6	no	—
228	M₁₆	—	158	4	no	—
229	C₂×C₈	—	316	6	no	—
230	C₂×C₈	C₂²⋊Dic₇₉	1	3	yes	—
231	C₂×C₈	C₂²⋊Dic₇₉	1	3	yes	—
232	C₂×C₈	C₂²⋊Dic₇₉	1	6	yes	—
233	C₂×C₈	C₂²×Dic₇₉	1	3	yes	—
234	C₂×C₈	C₂²×Dic₇₉	1	3	yes	—
235	C₂×C₈	C₂×C₂²⋊D₇₉	1	3	yes	—
236	C₂×C₈	C₂×C₂²⋊D₇₉	1	3	yes	—
237	C₂×C₈	C₂×C₂²⋊D₇₉	1	3	yes	—
238	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
239	C₁₆	—	158	4	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	8	4	no	—
243	C₂³	#14579082	1	3	yes	—
244	C₂³	#14579082	1	1	yes	characteristic
245	C₂³	C₂×#11929612	1	3	yes	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	6	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	12	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	6	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	—	4	12	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	#14579082	1	3	yes	—
256	C₂×C₄	#14579082	1	3	yes	—
257	C₂×C₄	#14579082	1	6	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	3	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	3	yes	—
260	C₂×C₄	C₂×#11929609	1	3	yes	—
261	C₂×C₄	C₂×#11929609	1	3	yes	—
262	C₂×C₄	C₂×#11929612	1	3	yes	—
263	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
264	C₈	—	316	4	no	—
265	C₈	C₂×C₂²⋊Dic₇₉	1	3	yes	—
266	C₈	C₂×C₂²⋊Dic₇₉	1	3	yes	—
267	C₈	C₂³×Dic₇₉	1	1	yes	characteristic
268	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
269	C₂²	—	2	6	no	—
270	C₂²	—	2	6	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	8	12	no	—
273	C₂²	#19098370	1	4	yes	direct product factor
274	C₂²	C₂×#14579082	1	3	yes	—
275	C₂²	C₂×#14579082	1	3	yes	—
276	C₂²	C₂²×#11929612	1	1	yes	characteristic
277	C₄	—	4	4	no	—
278	C₄	C₂×#14579082	1	3	yes	—
279	C₄	C₂×#14579082	1	3	yes	—
280	C₄	C₂²×C₂²⋊Dic₇₉	1	1	yes	characteristic
281	C₄	C₂²×#11929609	1	1	yes	characteristic
282	C₂	—	2	3	no	—
283	C₂	—	2	1	no	—
284	C₂	—	8	4	no	—
285	C₂	C₂×#19098370	1	6	yes	direct product factor
286	C₂	C₂²×#14579082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11929609
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting, prime radical	C₂²×#135×C₇₉	C₂
socle	C₂³×C₇₉	#102
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂²×#135	D₇₉
hypocenter	C₇₉	C₂²×#359
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇₉⋊C₁₆	C₂	semidirect
2. C₂²×#14579097	C₂	semidirect
3. C₂×#19098370	C₂	direct (N × H)
8. C₂²×C₇₉⋊C₁₆	C₂²	semidirect
10. C₂×#14579097	C₂²	semidirect
11. #19098370	C₂²	direct (N × H)
26. C₂×C₇₉⋊C₁₆	C₂³	semidirect
28. #14579097	C₂³	semidirect
129. C₂²×C₇₉	#359	semidirect
142. C₂×C₇₉	C₂×#359	semidirect
148. C₇₉	C₂²×#359	semidirect
273. C₂²	#19098370	direct (N × H)
285. C₂	C₂×#19098370	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₉⋊C₁₆	2
C₂²×#14579097	2
C₂×#19098370	2
C₂²×#135×C₇₉	2
C₂²×#359	79

Subgroups — C22×#19098370 #34441192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19098370 #34441192