Subgroups — #28371979

Order 13952.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295583	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	2	yes	maximal, complemented
3	#20295154	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₁₈	C₂	1	2	yes	maximal, complemented
5	#20295054	C₂	1	2	yes	maximal
6	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
7	#20294984	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11724784	C₂²	1	2	yes	—
10	#16812699	C₄	1	1	yes	characteristic
11	#16812699	C₄	1	1	yes	characteristic
12	#16812691	C₄	1	2	yes	—
13	#16812678	C₄	1	2	yes	—
14	#16812678	C₄	1	2	yes	—
15	C₄×Dic₂₁₈	C₂²	1	2	yes	—
16	C₄×Dic₂₁₈	C₂²	1	2	yes	—
17	#16812658	C₂²	1	2	yes	—
18	C₄⋊#11724784	C₂²	1	1	yes	characteristic
19	C₄⋊#11724784	C₂²	1	2	yes	—
20	C₄⋊#11724784	C₂²	1	1	yes	characteristic
21	#16812628	C₂²	1	4	yes	—
22	#16812628	C₂²	1	4	yes	—
23	#16812609	C₂²	1	4	yes	—
24	C₈×Dic₁₀₉	C₂²	1	4	yes	—
25	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
28	#97×C₁₀₉	C₂²	1	2	yes	—
29	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
30	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
32	C₂×#11724784	—	2	4	no	—
33	C₂×#11724784	C₂³	1	2	yes	—
34	C₂×#11724784	C₂³	1	2	yes	—
35	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
36	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
39	#13182008	C₂×C₄	1	4	yes	—
40	#13182008	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₀₉	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	M₁₆×C₁₀₉	—	2	2	no	—
47	M₁₆×C₁₀₉	Q₈	1	4	yes	complemented
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	4	1	no	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₀₉	#41	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
67	Dic₂₁₈	—	2	4	no	—
68	Dic₂₁₈	—	4	2	no	—
69	C₈×C₁₀₉	—	2	2	no	—
70	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
71	#11724784	—	2	2	no	—
72	#11724784	—	2	2	no	—
73	C₂²×C₁₀₉	—	2	1	no	—
74	C₂²×C₁₀₉	—	2	1	no	—
75	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
76	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₀₉	—	2	1	no	—
78	C₄×C₁₀₉	—	2	1	no	—
79	C₄×C₁₀₉	—	2	1	no	—
80	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
81	C₄×C₁₀₉	#131	1	2	yes	—
82	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
84	Dic₁₀₉	—	2	4	no	—
85	Dic₁₀₉	—	2	4	no	—
86	C₂×C₁₀₉	—	2	2	no	—
87	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
89	C₂×C₁₀₉	#568	1	1	yes	characteristic
90	#2962	—	109	1	no	maximal, Sylow
91	C₁₀₉	#2962	1	1	yes	characteristic, complemented, Sylow
92	#533	—	109	1	no	—
93	#446	—	109	4	no	—
94	#446	—	109	2	no	—
95	C₈×Q₈	—	109	2	no	—
96	#424	—	109	2	no	—
97	C₂×C₄⋊C₈	—	109	1	no	—
98	#405	—	109	2	no	—
99	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
102	C₂²×C₈	—	109	2	no	—
103	C₄⋊Q₈	—	109	1	no	—
104	C₄⋊Q₈	—	109	1	no	—
105	#125	—	109	2	no	—
106	C₂²⋊Q₈	—	109	2	no	—
107	C₂²⋊Q₈	—	109	2	no	—
108	C₄×Q₈	—	109	2	no	—
109	C₄×Q₈	—	109	2	no	—
110	#117	—	109	2	no	—
111	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
112	C₄⋊C₈	—	109	4	no	—
113	C₄⋊C₈	—	109	4	no	—
114	C₄⋊C₈	—	109	1	no	—
115	C₄⋊C₈	—	109	1	no	—
116	C₄⋊C₈	—	109	2	no	—
117	#97	—	109	4	no	—
118	#97	C₂×D₁₀₉	1	2	yes	—
119	C₄×C₈	—	109	4	no	—
120	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
121	C₂×Q₈	—	109	2	no	—
122	C₂×Q₈	—	109	2	no	—
123	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₂₁₈	1	4	yes	complemented
128	C₂×C₈	—	109	2	no	—
129	C₂×C₈	—	109	2	no	—
130	C₂×C₈	—	218	4	no	—
131	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
132	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
133	C₄⋊C₄	—	109	4	no	—
134	C₄⋊C₄	—	109	4	no	—
135	C₄⋊C₄	—	109	2	no	—
136	C₄⋊C₄	—	109	2	no	—
137	C₄⋊C₄	—	109	2	no	—
138	C₄⋊C₄	—	109	2	no	—
139	C₂²⋊C₄	—	109	4	no	—
140	C₄²	—	109	4	no	—
141	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
142	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
143	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
144	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
146	Q₈	—	436	2	no	—
147	Q₈	—	218	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	109	4	no	—
151	C₂×C₄	—	109	4	no	—
152	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
158	C₂×C₄	#13182029	1	2	yes	—
159	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	218	2	no	—
162	C₈	—	218	2	no	—
163	C₈	C₂×Dic₂₁₈	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₂₁₈	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	218	4	no	—
172	C₄	—	218	4	no	—
173	C₄	C₄×Dic₂₁₈	1	2	yes	—
174	C₄	#16812707	1	2	yes	—
175	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
176	C₄	C₂×#13182029	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₂₁₈	1	1	yes	characteristic
179	C₂	C₂×#16812707	1	1	yes	characteristic
180	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×M₁₆×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#2962
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. #20295154	C₂	semidirect
4. C₈×Dic₂₁₈	C₂	semidirect
47. M₁₆×C₁₀₉	Q₈	semidirect
91. C₁₀₉	#2962	semidirect
127. M₁₆	Dic₂₁₈	semidirect

Maximal subgroups

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

subgroup	index
#20295583	2
#20295156	2
#20295154	2
C₈×Dic₂₁₈	2
#20295054	2
C₂×C₄⋊#11724784	2
#20294984	2
C₄×M₁₆×C₁₀₉	2
#2962	109