Subgroups — #44487604

Order 30592.

552 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#21439960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₄₇₈	C₂	1	2	yes	maximal
3	#29504678	C₂	1	2	yes	maximal
4	#29504668	C₂	1	2	yes	maximal
5	#29504655	C₂	1	4	yes	maximal
6	C₄⋊C₄×Dic₂₃₉	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₄₇₈	C₂²	1	2	yes	—
10	#21439960	C₄	1	8	yes	complemented
11	C₄×Dic₄₇₈	—	2	8	no	—
12	C₂×C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₂₃₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₂₃₉	C₂²	1	4	yes	—
19	C₂₃₉⋊#95	C₂²	1	2	yes	—
20	C₂₃₉⋊#95	C₂²	1	4	yes	—
21	C₂₃₉⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
26	#95×C₂₃₉	C₂²	1	2	yes	—
27	C₂×Dic₄₇₈	—	2	4	no	—
28	C₂×Dic₄₇₈	—	2	4	no	—
29	C₂×Dic₄₇₈	C₂×C₄	1	8	yes	—
30	C₂²×Dic₂₃₉	C₂³	1	2	yes	—
31	C₂²×Dic₂₃₉	C₂³	1	4	yes	—
32	C₂²×Dic₂₃₉	C₂³	1	2	yes	—
33	C₄⋊Dic₂₃₉	—	2	2	no	—
34	C₄⋊Dic₂₃₉	C₂×C₄	1	8	yes	—
35	C₄⋊Dic₂₃₉	C₂×C₄	1	8	yes	—
36	C₄⋊Dic₂₃₉	D₄	1	4	yes	—
37	Dic₂₃₉⋊C₄	—	2	8	no	—
38	C₄×Dic₂₃₉	—	2	8	no	—
39	C₄×Dic₂₃₉	—	2	4	no	—
40	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃₉	—	2	2	no	—
46	C₄⋊C₄×C₂₃₉	Q₈	1	4	yes	complemented
47	C₄²×C₂₃₉	—	2	4	no	—
48	C₄²×C₂₃₉	C₂×C₄	1	4	yes	—
49	Dic₄₇₈	—	2	16	no	—
50	C₂×Dic₂₃₉	—	2	4	no	—
51	C₂×Dic₂₃₉	—	2	2	no	—
52	C₂×Dic₂₃₉	—	2	4	no	—
53	C₂×Dic₂₃₉	—	2	2	no	—
54	C₂×Dic₂₃₉	—	2	2	no	—
55	C₂×Dic₂₃₉	—	2	2	no	—
56	C₂×Dic₂₃₉	—	2	2	no	—
57	C₂×Dic₂₃₉	C₂²×C₄	1	8	yes	—
58	C₂×Dic₂₃₉	C₂×D₄	1	4	yes	—
59	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₂₃₉	—	2	2	no	—
61	C₂×C₄×C₂₃₉	—	2	1	no	—
62	C₂×C₄×C₂₃₉	—	2	1	no	—
63	C₂×C₄×C₂₃₉	—	2	2	no	—
64	C₂×C₄×C₂₃₉	—	2	2	no	—
65	C₂×C₄×C₂₃₉	—	2	2	no	—
66	C₂×C₄×C₂₃₉	—	2	2	no	—
67	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₃₉	C₂×Q₈	1	4	yes	—
72	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₂₃₉	#41	1	2	yes	—
74	C₂×C₄×C₂₃₉	#41	1	2	yes	—
75	C₂²×C₂₃₉	C₂³×C₄	1	1	yes	characteristic
76	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₂₃₉	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₂₃₉	C₂×#41	1	1	yes	characteristic
79	C₂²×C₂₃₉	C₂×#41	1	1	yes	characteristic
80	C₂²×C₂₃₉	2₋¹⁺⁴	1	1	yes	characteristic
81	C₂²×C₂₃₉	2₋¹⁺⁴	1	1	yes	characteristic
82	C₄×C₂₃₉	—	2	2	no	—
83	C₄×C₂₃₉	—	2	4	no	—
84	C₄×C₂₃₉	—	4	2	no	—
85	C₄×C₂₃₉	C₄×D₄	1	4	yes	—
86	C₄×C₂₃₉	C₄×Q₈	1	4	yes	complemented
87	Dic₂₃₉	—	2	8	no	—
88	Dic₂₃₉	—	2	4	no	—
89	Dic₂₃₉	—	4	2	no	—
90	C₂×C₂₃₉	C₂×C₄×D₄	1	1	yes	characteristic
91	C₂×C₂₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₂₃₉	#519	1	1	yes	characteristic
93	C₂×C₂₃₉	#547	1	1	yes	characteristic
94	C₂×C₂₃₉	D₄×Q₈	1	1	yes	characteristic
95	C₂×C₂₃₉	#556	1	1	yes	characteristic
96	C₂×C₂₃₉	#557	1	1	yes	characteristic
97	C₂₃₉	#2365	1	1	yes	characteristic, complemented, Sylow
98	#2365	—	239	1	no	maximal, Sylow
99	C₂×C₄⋊Q₈	—	239	1	no	—
100	C₂×C₄×Q₈	—	239	2	no	—
101	#391	—	239	2	no	—
102	#389	—	239	2	no	—
103	#387	—	239	4	no	—
104	C₄×C₄⋊C₄	—	239	2	no	—
105	C₄×C₄⋊C₄	—	239	1	no	—
106	C₄×C₄⋊C₄	D₂₃₉	1	1	yes	characteristic
107	C₂²×Q₈	—	239	2	no	—
108	C₄⋊Q₈	—	239	8	no	—
109	C₄×Q₈	—	478	8	no	—
110	C₂×C₄⋊C₄	—	239	4	no	—
111	C₂×C₄⋊C₄	—	239	1	no	—
112	C₂×C₄⋊C₄	—	239	1	no	—
113	C₂×C₄⋊C₄	—	239	2	no	—
114	C₂×C₄⋊C₄	—	239	2	no	—
115	C₂×C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂×D₂₃₉	1	1	yes	characteristic
117	C₂×C₄²	—	239	4	no	—
118	C₂×C₄²	—	239	2	no	—
119	C₂×C₄²	C₂×D₂₃₉	1	2	yes	—
120	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
121	#95	—	239	2	no	—
122	#95	—	239	4	no	—
123	#95	—	239	4	no	—
124	#95	C₂×D₂₃₉	1	2	yes	—
125	C₂×Q₈	—	478	4	no	—
126	C₂×Q₈	—	478	4	no	—
127	C₂×Q₈	—	239	8	no	—
128	C₂²×C₄	—	239	2	no	—
129	C₂²×C₄	—	239	4	no	—
130	C₂²×C₄	—	239	2	no	—
131	C₂²×C₄	C₂²×D₂₃₉	1	2	yes	—
132	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₂₃₉	1	2	yes	—
134	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	478	8	no	—
138	C₄⋊C₄	—	478	2	no	—
139	C₄⋊C₄	—	239	8	no	—
140	C₄⋊C₄	—	239	8	no	—
141	C₄⋊C₄	—	239	4	no	—
142	C₄⋊C₄	Dic₄₇₈	1	4	yes	complemented
143	C₄²	—	2	4	no	—
144	C₄²	—	478	8	no	—
145	C₄²	—	478	4	no	—
146	C₄²	C₄×D₂₃₉	1	4	yes	—
147	C₂³	C₂³×D₂₃₉	1	1	yes	characteristic
148	Q₈	—	478	16	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	478	2	no	—
157	C₂×C₄	—	478	2	no	—
158	C₂×C₄	—	478	2	no	—
159	C₂×C₄	—	478	2	no	—
160	C₂×C₄	—	478	4	no	—
161	C₂×C₄	—	478	2	no	—
162	C₂×C₄	—	478	4	no	—
163	C₂×C₄	—	239	8	no	—
164	C₂×C₄	—	239	4	no	—
165	C₂×C₄	C₂×C₄×D₂₃₉	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₂₃₉	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₂₃₉	1	2	yes	—
168	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
169	C₂×C₄	C₂×Dic₄₇₈	1	4	yes	—
170	C₂×C₄	C₂×Dic₄₇₈	1	2	yes	—
171	C₂×C₄	#16838548	1	2	yes	—
172	C₂×C₄	#16838549	1	2	yes	—
173	C₂²	C₂²×C₄×D₂₃₉	1	1	yes	characteristic
174	C₂²	C₂×D₄×D₂₃₉	1	1	yes	characteristic
175	C₂²	C₂²×Dic₄₇₈	1	1	yes	characteristic
176	C₂²	C₂×#16838548	1	1	yes	characteristic
177	C₂²	C₂×#16838549	1	1	yes	characteristic
178	C₂²	#21440007	1	1	yes	characteristic
179	C₂²	#21440007	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	956	2	no	—
184	C₄	—	478	4	no	—
185	C₄	—	478	8	no	—
186	C₄	#21439922	1	4	yes	—
187	C₄	C₄×Dic₄₇₈	1	4	yes	complemented
188	C₂	C₂×#21439922	1	1	yes	characteristic
189	C₂	C₂×C₄×Dic₄₇₈	1	1	yes	characteristic
190	C₂	#29505249	1	1	yes	characteristic
191	C₂	#29505427	1	1	yes	characteristic
192	C₂	D₄×Dic₄₇₈	1	1	yes	characteristic
193	C₂	#29505495	1	1	yes	characteristic
194	C₂	#29505501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃₉
commutator	C₂²×C₂₃₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₂₃₉	C₂
socle, prime radical, cosocle	C₂³×C₂₃₉	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₂₃₉
hypocenter	C₂₃₉	#2365
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #21439960	C₄	semidirect
46. C₄⋊C₄×C₂₃₉	Q₈	semidirect
86. C₄×C₂₃₉	C₄×Q₈	semidirect
97. C₂₃₉	#2365	semidirect
142. C₄⋊C₄	Dic₄₇₈	semidirect
187. C₄	C₄×Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#21439960	2
C₂×C₄×Dic₄₇₈	2
#29504678	2
#29504668	2
#29504655	2
C₄⋊C₄×Dic₂₃₉	2
C₄×C₄⋊Dic₂₃₉	2
C₄×C₄⋊C₄×C₂₃₉	2
#2365	239