Subgroups — #28379823

Order 13952.

348 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296078	C₂	1	2	yes	maximal, complemented
2	#20296078	C₂	1	2	yes	maximal, complemented
3	#20296077	C₂	1	2	yes	maximal, complemented
4	#20296076	C₂	1	2	yes	maximal, complemented
5	#20296032	C₂	1	2	yes	maximal
6	#20296029	C₂	1	1	yes	maximal, characteristic
7	#20295990	C₂	1	2	yes	maximal
8	C₂×#16812754	C₂	1	1	yes	maximal, characteristic
9	#20295578	C₂	1	1	yes	maximal, characteristic
10	C₂×#13182038	C₂²	1	2	yes	—
11	C₂²×#11724786	C₂²	1	1	yes	characteristic
12	C₂²×#11724786	C₂²	1	1	yes	characteristic
13	#16812766	C₂²	1	2	yes	—
14	#16812766	C₂²	1	2	yes	—
15	#16812766	C₂²	1	2	yes	—
16	C₄⋊#11724786	C₂²	1	1	yes	characteristic
17	C₄⋊#11724786	C₂²	1	1	yes	characteristic
18	C₄⋊#11724786	C₂²	1	2	yes	—
19	C₄⋊#11724786	C₂²	1	2	yes	—
20	#16812754	C₂²	1	2	yes	—
21	#16812754	C₂²	1	2	yes	—
22	#16812754	C₂²	1	2	yes	—
23	#16812754	C₂²	1	2	yes	—
24	C₄×#11724786	C₂²	1	2	yes	—
25	C₄×#11724786	C₂²	1	2	yes	—
26	#16812698	C₄	1	1	yes	characteristic
27	#16812697	C₄	1	1	yes	characteristic
28	#16812689	C₄	1	2	yes	—
29	#16812678	C₄	1	1	yes	characteristic
30	#16812676	C₄	1	2	yes	—
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	#16812664	C₂²	1	2	yes	—
33	#16812664	C₂²	1	1	yes	characteristic
34	#16812655	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
37	#13182038	—	2	4	no	—
38	C₂×#11724786	—	2	2	no	—
39	C₂×#11724786	Q₈	1	4	yes	complemented
40	C₂×#11724786	C₂³	1	2	yes	—
41	C₂×#11724786	C₂³	1	2	yes	—
42	C₂×#11724786	C₂³	1	2	yes	—
43	C₂×#11724786	C₂³	1	2	yes	—
44	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
45	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
50	#13182008	C₂×C₄	1	2	yes	—
51	#13182008	C₂×C₄	1	2	yes	—
52	#13182008	C₂×C₄	1	2	yes	—
53	#13182008	C₂×C₄	1	2	yes	—
54	#13182008	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
59	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₀₉	C₂³	1	2	yes	—
61	C₄×Dic₁₀₉	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
65	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
67	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	Q₈×C₁₀₉	—	4	1	no	—
69	C₂×C₄×C₁₀₉	—	4	1	no	—
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₀₉	—	4	1	no	—
75	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₀₉	C₂×Q₈	1	2	yes	—
81	C₂×Dic₁₀₉	#41	1	2	yes	—
82	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
83	Dic₂₁₈	—	4	2	no	—
84	Dic₂₁₈	—	4	1	no	—
85	#11724786	—	2	2	no	—
86	#11724786	—	2	2	no	—
87	#11724786	—	2	2	no	—
88	#11724786	C₂×Q₈	1	4	yes	—
89	C₂²×C₁₀₉	—	2	1	no	—
90	C₂²×C₁₀₉	—	2	1	no	—
91	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
92	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	2	no	—
95	C₄×C₁₀₉	—	2	1	no	—
96	C₄×C₁₀₉	—	2	2	no	—
97	Dic₁₀₉	—	2	1	no	—
98	Dic₁₀₉	—	2	2	no	—
99	Dic₁₀₉	—	2	2	no	—
100	Dic₁₀₉	—	2	1	no	—
101	Dic₁₀₉	C₄×Q₈	1	2	yes	—
102	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
103	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
104	C₂×C₁₀₉	—	2	2	no	—
105	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
106	C₂×C₁₀₉	#568	1	1	yes	characteristic
107	C₂×C₁₀₉	#568	1	1	yes	characteristic
108	#2961	—	109	1	no	maximal, Sylow
109	C₁₀₉	#2961	1	1	yes	characteristic, complemented, Sylow
110	#533	—	109	1	no	—
111	#446	—	109	2	no	—
112	#446	—	109	2	no	—
113	#446	—	109	2	no	—
114	#446	—	109	2	no	—
115	#424	—	109	2	no	—
116	#423	—	109	1	no	—
117	#405	—	109	2	no	—
118	C₂×#97	—	109	1	no	—
119	C₂×M₁₆	—	109	2	no	—
120	C₂²×C₈	—	109	1	no	—
121	C₂²×C₈	—	109	1	no	—
122	C₄⋊Q₈	—	109	1	no	—
123	C₄⋊Q₈	—	109	1	no	—
124	#125	—	109	2	no	—
125	C₂²⋊Q₈	—	109	1	no	—
126	C₂²⋊Q₈	—	109	2	no	—
127	C₂²⋊Q₈	#89137	1	1	yes	characteristic
128	C₄×Q₈	—	109	1	no	—
129	C₄×Q₈	—	109	1	no	—
130	C₄×Q₈	—	109	2	no	—
131	#117	—	109	2	no	—
132	C₂×C₄²	—	109	1	no	—
133	C₄⋊C₈	—	109	1	no	—
134	C₄⋊C₈	—	109	1	no	—
135	C₄⋊C₈	—	109	2	no	—
136	C₄⋊C₈	—	109	2	no	—
137	C₄⋊C₈	—	109	2	no	—
138	C₄⋊C₈	—	109	2	no	—
139	C₄⋊C₈	—	109	2	no	—
140	#97	—	109	2	no	—
141	#97	—	109	2	no	—
142	#97	—	109	2	no	—
143	#97	—	109	2	no	—
144	C₄×C₈	—	109	2	no	—
145	C₄×C₈	—	109	2	no	—
146	C₂×Q₈	—	109	2	no	—
147	C₂×Q₈	—	109	1	no	—
148	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
149	C₂²×C₄	—	109	1	no	—
150	C₂²×C₄	—	109	1	no	—
151	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
152	M₁₆	—	218	4	no	—
153	C₂×C₈	—	109	4	no	—
154	C₂×C₈	—	109	2	no	—
155	C₂×C₈	—	109	2	no	—
156	C₂×C₈	—	109	2	no	—
157	C₂×C₈	—	109	2	no	—
158	C₂×C₈	—	218	2	no	—
159	C₄⋊C₄	—	109	2	no	—
160	C₄⋊C₄	—	109	2	no	—
161	C₄⋊C₄	—	109	2	no	—
162	C₄⋊C₄	—	109	2	no	—
163	C₄⋊C₄	—	109	2	no	—
164	C₄⋊C₄	—	109	2	no	—
165	C₄⋊C₄	—	109	1	no	—
166	C₄⋊C₄	C₂×#89137	1	2	yes	—
167	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
168	C₂²⋊C₄	—	109	2	no	—
169	C₂²⋊C₄	C₂×#89137	1	2	yes	—
170	C₄²	—	109	1	no	—
171	C₄²	—	109	1	no	—
172	C₄²	—	109	2	no	—
173	C₄²	—	109	2	no	—
174	C₄²	—	109	1	no	—
175	C₄²	—	109	1	no	—
176	C₂³	C₂²×#89137	1	1	yes	characteristic
177	Q₈	—	4	1	no	—
178	Q₈	—	436	1	no	—
179	Q₈	—	436	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	109	2	no	—
182	C₂×C₄	—	109	2	no	—
183	C₂×C₄	—	109	1	no	—
184	C₂×C₄	—	109	1	no	—
185	C₂×C₄	—	109	1	no	—
186	C₂×C₄	—	109	1	no	—
187	C₂×C₄	—	109	2	no	—
188	C₂×C₄	—	109	2	no	—
189	C₂×C₄	—	436	1	no	—
190	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
191	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
192	C₂×C₄	C₂²×#89137	1	2	yes	—
193	C₂×C₄	C₂²×#89137	1	2	yes	—
194	C₈	—	109	4	no	—
195	C₈	—	218	2	no	—
196	C₈	—	218	2	no	—
197	C₈	—	218	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	Q₈×#89137	1	2	yes	—
201	C₂²	C₂³×#89137	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	109	1	no	—
207	C₄	—	109	1	no	—
208	C₄	—	109	2	no	—
209	C₄	—	218	2	no	—
210	C₄	—	218	1	no	—
211	C₄	—	218	1	no	—
212	C₄	—	218	2	no	—
213	C₂	—	2	2	no	—
214	C₂	C₂×Q₈×#89137	1	1	yes	characteristic
215	C₂	#20296104	1	1	yes	characteristic
216	C₂	#20296105	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₂²⋊Q₈	#89137
hypocenter	C₁₀₉	#2961
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20296078	C₂	semidirect
2. #20296078	C₂	semidirect
3. #20296077	C₂	semidirect
4. #20296076	C₂	semidirect
39. C₂×#11724786	Q₈	semidirect
109. C₁₀₉	#2961	semidirect

Maximal subgroups

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

subgroup	index
#20296078	2
#20296077	2
#20296076	2
#20296032	2
#20296029	2
#20295990	2
C₂×#16812754	2
#20295578	2
#2961	109