Subgroups — #35197345

Order 20224.

6578 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777210	C₂	1	16	yes	maximal, complemented
2	#23777206	C₂	1	32	yes	maximal, complemented
3	#23776805	C₂	1	12	yes	maximal
4	C₂×#19099041	C₂	1	2	yes	maximal
5	C₂×#519×C₇₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₅₈	C₂²	1	64	yes	complemented
7	#19099292	C₂²	1	64	yes	complemented
8	#19099289	C₂²	1	128	yes	complemented
9	#19099194	C₂²	1	48	yes	—
10	#19099191	C₂²	1	96	yes	—
11	#19099189	C₂²	1	16	yes	—
12	#19099120	C₂²	1	48	yes	—
13	#19099116	C₂²	1	96	yes	—
14	#19099041	C₂²	1	16	yes	—
15	#19099041	C₂²	1	16	yes	—
16	C₂×Q₈×Dic₇₉	C₂²	1	16	yes	—
17	C₂×#14579120	C₂²	1	12	yes	—
18	C₂³×Q₈×C₇₉	C₂²	1	1	yes	characteristic
19	#519×C₇₉	C₂²	1	16	yes	—
20	C₂×C₄×Q₈×C₇₉	C₂²	1	8	yes	—
21	C₂×#117×C₇₉	C₂²	1	6	yes	—
22	#14579161	C₂³	1	96	yes	—
23	#14579160	C₂³	1	192	yes	—
24	#14579153	C₂³	1	96	yes	—
25	#14579152	C₂³	1	192	yes	—
26	#14579140	C₂³	1	64	yes	—
27	#14579138	C₂³	1	128	yes	—
28	Q₈×Dic₇₉	C₂³	1	128	yes	—
29	C₄×Dic₁₅₈	C₂³	1	192	yes	—
30	#14579120	C₂³	1	48	yes	—
31	#14579120	C₂³	1	48	yes	—
32	C₂×C₄⋊Dic₇₉	C₂³	1	24	yes	—
33	C₂×C₂²⋊Dic₇₉	C₂³	1	8	yes	—
34	C₂×C₄×Dic₇₉	C₂³	1	24	yes	—
35	C₂²×Q₈×C₇₉	Q₈	1	16	yes	complemented
36	C₂²×Q₈×C₇₉	C₂³	1	4	yes	—
37	C₂²×Q₈×C₇₉	C₂³	1	8	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
39	C₄×Q₈×C₇₉	C₂³	1	64	yes	—
40	#117×C₇₉	C₂³	1	48	yes	—
41	C₂×C₄⋊C₄×C₇₉	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	12	yes	—
44	C₂×Dic₁₅₈	C₂⁴	1	64	yes	—
45	C₂²×Dic₇₉	C₂⁴	1	16	yes	—
46	C₄⋊Dic₇₉	C₂⁴	1	96	yes	—
47	C₄⋊Dic₇₉	C₂⁴	1	64	yes	—
48	Dic₇₉⋊C₄	C₂⁴	1	128	yes	—
49	C₂²⋊Dic₇₉	C₂⁴	1	16	yes	—
50	C₂²⋊Dic₇₉	C₂⁴	1	16	yes	—
51	C₄×Dic₇₉	C₂⁴	1	96	yes	—
52	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₇₉	C₂×Q₈	1	32	yes	—
54	C₂×Q₈×C₇₉	C₂×Q₈	1	64	yes	—
55	C₂×Q₈×C₇₉	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₇₉	C₂×Q₈	1	24	yes	—
57	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
60	C₄⋊C₄×C₇₉	C₂⁴	1	48	yes	—
61	C₂²⋊C₄×C₇₉	C₂⁴	1	16	yes	—
62	C₄²×C₇₉	C₂⁴	1	48	yes	—
63	C₂³×C₇₉	—	4	2	no	—
64	C₂³×C₇₉	C₂²×Q₈	1	4	yes	—
65	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
66	C₂³×C₇₉	C₂⁵	1	2	yes	—
67	Q₈×C₇₉	C₂²×Q₈	1	64	yes	—
68	C₂×C₄×C₇₉	—	4	8	no	—
69	C₂×C₄×C₇₉	C₂²×Q₈	1	24	yes	—
70	C₂×C₄×C₇₉	C₂²×Q₈	1	48	yes	—
71	C₂×C₄×C₇₉	C₂⁵	1	16	yes	—
72	C₂×C₄×C₇₉	C₂⁵	1	12	yes	—
73	C₂×Dic₇₉	—	4	16	no	—
74	C₂×Dic₇₉	C₂⁵	1	32	yes	—
75	Dic₁₅₈	—	4	64	no	—
76	C₂²×C₇₉	—	2	2	no	—
77	C₂²×C₇₉	—	2	4	no	—
78	C₂²×C₇₉	—	4	4	no	—
79	C₂²×C₇₉	C₂³×Q₈	1	2	yes	—
80	C₂²×C₇₉	C₂³×Q₈	1	4	yes	—
81	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
82	C₄×C₇₉	—	2	16	no	—
83	C₄×C₇₉	C₂³×Q₈	1	24	yes	—
84	Dic₇₉	—	2	32	no	—
85	#62849	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	—	2	2	no	—
87	C₂×C₇₉	—	2	4	no	—
88	C₂×C₇₉	C₂⁴×Q₈	1	1	yes	characteristic
89	C₂×C₇₉	2₋¹⁺⁶	1	2	yes	—
90	#3502	—	79	32	no	—
91	#3502	—	79	16	no	—
92	#3458	—	79	12	no	—
93	C₂×#519	—	79	2	no	—
94	C₂×#519	D₇₉	1	1	yes	characteristic
95	C₇₉	#62849	1	1	yes	characteristic, complemented, Sylow
96	C₂³×Q₈	C₂×D₇₉	1	1	yes	characteristic
97	Q₈²	—	79	64	no	—
98	#557	—	79	128	no	—
99	#557	—	79	64	no	—
100	#544	—	79	96	no	—
101	#544	—	79	48	no	—
102	#543	—	79	16	no	—
103	#533	—	79	96	no	—
104	#533	—	79	48	no	—
105	#519	—	79	16	no	—
106	#519	—	79	16	no	—
107	#519	C₂×D₇₉	1	16	yes	—
108	C₂×C₄×Q₈	—	79	16	no	—
109	C₂×C₄×Q₈	C₂×D₇₉	1	8	yes	—
110	C₂×#117	—	79	12	no	—
111	C₂×#117	C₂×D₇₉	1	6	yes	—
112	C₂²×Q₈	Dic₁₅₈	1	16	yes	complemented
113	C₂²×Q₈	C₂²×D₇₉	1	8	yes	—
114	C₂²×Q₈	C₂²×D₇₉	1	4	yes	—
115	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
116	C₄⋊Q₈	—	79	192	no	—
117	C₄⋊Q₈	—	79	96	no	—
118	#125	—	79	192	no	—
119	#125	—	79	96	no	—
120	C₂²⋊Q₈	—	79	64	no	—
121	C₂²⋊Q₈	—	79	128	no	—
122	C₄×Q₈	—	79	128	no	—
123	C₄×Q₈	—	79	192	no	—
124	C₄×Q₈	C₂²×D₇₉	1	64	yes	—
125	#117	—	79	48	no	—
126	#117	—	79	48	no	—
127	#117	C₂²×D₇₉	1	48	yes	—
128	C₂×C₄⋊C₄	—	79	24	no	—
129	C₂×C₄⋊C₄	C₂²×D₇₉	1	12	yes	—
130	C₂×C₂²⋊C₄	—	79	8	no	—
131	C₂×C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
132	C₂×C₄²	—	79	24	no	—
133	C₂×C₄²	C₂²×D₇₉	1	12	yes	—
134	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
135	C₂×Q₈	—	79	64	no	—
136	C₂×Q₈	C₂×Dic₁₅₈	1	64	yes	—
137	C₂×Q₈	C₂×Dic₁₅₈	1	32	yes	—
138	C₂×Q₈	C₂³×D₇₉	1	16	yes	—
139	C₂²×C₄	—	79	16	no	—
140	C₂²×C₄	C₂×Dic₁₅₈	1	24	yes	—
141	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
142	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
143	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
144	C₄⋊C₄	—	79	128	no	—
145	C₄⋊C₄	—	79	96	no	—
146	C₄⋊C₄	—	79	64	no	—
147	C₄⋊C₄	C₂³×D₇₉	1	48	yes	—
148	C₂²⋊C₄	—	79	16	no	—
149	C₂²⋊C₄	—	79	16	no	—
150	C₂²⋊C₄	C₂³×D₇₉	1	16	yes	—
151	C₄²	—	79	96	no	—
152	C₄²	C₂³×D₇₉	1	48	yes	—
153	C₂³	—	4	2	no	—
154	C₂³	C₂²×Dic₁₅₈	1	4	yes	—
155	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
156	C₂³	C₂⁴×D₇₉	1	2	yes	—
157	Q₈	—	316	64	no	—
158	Q₈	C₂²×Dic₁₅₈	1	64	yes	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	316	16	no	—
161	C₂×C₄	—	79	32	no	—
162	C₂×C₄	C₂²×Dic₁₅₈	1	48	yes	—
163	C₂×C₄	C₂²×Dic₁₅₈	1	24	yes	—
164	C₂×C₄	C₂⁴×D₇₉	1	16	yes	—
165	C₂×C₄	C₂⁴×D₇₉	1	12	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	C₂³×Dic₁₅₈	1	2	yes	—
170	C₂²	C₂³×Dic₁₅₈	1	4	yes	—
171	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
172	C₄	—	2	16	no	—
173	C₄	—	158	32	no	—
174	C₄	C₂³×Dic₁₅₈	1	24	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	4	no	—
177	C₂	C₂⁴×Dic₁₅₈	1	1	yes	characteristic
178	C₂	#23778212	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁶
Fitting	C₂×#519×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×#519	D₇₉
hypocenter	C₇₉	#62849
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
1. #23777210	C₂	semidirect
2. #23777206	C₂	semidirect
6. Q₈×Dic₁₅₈	C₂²	semidirect
7. #19099292	C₂²	semidirect
8. #19099289	C₂²	semidirect
35. C₂²×Q₈×C₇₉	Q₈	semidirect
95. C₇₉	#62849	semidirect
112. C₂²×Q₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23777210	2
#23777206	2
#23776805	2
C₂×#19099041	2
C₂×#519×C₇₉	2
#62849	79