Subgroups — #28345524

Order 13696.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293174	C₂	1	1	yes	maximal, characteristic
2	#20292706	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292705	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292702	C₂	1	2	yes	maximal, complemented
5	#20292692	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₁₀₇	C₂	1	2	yes	maximal, complemented
7	#20292679	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292678	C₂	1	1	yes	maximal, characteristic, complemented
9	#20292593	C₂	1	2	yes	maximal, complemented
10	#20292592	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133818	C₂²	1	2	yes	—
13	C₂×#13133802	C₂²	1	2	yes	—
14	C₂×#13133801	C₂²	1	2	yes	—
15	C₂×C₈×D₁₀₇	C₂²	1	2	yes	—
16	#16812108	C₄	1	1	yes	characteristic
17	#16812105	C₄	1	1	yes	characteristic
18	#16812095	C₄	1	2	yes	—
19	#16812084	C₄	1	1	yes	characteristic
20	#16812082	C₄	1	2	yes	—
21	#16812080	C₄	1	1	yes	characteristic
22	C₄×D₄₂₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
24	C₄²×D₁₀₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₀₇)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₀₇)⋊C₈	C₂²	1	2	yes	—
34	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₀₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
37	C₄×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₀₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
41	#13133818	—	4	2	no	—
42	#13133818	—	4	2	no	—
43	C₂×Dic₂₁₄	C₂×C₄	1	2	yes	—
44	C₂×D₄₂₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄₂₈	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₀₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₀₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₀₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₁₀₇	C₂³	1	2	yes	—
53	#13133802	—	4	2	no	—
54	#13133801	—	4	2	no	—
55	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₀₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
58	C₈×D₁₀₇	—	4	2	no	—
59	C₄⋊Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
60	Dic₁₀₇⋊C₄	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₀₇	C₂×C₄	1	2	yes	—
62	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₁₀₇)⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₀₇	—	4	2	no	—
70	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₀₇	C₂³	1	2	yes	—
73	C₄⋊C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₀₇	C₂×C₄	1	2	yes	—
75	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
76	C₂³×C₁₀₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₀₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₀₇	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₁₀₇	—	2	2	no	—
80	C₂×C₄×C₁₀₇	—	2	2	no	—
81	C₂×C₄×C₁₀₇	—	2	2	no	—
82	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₀₇	—	2	4	no	—
87	C₂²⋊D₁₀₇	—	4	2	no	—
88	C₂²⋊D₁₀₇	—	4	2	no	—
89	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₀₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₀₇	C₂²×C₄	1	1	yes	characteristic
93	D₄₂₈	—	2	2	no	—
94	D₄₂₈	—	4	2	no	—
95	C₄×D₁₀₇	—	2	2	no	—
96	C₄×D₁₀₇	—	2	2	no	—
97	C₄×D₁₀₇	—	2	2	no	—
98	C₄×D₁₀₇	—	2	1	no	—
99	C₄×D₁₀₇	—	2	1	no	—
100	C₄×D₁₀₇	—	2	2	no	—
101	Dic₂₁₄	—	4	2	no	—
102	C₈×C₁₀₇	—	2	1	no	—
103	C₈×C₁₀₇	—	2	2	no	—
104	C₈×C₁₀₇	—	2	1	no	—
105	C₁₀₇⋊C₈	—	2	1	no	—
106	C₁₀₇⋊C₈	—	2	2	no	—
107	C₁₀₇⋊C₈	—	2	1	no	—
108	C₂²×C₁₀₇	—	2	2	no	—
109	C₂²×C₁₀₇	—	2	2	no	—
110	C₂²×C₁₀₇	—	2	2	no	—
111	C₂²×C₁₀₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₀₇	—	2	1	no	—
113	C₂×D₁₀₇	—	2	2	no	—
114	C₂×D₁₀₇	—	2	1	no	—
115	C₂×D₁₀₇	—	2	1	no	—
116	C₂×D₁₀₇	—	2	2	no	—
117	C₂×D₁₀₇	—	2	2	no	—
118	C₄×C₁₀₇	—	2	2	no	—
119	C₄×C₁₀₇	—	2	1	no	—
120	C₄×C₁₀₇	#131	1	2	yes	—
121	C₄×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₁₀₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₁₀₇	—	2	1	no	—
124	Dic₁₀₇	—	2	2	no	—
125	Dic₁₀₇	—	2	1	no	—
126	Dic₁₀₇	#131	1	2	yes	—
127	C₂×C₁₀₇	—	4	2	no	—
128	C₂×C₁₀₇	#518	1	1	yes	characteristic
129	C₂×C₁₀₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₀₇	C₂×#131	1	1	yes	characteristic
131	D₁₀₇	—	4	1	no	—
132	D₁₀₇	—	4	2	no	—
133	#2979	—	107	1	no	maximal, Sylow
134	C₁₀₇	#2979	1	1	yes	characteristic, complemented, Sylow
135	#532	—	107	1	no	—
136	#436	—	107	1	no	—
137	#436	—	107	2	no	—
138	#436	—	107	1	no	—
139	C₈×D₄	—	107	2	no	—
140	C₈×D₄	—	107	1	no	—
141	C₈×D₄	D₁₀₇	1	1	yes	characteristic, complemented
142	#431	—	107	1	no	—
143	#431	—	107	1	no	—
144	#408	—	107	2	no	—
145	#408	—	107	2	no	—
146	C₂×#41	—	107	2	no	—
147	C₂×M₁₆	—	107	2	no	—
148	C₂×M₁₆	—	107	2	no	—
149	C₂²×C₈	—	107	2	no	—
150	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
151	C₄⋊Q₈	—	107	1	no	—
152	#127	—	107	1	no	—
153	#124	—	107	2	no	—
154	#121	—	107	1	no	—
155	#121	—	107	1	no	—
156	#121	—	107	2	no	—
157	C₄×D₄	—	107	2	no	—
158	C₄×D₄	—	107	1	no	—
159	C₄×D₄	C₂×D₁₀₇	1	1	yes	characteristic
160	C₂×C₄²	—	107	1	no	—
161	C₄⋊C₈	—	107	1	no	—
162	C₄⋊C₈	—	107	1	no	—
163	C₄⋊C₈	—	107	1	no	—
164	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
165	C₂²⋊C₈	—	107	2	no	—
166	C₂²⋊C₈	—	107	1	no	—
167	C₂²⋊C₈	—	107	2	no	—
168	C₂²⋊C₈	—	107	1	no	—
169	C₂²⋊C₈	—	107	2	no	—
170	C₂²⋊C₈	—	107	2	no	—
171	C₂²⋊C₈	C₂×D₁₀₇	1	2	yes	—
172	C₄×C₈	—	107	2	no	—
173	C₄×C₈	—	107	1	no	—
174	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
175	#41	—	428	2	no	—
176	#41	—	428	2	no	—
177	C₂×Q₈	—	107	2	no	—
178	C₂×D₄	—	107	2	no	—
179	C₂×D₄	—	107	2	no	—
180	C₂×D₄	—	107	2	no	—
181	C₂×D₄	—	107	2	no	—
182	C₂×D₄	—	107	1	no	—
183	C₂×D₄	C₄×D₁₀₇	1	1	yes	characteristic
184	C₂²×C₄	—	107	1	no	—
185	C₂²×C₄	—	107	1	no	—
186	C₂²×C₄	—	107	1	no	—
187	C₂²×C₄	—	107	2	no	—
188	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
189	M₁₆	—	428	2	no	—
190	M₁₆	—	428	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	107	1	no	—
193	C₂×C₈	—	107	2	no	—
194	C₂×C₈	—	107	1	no	—
195	C₂×C₈	—	428	2	no	—
196	C₂×C₈	C₂²×D₁₀₇	1	2	yes	—
197	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
199	C₄⋊C₄	—	107	2	no	—
200	C₄⋊C₄	—	107	1	no	—
201	C₄⋊C₄	C₄×D₁₀₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	107	2	no	—
203	C₂²⋊C₄	—	107	2	no	—
204	C₂²⋊C₄	—	107	2	no	—
205	C₂²⋊C₄	C₄×D₁₀₇	1	2	yes	—
206	C₄²	—	107	1	no	—
207	C₄²	—	107	1	no	—
208	C₄²	—	107	1	no	—
209	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
210	C₂³	—	107	2	no	—
211	C₂³	—	107	1	no	—
212	C₂³	C₂×C₄×D₁₀₇	1	2	yes	—
213	Q₈	—	428	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	428	2	no	—
216	D₄	—	428	2	no	—
217	D₄	—	428	2	no	—
218	D₄	—	214	4	no	—
219	D₄	—	214	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	107	1	no	—
223	C₂×C₄	—	107	1	no	—
224	C₂×C₄	—	107	1	no	—
225	C₂×C₄	—	107	2	no	—
226	C₂×C₄	—	214	2	no	—
227	C₂×C₄	—	214	2	no	—
228	C₂×C₄	—	214	2	no	—
229	C₂×C₄	—	214	1	no	—
230	C₂×C₄	—	214	2	no	—
231	C₂×C₄	—	214	1	no	—
232	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	214	1	no	—
240	C₈	—	214	2	no	—
241	C₈	—	214	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	214	1	no	—
246	C₂²	—	214	2	no	—
247	C₂²	—	214	1	no	—
248	C₂²	—	214	2	no	—
249	C₂²	—	214	1	no	—
250	C₂²	—	214	2	no	—
251	C₂²	C₂²×C₄×D₁₀₇	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	107	2	no	—
255	C₄	—	214	1	no	—
256	C₄	—	214	1	no	—
257	C₄	—	214	2	no	—
258	C₄	#16812115	1	2	yes	—
259	C₄	#16812154	1	1	yes	characteristic
260	C₄	#16812155	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	428	1	no	—
263	C₂	—	428	2	no	—
264	C₂	#20293099	1	1	yes	characteristic
265	C₂	#131×D₁₀₇	1	1	yes	characteristic
266	C₂	C₂×#16812115	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂³×C₄
Fitting	C₈×D₄×C₁₀₇	C₂
prime radical	#20293174	C₂
cosocle	C₂×C₄×C₁₀₇	C₂⁴
hypercenter	C₈×D₄	D₁₀₇
hypocenter	C₁₀₇	#2979
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #20292706	C₂	semidirect
3. #20292705	C₂	semidirect
4. #20292702	C₂	semidirect
5. #20292692	C₂	semidirect
6. C₈×C₂²⋊D₁₀₇	C₂	semidirect
7. #20292679	C₂	semidirect
8. #20292678	C₂	semidirect
9. #20292593	C₂	semidirect
10. #20292592	C₂	semidirect
11. C₈×D₄×C₁₀₇	C₂	semidirect
134. C₁₀₇	#2979	semidirect
141. C₈×D₄	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293174	2
#20292706	2
#20292705	2
#20292702	2
#20292692	2
C₈×C₂²⋊D₁₀₇	2
#20292679	2
#20292678	2
#20292593	2
#20292592	2
C₈×D₄×C₁₀₇	2
#2979	107