Subgroups — #28371694

Order 13952.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295084	C₂	1	2	yes	maximal, complemented
3	#20295084	C₂	1	2	yes	maximal, complemented
4	#20295080	C₂	1	2	yes	maximal, complemented
5	#20295076	C₂	1	2	yes	maximal, complemented
6	#20295049	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16812628	C₂	1	1	yes	maximal, characteristic, complemented
8	#20294993	C₂	1	1	yes	maximal, characteristic
9	#20294984	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294981	C₂	1	1	yes	maximal, characteristic, complemented
11	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#13182013	C₂²	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	Q₈×Dic₁₀₉	C₄	1	1	yes	characteristic
16	D₄×Dic₁₀₉	C₄	1	2	yes	—
17	D₄×Dic₁₀₉	C₄	1	1	yes	characteristic
18	D₄×Dic₁₀₉	C₄	1	2	yes	—
19	D₄×Dic₁₀₉	C₄	1	1	yes	characteristic
20	#16812658	C₂²	1	1	yes	characteristic
21	#16812658	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
24	#16812628	C₂²	1	1	yes	characteristic
25	#16812628	C₂²	1	1	yes	characteristic
26	#16812628	C₂²	1	1	yes	characteristic
27	#16812628	C₂²	1	1	yes	characteristic
28	#16812628	C₂²	1	2	yes	—
29	#16812628	C₂²	1	2	yes	—
30	C₂²⋊#11724784	C₂²	1	2	yes	—
31	C₂²⋊#11724784	C₂²	1	2	yes	—
32	#16812609	C₂²	1	2	yes	—
33	#16812609	C₂²	1	2	yes	—
34	C₈×Dic₁₀₉	C₂²	1	2	yes	—
35	C₈×Dic₁₀₉	C₂²	1	2	yes	—
36	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
41	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
45	#13182013	—	2	2	no	—
46	C₂×#11724784	—	2	2	no	—
47	C₂×#11724784	C₂³	1	1	yes	characteristic
48	C₂×#11724784	C₂³	1	2	yes	—
49	C₂×#11724784	C₂³	1	1	yes	characteristic
50	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₀₉	C₂³	1	2	yes	—
62	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
63	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
64	#41×C₁₀₉	—	2	2	no	—
65	#41×C₁₀₉	—	2	2	no	—
66	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
67	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
68	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	M₁₆×C₁₀₉	—	2	2	no	—
72	C₂×C₈×C₁₀₉	—	2	2	no	—
73	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
76	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	Q₈×C₁₀₉	—	2	2	no	—
79	D₄×C₁₀₉	—	2	4	no	—
80	D₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	#41	1	2	yes	—
86	C₂×C₄×C₁₀₉	#41	1	2	yes	—
87	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₀₉	—	2	2	no	—
89	C₂×Dic₁₀₉	—	2	4	no	—
90	C₂×Dic₁₀₉	—	4	1	no	—
91	C₂×Dic₁₀₉	—	4	2	no	—
92	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
98	C₈×C₁₀₉	—	2	1	no	—
99	C₈×C₁₀₉	—	2	1	no	—
100	C₈×C₁₀₉	—	2	2	no	—
101	#11724784	—	2	1	no	—
102	#11724784	—	2	1	no	—
103	#11724784	—	2	2	no	—
104	C₂²×C₁₀₉	—	2	1	no	—
105	C₂²×C₁₀₉	—	2	1	no	—
106	C₂²×C₁₀₉	—	2	2	no	—
107	C₂²×C₁₀₉	—	2	2	no	—
108	C₂²×C₁₀₉	—	2	2	no	—
109	C₂²×C₁₀₉	#117	1	2	yes	—
110	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₁₀₉	—	2	2	no	—
112	C₄×C₁₀₉	#117	1	2	yes	—
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
115	Dic₁₀₉	—	2	1	no	—
116	Dic₁₀₉	—	2	2	no	—
117	Dic₁₀₉	—	2	1	no	—
118	Dic₁₀₉	—	2	1	no	—
119	Dic₁₀₉	—	2	2	no	—
120	Dic₁₀₉	#131	1	2	yes	—
121	C₂×C₁₀₉	—	2	2	no	—
122	C₂×C₁₀₉	—	4	2	no	—
123	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
124	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
125	C₂×C₁₀₉	#568	1	1	yes	characteristic
126	#2925	—	109	1	no	maximal, Sylow
127	C₁₀₉	#2925	1	1	yes	characteristic, complemented, Sylow
128	C₄×#41	—	109	1	no	—
129	#433	—	109	2	no	—
130	#433	—	109	2	no	—
131	#432	—	109	2	no	—
132	#431	—	109	2	no	—
133	#423	—	109	1	no	—
134	C₂×C₄⋊C₈	—	109	1	no	—
135	#408	—	109	1	no	—
136	#408	D₁₀₉	1	1	yes	characteristic
137	#405	—	109	1	no	—
138	#405	—	109	1	no	—
139	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
140	C₂×M₁₆	—	109	1	no	—
141	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
142	C₂²×C₈	—	109	1	no	—
143	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
144	C₄×Q₈	—	109	1	no	—
145	C₄×Q₈	—	109	1	no	—
146	C₄×D₄	—	109	2	no	—
147	C₄×D₄	—	109	1	no	—
148	C₄×D₄	—	109	2	no	—
149	C₄×D₄	—	109	1	no	—
150	#117	—	109	1	no	—
151	#117	—	109	2	no	—
152	C₂×C₄²	—	109	1	no	—
153	C₂×C₄²	—	109	2	no	—
154	C₄⋊C₈	—	109	1	no	—
155	C₄⋊C₈	—	109	1	no	—
156	C₄⋊C₈	—	109	1	no	—
157	C₄⋊C₈	—	109	1	no	—
158	C₄⋊C₈	—	109	2	no	—
159	C₄⋊C₈	—	109	2	no	—
160	C₂²⋊C₈	—	109	2	no	—
161	C₂²⋊C₈	—	109	2	no	—
162	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
163	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
164	#97	—	109	2	no	—
165	#97	—	109	2	no	—
166	C₄×C₈	—	109	2	no	—
167	C₄×C₈	—	109	2	no	—
168	#41	—	2	2	no	—
169	#41	—	2	2	no	—
170	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
171	C₂×D₄	C₄×D₁₀₉	1	2	yes	—
172	C₂×D₄	C₄×D₁₀₉	1	1	yes	characteristic
173	C₂²×C₄	—	109	2	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	2	no	—
176	C₂²×C₄	—	109	1	no	—
177	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
178	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	218	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	109	1	no	—
183	C₂×C₈	—	109	2	no	—
184	C₂×C₈	—	109	1	no	—
185	C₂×C₈	—	218	2	no	—
186	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
188	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
189	C₄⋊C₄	—	109	2	no	—
190	C₄⋊C₄	—	109	2	no	—
191	C₄⋊C₄	—	109	1	no	—
192	C₄⋊C₄	—	109	1	no	—
193	C₂²⋊C₄	—	109	2	no	—
194	C₂²⋊C₄	—	109	2	no	—
195	C₂²⋊C₄	—	109	2	no	—
196	C₄²	—	109	2	no	—
197	C₄²	—	109	1	no	—
198	C₄²	—	109	2	no	—
199	C₄²	—	109	1	no	—
200	C₄²	—	109	2	no	—
201	C₄²	—	109	1	no	—
202	C₄²	—	109	1	no	—
203	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
204	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	109	1	no	—
211	C₂×C₄	—	109	1	no	—
212	C₂×C₄	—	109	2	no	—
213	C₂×C₄	—	109	1	no	—
214	C₂×C₄	—	109	2	no	—
215	C₂×C₄	—	109	1	no	—
216	C₂×C₄	—	436	1	no	—
217	C₂×C₄	—	436	2	no	—
218	C₂×C₄	—	218	2	no	—
219	C₂×C₄	—	218	4	no	—
220	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
222	C₂×C₄	#13182028	1	2	yes	—
223	C₂×C₄	#13182028	1	2	yes	—
224	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	218	1	no	—
229	C₈	—	218	1	no	—
230	C₈	—	218	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#16812655	1	2	yes	—
237	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	109	2	no	—
240	C₄	—	218	2	no	—
241	C₄	—	218	1	no	—
242	C₄	—	218	1	no	—
243	C₄	—	218	2	no	—
244	C₄	—	218	1	no	—
245	C₄	#16812655	1	2	yes	—
246	C₄	C₂×#13182028	1	1	yes	characteristic
247	C₄	C₂×#13182028	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#16812655	1	1	yes	characteristic
251	C₂	#131×D₁₀₉	1	1	yes	characteristic
252	C₂	#20295791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#408×C₁₀₉	C₂
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#408	D₁₀₉
hypocenter	C₁₀₉	#2925
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #20295084	C₂	semidirect
3. #20295084	C₂	semidirect
4. #20295080	C₂	semidirect
5. #20295076	C₂	semidirect
6. #20295049	C₂	semidirect
7. C₂×#16812628	C₂	semidirect
9. #20294984	C₂	semidirect
10. #20294981	C₂	semidirect
127. C₁₀₉	#2925	semidirect

Maximal subgroups

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

subgroup	index
#41×Dic₁₀₉	2
#20295084	2
#20295080	2
#20295076	2
#20295049	2
C₂×#16812628	2
#20294993	2
#20294984	2
#20294981	2
#408×C₁₀₉	2
#2925	109