Subgroups — C₂×C₄×#14579097 #34286365

Order 20224.

1086 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579097	C₂	1	2	yes	maximal
2	C₂×C₄×#11929608	C₂	1	2	yes	maximal
3	C₄×#14579097	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098328	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	4	yes	—
7	C₂×#14579097	C₄	1	16	yes	direct product factor
8	C₂×#14579097	C₂²	1	8	yes	—
9	C₄×#11929608	—	2	4	no	—
10	C₄×#11929608	C₂²	1	8	yes	—
11	C₂×#14579072	—	2	2	no	—
12	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
13	#19098328	C₂²	1	8	yes	—
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₉	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₇₉	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	4	yes	—
20	C₂×#11929608	—	2	4	no	—
21	C₂×#11929608	—	2	8	no	—
22	C₂×#11929608	C₂×C₄	1	16	yes	—
23	C₂×#11929608	C₂³	1	8	yes	—
24	C₂²×C₇₉⋊C₈	—	2	4	no	—
25	#14579097	C₂×C₄	1	32	yes	direct product factor
26	#14579072	—	2	8	no	—
27	C₄×C₇₉⋊C₈	—	2	8	no	—
28	C₂³×C₄×C₇₉	Q₈	1	2	yes	—
29	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	—	2	4	no	—
31	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	2	yes	—
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
44	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
45	C₄×C₈×C₇₉	C₂×C₄	1	16	yes	—
46	#11929608	—	2	16	no	—
47	#11929608	—	2	8	no	—
48	#11929608	C₂²×C₄	1	16	yes	—
49	C₂×C₇₉⋊C₈	—	2	16	no	—
50	C₂×C₇₉⋊C₈	—	2	8	no	—
51	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	—	2	8	no	—
53	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	#41	1	4	yes	—
64	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₇₉	—	2	4	no	—
66	C₂×C₈×C₇₉	—	2	4	no	—
67	C₂×C₈×C₇₉	—	2	4	no	—
68	C₂×C₈×C₇₉	C₄²	1	8	yes	—
69	C₂×C₈×C₇₉	C₄²	1	8	yes	—
70	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
73	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
74	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
75	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
77	C₄²×C₇₉	C₂×D₄	1	2	yes	—
78	C₄²×C₇₉	C₂×D₄	1	2	yes	—
79	C₂³×C₇₉	—	2	4	no	—
80	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₇₉	C₄×Q₈	1	4	yes	—
83	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	—	2	8	no	—
86	C₂×C₄×C₇₉	—	2	8	no	—
87	C₂×C₄×C₇₉	#108	1	8	yes	complemented
88	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₇₉	#117	1	2	yes	—
95	C₂×C₄×C₇₉	#117	1	2	yes	—
96	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
103	C₈×C₇₉	—	2	4	no	—
104	C₈×C₇₉	C₂×C₄²	1	8	yes	—
105	C₇₉⋊C₈	—	2	16	no	—
106	C₂²×C₇₉	—	2	2	no	—
107	C₂²×C₇₉	—	2	2	no	—
108	C₂²×C₇₉	—	2	8	no	—
109	C₂²×C₇₉	C₄×C₄⋊C₄	1	2	yes	—
110	C₂²×C₇₉	C₄×C₄⋊C₄	1	2	yes	—
111	C₂²×C₇₉	C₂×#108	1	4	yes	—
112	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₇₉	C₂×C₄×Q₈	1	2	yes	—
114	C₄×C₇₉	—	2	4	no	—
115	C₄×C₇₉	C₄×C₄⋊C₄	1	2	yes	—
116	C₄×C₇₉	C₄×C₄⋊C₄	1	2	yes	—
117	C₄×C₇₉	C₂×#108	1	8	yes	complemented
118	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
119	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
120	C₄×C₇₉	C₂×C₄×D₄	1	2	yes	—
121	C₂×C₄×#108	—	79	1	no	maximal, Sylow
122	C₂×C₇₉	—	2	2	no	—
123	C₂×C₇₉	—	2	2	no	—
124	C₂×C₇₉	C₄×#108	1	4	yes	complemented
125	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₇₉	C₂²×#108	1	2	yes	—
127	C₂²×#108	—	79	2	no	—
128	C₂×C₄×M₁₆	—	79	2	no	—
129	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic
130	C₄×#108	—	79	8	no	—
131	C₂×#341	—	79	2	no	—
132	C₇₉	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
133	C₂²×M₁₆	—	79	4	no	—
134	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
136	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
137	C₂×#108	—	79	8	no	—
138	C₂×#108	—	79	16	no	—
139	C₄×M₁₆	—	79	8	no	—
140	C₄×M₁₆	—	158	4	no	—
141	C₂×#97	—	158	2	no	—
142	C₂×C₄×C₈	—	158	2	no	—
143	C₂×C₄×C₈	Dic₇₉	1	4	yes	—
144	C₂×C₄×C₈	Dic₇₉	1	4	yes	—
145	C₂×C₄×C₈	C₂×D₇₉	1	4	yes	—
146	#341	—	79	8	no	—
147	C₂³×C₄	Dic₁₅₈	1	2	yes	—
148	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
149	C₂×M₁₆	—	79	16	no	—
150	C₂×M₁₆	—	79	8	no	—
151	C₂×M₁₆	—	158	4	no	—
152	C₂×M₁₆	—	158	8	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	158	4	no	—
155	C₂²×C₈	C₄×D₇₉	1	8	yes	—
156	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
157	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
158	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
159	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
160	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
161	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
162	C₂×C₄²	Dic₁₅₈	1	2	yes	—
163	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
164	C₂×C₄²	D₃₁₆	1	2	yes	—
165	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
167	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
168	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
169	#108	—	79	32	no	—
170	#97	—	158	8	no	—
171	C₄×C₈	—	158	8	no	—
172	C₄×C₈	C₂×Dic₇₉	1	16	yes	—
173	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
176	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
177	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
184	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
185	C₂²×C₄	#11929624	1	4	yes	—
186	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
187	M₁₆	—	79	16	no	—
188	M₁₆	—	158	8	no	—
189	M₁₆	—	158	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	158	16	no	—
194	C₂×C₈	—	158	8	no	—
195	C₂×C₈	C₄×Dic₇₉	1	8	yes	—
196	C₂×C₈	C₄×Dic₇₉	1	8	yes	—
197	C₂×C₈	C₂×C₄×D₇₉	1	8	yes	—
198	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
199	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
200	C₄²	C₄⋊Dic₇₉	1	2	yes	—
201	C₄²	C₄⋊Dic₇₉	1	4	yes	—
202	C₄²	C₄⋊Dic₇₉	1	2	yes	—
203	C₄²	C₂²×Dic₇₉	1	4	yes	—
204	C₄²	C₂×D₃₁₆	1	2	yes	—
205	C₄²	C₂×D₃₁₆	1	2	yes	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
208	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
209	C₂³	C₄×Dic₁₅₈	1	4	yes	—
210	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	#14579097	1	8	yes	direct product factor
215	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
216	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
221	C₂×C₄	#14579120	1	2	yes	—
222	C₂×C₄	#14579120	1	2	yes	—
223	C₂×C₄	C₄×D₃₁₆	1	4	yes	—
224	C₂×C₄	C₄×D₃₁₆	1	4	yes	—
225	C₂×C₄	C₄×Dic₁₅₈	1	4	yes	—
226	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
227	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
228	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
229	C₂×C₄	C₂×#11929624	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	158	16	no	—
232	C₈	C₂×C₄×Dic₇₉	1	8	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	8	no	—
236	C₂²	C₄×C₄⋊Dic₇₉	1	2	yes	—
237	C₂²	C₄×C₄⋊Dic₇₉	1	2	yes	—
238	C₂²	C₂×#14579097	1	4	yes	—
239	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
240	C₂²	C₂×C₄×Dic₁₅₈	1	2	yes	—
241	C₄	—	2	4	no	—
242	C₄	C₄×C₄⋊Dic₇₉	1	2	yes	—
243	C₄	C₄×C₄⋊Dic₇₉	1	2	yes	—
244	C₄	C₂×#14579097	1	8	yes	direct product factor
245	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
246	C₄	C₂×#14579120	1	1	yes	characteristic
247	C₄	C₂×C₄×D₃₁₆	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₄×#14579097	1	4	yes	direct product factor
251	C₂	C₂×C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
252	C₂	C₂²×#14579097	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₃₁₆
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂×C₄×#108
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#14579097	C₂	direct (N × H)
7. C₂×#14579097	C₄	direct (N × H)
25. #14579097	C₂×C₄	direct (N × H)
87. C₂×C₄×C₇₉	#108	semidirect
117. C₄×C₇₉	C₂×#108	semidirect
124. C₂×C₇₉	C₄×#108	semidirect
132. C₇₉	C₂×C₄×#108	semidirect
214. C₂×C₄	#14579097	direct (N × H)
244. C₄	C₂×#14579097	direct (N × H)
250. C₂	C₄×#14579097	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579097	2
C₂×C₄×#11929608	2
C₄×#14579097	2
C₂×#19098328	2
C₂²×C₄×C₈×C₇₉	2
C₂×C₄×#108	79

Subgroups — C2×C4×#14579097 #34286365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14579097 #34286365