Subgroups — #28378725

Order 13952.

372 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#20295901	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	—	2	1	no	—
6	C₂×#13182039	C₄	1	2	yes	—
7	D₁₀₉⋊M₁₆	—	2	1	no	—
8	D₁₀₉⋊M₁₆	—	2	1	no	—
9	D₁₀₉⋊M₁₆	—	2	1	no	—
10	D₁₀₉⋊M₁₆	—	2	1	no	—
11	D₁₀₉⋊M₁₆	C₄	1	2	yes	—
12	D₁₀₉⋊M₁₆	C₄	1	2	yes	—
13	D₁₀₉⋊M₁₆	C₄	1	2	yes	—
14	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
15	C₂×#13182038	—	2	1	no	—
16	C₂×#13182038	—	2	1	no	—
17	C₂×D₁₀₉⋊C₈	—	2	1	no	—
18	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×#13182013	C₄	1	1	yes	characteristic
21	C₂×#13182012	—	2	1	no	—
22	M₁₆×D₁₀₉	—	2	1	no	—
23	M₁₆×D₁₀₉	—	2	1	no	—
24	M₁₆×D₁₀₉	C₄	1	2	yes	—
25	M₁₆×D₁₀₉	C₂²	1	2	yes	—
26	C₂×C₈×D₁₀₉	—	2	1	no	—
27	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
28	#13182039	—	2	2	no	—
29	#13182039	—	2	2	no	—
30	#13182039	—	2	2	no	—
31	#13182039	C₂×C₄	1	4	yes	—
32	#13182038	—	2	2	no	—
33	#13182038	—	2	2	no	—
34	#13182038	—	2	2	no	—
35	#13182038	—	2	2	no	—
36	D₁₀₉⋊C₈	—	2	2	no	—
37	D₁₀₉⋊C₈	—	2	2	no	—
38	D₁₀₉⋊C₈	—	2	2	no	—
39	D₁₀₉⋊C₈	C₂×C₄	1	4	yes	—
40	C₂×#11724786	—	2	2	no	—
41	C₂×#11724786	—	2	2	no	—
42	C₂³×D₁₀₉	Q₈	1	1	yes	characteristic
43	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
55	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
56	#13182013	—	2	1	no	—
57	#13182013	C₂×C₄	1	2	yes	—
58	#13182012	—	2	2	no	—
59	#13182012	—	2	2	no	—
60	C₂×#11724784	—	2	1	no	—
61	C₈×D₁₀₉	—	2	2	no	—
62	C₈×D₁₀₉	—	2	2	no	—
63	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
64	M₁₆×C₁₀₉	—	2	1	no	—
65	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
66	C₂×C₈×C₁₀₉	—	2	1	no	—
67	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×D₁₀₉	—	4	2	no	—
69	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
88	C₄×D₁₀₉	C₄²	1	1	yes	characteristic
89	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
97	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
98	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
103	#11724786	—	2	4	no	—
104	#11724786	—	2	4	no	—
105	C₈×C₁₀₉	—	2	2	no	—
106	#11724784	—	2	2	no	—
107	C₂²×C₁₀₉	—	4	1	no	—
108	C₂²×C₁₀₉	#95	1	1	yes	characteristic
109	C₂²×C₁₀₉	#95	1	1	yes	characteristic
110	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₁₀₉	—	2	2	no	—
112	C₂×D₁₀₉	—	2	2	no	—
113	C₂×D₁₀₉	—	2	2	no	—
114	C₂×D₁₀₉	—	4	1	no	—
115	C₂×D₁₀₉	—	4	1	no	—
116	C₂×D₁₀₉	—	4	1	no	—
117	C₂×D₁₀₉	#95	1	1	yes	characteristic
118	C₂×D₁₀₉	#95	1	1	yes	characteristic
119	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₁₀₉	#95	1	1	yes	characteristic
122	C₄×C₁₀₉	#95	1	1	yes	characteristic
123	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
124	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	Dic₁₀₉	#95	1	1	yes	characteristic
126	Dic₁₀₉	#95	1	1	yes	characteristic
127	Dic₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	Dic₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₁₀₉	—	2	1	no	—
130	C₂×C₁₀₉	—	2	1	no	—
131	C₂×C₁₀₉	—	2	1	no	—
132	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
133	D₁₀₉	—	2	1	no	—
134	D₁₀₉	—	2	1	no	—
135	D₁₀₉	—	2	1	no	—
136	D₁₀₉	#430	1	2	yes	—
137	#1738	—	109	1	no	maximal, Sylow
138	C₁₀₉	#1738	1	1	yes	characteristic, complemented, Sylow
139	C₂²×M₁₆	—	109	1	no	—
140	C₂²×M₁₆	—	109	1	no	—
141	C₂²×M₁₆	—	109	1	no	—
142	#341	—	109	4	no	—
143	C₂³×C₄	—	109	1	no	—
144	C₂×M₁₆	—	109	2	no	—
145	C₂×M₁₆	—	109	2	no	—
146	C₂×M₁₆	—	109	2	no	—
147	C₂×M₁₆	—	109	2	no	—
148	C₂×M₁₆	—	109	2	no	—
149	C₂×M₁₆	—	109	1	no	—
150	C₂×M₁₆	—	109	2	no	—
151	C₂×M₁₆	—	109	2	no	—
152	C₂×M₁₆	—	218	1	no	—
153	C₂×M₁₆	—	218	1	no	—
154	C₂×M₁₆	—	218	1	no	—
155	C₂×M₁₆	—	218	1	no	—
156	C₂×M₁₆	—	218	1	no	—
157	C₂×M₁₆	—	218	1	no	—
158	C₂×M₁₆	—	218	1	no	—
159	C₂×M₁₆	—	218	1	no	—
160	C₂×M₁₆	—	218	1	no	—
161	C₂×M₁₆	—	218	1	no	—
162	C₂×M₁₆	#89137	1	1	yes	characteristic
163	C₂²×C₈	—	109	2	no	—
164	C₂²×C₈	—	218	1	no	—
165	C₂²×C₈	—	218	1	no	—
166	C₂⁴	—	109	1	no	—
167	C₂²×C₄	—	109	1	no	—
168	C₂²×C₄	—	109	1	no	—
169	C₂²×C₄	—	109	1	no	—
170	C₂²×C₄	—	109	1	no	—
171	C₂²×C₄	—	109	1	no	—
172	C₂²×C₄	—	109	1	no	—
173	C₂²×C₄	—	109	1	no	—
174	C₂²×C₄	—	109	1	no	—
175	C₂²×C₄	—	109	1	no	—
176	C₂²×C₄	—	109	1	no	—
177	C₂²×C₄	—	109	1	no	—
178	C₂²×C₄	—	109	1	no	—
179	C₂²×C₄	—	109	1	no	—
180	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
181	M₁₆	—	2	1	no	—
182	M₁₆	—	109	4	no	—
183	M₁₆	—	109	2	no	—
184	M₁₆	—	218	2	no	—
185	M₁₆	—	218	2	no	—
186	M₁₆	—	218	2	no	—
187	M₁₆	—	218	2	no	—
188	M₁₆	—	218	2	no	—
189	M₁₆	—	218	2	no	—
190	M₁₆	—	218	2	no	—
191	M₁₆	—	218	2	no	—
192	M₁₆	—	218	2	no	—
193	M₁₆	—	218	1	no	—
194	M₁₆	C₂×#89137	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	109	4	no	—
197	C₂×C₈	—	218	2	no	—
198	C₂×C₈	—	218	2	no	—
199	C₂×C₈	—	218	2	no	—
200	C₂×C₈	—	218	2	no	—
201	C₂×C₈	—	218	2	no	—
202	C₂×C₈	—	218	2	no	—
203	C₂×C₈	—	218	2	no	—
204	C₂×C₈	—	218	1	no	—
205	C₂³	—	109	1	no	—
206	C₂³	—	109	1	no	—
207	C₂³	—	109	1	no	—
208	C₂³	—	109	1	no	—
209	C₂³	—	109	1	no	—
210	C₂³	—	109	1	no	—
211	C₂³	—	436	2	no	—
212	C₂³	C₄⋊#89137	1	1	yes	characteristic
213	C₂×C₄	—	109	1	no	—
214	C₂×C₄	—	109	1	no	—
215	C₂×C₄	—	109	1	no	—
216	C₂×C₄	—	109	1	no	—
217	C₂×C₄	—	109	1	no	—
218	C₂×C₄	—	109	1	no	—
219	C₂×C₄	—	109	1	no	—
220	C₂×C₄	—	109	1	no	—
221	C₂×C₄	—	109	1	no	—
222	C₂×C₄	—	109	1	no	—
223	C₂×C₄	—	109	1	no	—
224	C₂×C₄	—	109	1	no	—
225	C₂×C₄	—	109	1	no	—
226	C₂×C₄	—	109	1	no	—
227	C₂×C₄	—	109	1	no	—
228	C₂×C₄	—	109	1	no	—
229	C₂×C₄	—	109	1	no	—
230	C₂×C₄	—	109	1	no	—
231	C₂×C₄	—	109	1	no	—
232	C₂×C₄	—	109	1	no	—
233	C₂×C₄	—	109	1	no	—
234	C₂×C₄	—	109	1	no	—
235	C₂×C₄	C₄×#89137	1	1	yes	characteristic
236	C₂×C₄	C₄×#89137	1	1	yes	characteristic
237	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
238	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
239	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
240	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	218	4	no	—
243	C₈	—	218	4	no	—
244	C₈	—	218	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	109	1	no	—
247	C₂²	—	109	1	no	—
248	C₂²	—	109	1	no	—
249	C₂²	—	109	1	no	—
250	C₂²	—	436	1	no	—
251	C₂²	—	436	1	no	—
252	C₂²	—	436	1	no	—
253	C₂²	—	218	2	no	—
254	C₂²	—	218	2	no	—
255	C₂²	—	218	2	no	—
256	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
257	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
259	C₄	—	109	1	no	—
260	C₄	—	109	1	no	—
261	C₄	—	109	1	no	—
262	C₄	—	109	1	no	—
263	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
264	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
265	C₄	C₂×C₄×#89137	1	1	yes	characteristic
266	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	109	2	no	—
271	C₂	—	218	1	no	—
272	C₂	—	218	1	no	—
273	C₂	—	218	1	no	—
274	C₂	C₂×(C₂×#89137)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×#95
prime radical	C₂³×D₁₀₉	Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#1738
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20295901	C₂	semidirect
138. C₁₀₉	#1738	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₀₉⋊M₁₆	2
#20295901	2
C₂×M₁₆×D₁₀₉	2
#1738	109