Subgroups — #16836294

Order 3712.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14272215	C₂	1	4	yes	maximal, complemented
2	#14272214	C₂	1	2	yes	maximal, complemented
3	#14272212	C₂	1	2	yes	maximal, complemented
4	#14272168	C₂	1	2	yes	maximal
5	#14272167	C₂	1	1	yes	maximal, characteristic
6	C₄×#90796	C₂	1	2	yes	maximal
7	C₄×#90796	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×D₂₉	C₄	1	1	yes	characteristic
10	C₂×#90796	C₂²	1	2	yes	—
11	C₂×#90796	C₂²	1	1	yes	characteristic
12	C₂×#90796	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
14	Q₈×Dic₂₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
16	C₄⋊C₄×D₂₉	C₄	1	2	yes	—
17	#11745249	C₂²	1	1	yes	characteristic
18	#11745249	C₂²	1	2	yes	—
19	C₄²×D₂₉	C₂²	1	1	yes	characteristic
20	C₄²×D₂₉	C₂²	1	2	yes	—
21	C₄×Dic₅₈	C₄	1	2	yes	—
22	C₄×Dic₅₈	C₄	1	1	yes	characteristic
23	#11745220	C₂²	1	4	yes	—
24	#11745220	C₂²	1	4	yes	—
25	#11745218	C₂²	1	4	yes	—
26	#11745218	C₂²	1	2	yes	—
27	C₄⋊#6599	C₂²	1	2	yes	—
28	C₄⋊#6599	C₂²	1	2	yes	—
29	C₄×#6599	C₂²	1	2	yes	—
30	C₄×#6599	C₂²	1	4	yes	—
31	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
32	Q₈×D₂₉	—	2	4	no	—
33	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
38	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
39	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
40	C₂×#6599	C₂³	1	4	yes	—
41	C₂×#6599	C₂³	1	2	yes	—
42	C₂×#6599	C₂³	1	2	yes	—
43	#90796	—	2	4	no	—
44	#90796	—	2	2	no	—
45	#90796	Q₈	1	4	yes	complemented
46	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₉	C₂×C₄	1	2	yes	—
49	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
52	#90779	C₂×C₄	1	2	yes	—
53	#90779	C₂×C₄	1	2	yes	—
54	#90779	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₂₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₉	C₂³	1	2	yes	—
59	C₄×Dic₂₉	C₂³	1	2	yes	—
60	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
61	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₂₉	—	2	2	no	—
63	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
70	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
72	C₂×Dic₂₉	C₂⁴	1	1	yes	characteristic
73	C₄×D₂₉	—	2	2	no	—
74	C₄×D₂₉	—	2	4	no	—
75	C₄×D₂₉	—	4	1	no	—
76	C₄×D₂₉	—	4	2	no	—
77	C₄×D₂₉	C₂×Q₈	1	2	yes	—
78	C₄×D₂₉	#41	1	2	yes	—
79	Dic₅₈	—	2	4	no	—
80	Dic₅₈	M₁₆	1	4	yes	complemented
81	#6599	—	2	4	no	—
82	#6599	—	2	2	no	—
83	#6599	C₂×Q₈	1	4	yes	complemented
84	Q₈×M₁₆	—	29	1	no	maximal, Sylow
85	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
86	C₂×D₂₉	—	2	1	no	—
87	C₂×D₂₉	—	2	1	no	—
88	C₂×D₂₉	C₄×Q₈	1	2	yes	—
89	C₄×C₂₉	—	2	2	no	—
90	C₄×C₂₉	—	2	2	no	—
91	C₄×C₂₉	—	2	1	no	—
92	C₄×C₂₉	C₄×Q₈	1	2	yes	—
93	C₄×C₂₉	C₂×M₁₆	1	2	yes	—
94	Dic₂₉	—	2	1	no	—
95	Dic₂₉	—	2	1	no	—
96	Dic₂₉	—	2	2	no	—
97	Dic₂₉	C₂×M₁₆	1	4	yes	—
98	Dic₂₉	C₂²×Q₈	1	1	yes	characteristic
99	Dic₂₉	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×Q₈	—	29	1	no	—
101	#446	—	29	2	no	—
102	#446	—	29	4	no	—
103	C₈×Q₈	—	29	2	no	—
104	#423	—	29	1	no	—
105	#423	—	29	2	no	—
106	C₄×M₁₆	—	29	1	no	—
107	C₄×M₁₆	—	29	2	no	—
108	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
109	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
110	C₂×C₂₉	#568	1	1	yes	characteristic
111	D₂₉	—	2	2	no	—
112	C₂²×Q₈	—	29	1	no	—
113	C₂×M₁₆	—	29	1	no	—
114	C₂×M₁₆	—	29	2	no	—
115	C₂×M₁₆	—	29	1	no	—
116	C₄×Q₈	—	29	1	no	—
117	C₄×Q₈	—	29	1	no	—
118	C₄×Q₈	—	29	2	no	—
119	C₄×Q₈	—	29	1	no	—
120	C₄×Q₈	—	29	2	no	—
121	C₄×Q₈	#1181	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	29	2	no	—
123	C₂×C₄⋊C₄	—	29	1	no	—
124	C₂×C₄²	—	29	1	no	—
125	C₂×C₄²	—	29	2	no	—
126	C₄⋊C₈	—	29	2	no	—
127	C₄⋊C₈	—	29	4	no	—
128	C₄⋊C₈	—	29	2	no	—
129	C₄⋊C₈	—	29	4	no	—
130	#97	—	29	4	no	—
131	#97	—	29	2	no	—
132	C₄×C₈	—	29	2	no	—
133	C₄×C₈	—	29	4	no	—
134	C₂₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	29	1	no	—
136	C₂×Q₈	—	29	2	no	—
137	C₂×Q₈	—	58	4	no	—
138	C₂×Q₈	C₂×#1181	1	1	yes	characteristic
139	C₂²×C₄	—	29	1	no	—
140	C₂²×C₄	—	29	2	no	—
141	C₂²×C₄	—	29	1	no	—
142	C₂²×C₄	—	29	1	no	—
143	C₂²×C₄	—	29	2	no	—
144	M₁₆	—	29	4	no	—
145	M₁₆	—	58	4	no	—
146	M₁₆	—	58	2	no	—
147	C₂×C₈	—	29	2	no	—
148	C₂×C₈	—	29	4	no	—
149	C₂×C₈	—	29	2	no	—
150	C₄⋊C₄	—	29	2	no	—
151	C₄⋊C₄	—	29	2	no	—
152	C₄⋊C₄	—	29	2	no	—
153	C₄⋊C₄	—	29	2	no	—
154	C₄⋊C₄	—	29	1	no	—
155	C₄⋊C₄	C₂×#1181	1	2	yes	—
156	C₄⋊C₄	C₂×#1181	1	1	yes	characteristic
157	C₄²	—	29	2	no	—
158	C₄²	—	29	1	no	—
159	C₄²	—	29	2	no	—
160	C₄²	—	29	1	no	—
161	C₄²	—	29	2	no	—
162	C₄²	—	29	1	no	—
163	C₄²	C₂×#1181	1	2	yes	—
164	C₄²	C₂×#1181	1	1	yes	characteristic
165	C₂³	—	29	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	29	4	no	—
168	Q₈	—	58	4	no	—
169	C₂×C₄	—	29	1	no	—
170	C₂×C₄	—	29	2	no	—
171	C₂×C₄	—	29	2	no	—
172	C₂×C₄	—	29	2	no	—
173	C₂×C₄	—	29	1	no	—
174	C₂×C₄	—	29	2	no	—
175	C₂×C₄	—	29	1	no	—
176	C₂×C₄	—	58	4	no	—
177	C₂×C₄	—	58	2	no	—
178	C₂×C₄	—	116	2	no	—
179	C₂×C₄	—	116	1	no	—
180	C₂×C₄	C₂²×#1181	1	2	yes	—
181	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
182	C₂×C₄	C₂²×#1181	1	2	yes	—
183	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
184	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
185	C₈	—	29	4	no	—
186	C₈	—	58	4	no	—
187	C₈	—	58	2	no	—
188	C₂²	—	29	2	no	—
189	C₂²	—	58	1	no	—
190	C₂²	—	58	1	no	—
191	C₂²	C₂³×#1181	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	29	1	no	—
196	C₄	—	29	1	no	—
197	C₄	—	29	4	no	—
198	C₄	—	58	1	no	—
199	C₄	—	58	1	no	—
200	C₄	—	58	2	no	—
201	C₄	D₂₉⋊M₁₆	1	2	yes	—
202	C₄	Q₈×#1181	1	2	yes	—
203	C₂	—	58	2	no	—
204	C₂	C₂×Q₈×#1181	1	1	yes	characteristic
205	C₂	C₂×D₂₉⋊M₁₆	1	1	yes	characteristic
206	C₂	#14272242	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#1181
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₄×Q₈×C₂₉	C₄
prime radical	C₂²×D₂₉	C₂²×C₄
cosocle	C₂×Dic₂₉	C₂⁴
hypercenter	C₄×Q₈	#1181
hypocenter	C₂₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14272215	C₂	semidirect
2. #14272214	C₂	semidirect
3. #14272212	C₂	semidirect
45. #90796	Q₈	semidirect
80. Dic₅₈	M₁₆	semidirect
83. #6599	C₂×Q₈	semidirect
134. C₂₉	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#14272215	2
#14272214	2
#14272212	2
#14272168	2
#14272167	2
C₄×#90796	2
C₄×Q₈×D₂₉	2
Q₈×M₁₆	29