Subgroups — C₂×C₁₀₇×#2188 #41059477

Order 27392.

1126 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₃₂×C₁₀₇	C₂	1	2	yes	maximal, complemented
3	C₂×#467×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#2188	C₂	1	8	yes	maximal, direct product factor
5	C₂×#358×C₁₀₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₀₇	—	2	1	no	—
8	C₂²×Q₁₆×C₁₀₇	—	2	8	no	—
9	C₂²×Q₁₆×C₁₀₇	—	2	1	no	—
10	C₂²×Q₁₆×C₁₀₇	—	2	2	no	—
11	C₂²×Q₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₁₀₇	C₂²	1	4	yes	—
13	C₂²×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₀₇	—	2	1	no	—
16	C₂×Q₃₂×C₁₀₇	—	2	8	no	—
17	C₂×Q₃₂×C₁₀₇	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₁₀₇	C₂²	1	2	yes	—
19	#467×C₁₀₇	—	2	2	no	—
20	#467×C₁₀₇	C₂²	1	4	yes	—
21	C₂×#107×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₁₀₇	—	2	1	no	—
23	#358×C₁₀₇	C₂²	1	8	yes	complemented
24	C₂²⋊C₁₆×C₁₀₇	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₀₇	—	2	8	no	—
26	C₂²×Q₈×C₁₀₇	—	2	4	no	—
27	C₂²×Q₈×C₁₀₇	—	2	2	no	—
28	C₂²×Q₈×C₁₀₇	—	2	1	no	—
29	C₂²×Q₈×C₁₀₇	—	4	4	no	—
30	C₂²×Q₈×C₁₀₇	—	4	2	no	—
31	C₂²×Q₈×C₁₀₇	—	4	1	no	—
32	C₂³×C₄×C₁₀₇	—	4	1	no	—
33	C₂³×C₄×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂×Q₁₆×C₁₀₇	—	2	8	no	—
35	C₂×Q₁₆×C₁₀₇	—	2	16	no	—
36	C₂×Q₁₆×C₁₀₇	—	2	16	no	—
37	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
38	C₂×Q₁₆×C₁₀₇	—	2	2	no	—
39	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
40	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
41	C₂×Q₁₆×C₁₀₇	—	2	1	no	—
42	C₂×Q₁₆×C₁₀₇	D₄	1	8	yes	—
43	C₂×Q₁₆×C₁₀₇	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₁₀₇	C₂³	1	4	yes	—
45	C₂²×C₈×C₁₀₇	—	2	4	no	—
46	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
48	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
50	C₂²⋊Q₈×C₁₀₇	—	2	4	no	—
51	C₂²⋊Q₈×C₁₀₇	—	4	2	no	—
52	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₀₇	—	4	1	no	—
54	C₂×C₂²⋊C₄×C₁₀₇	—	4	1	no	—
55	Q₃₂×C₁₀₇	—	2	16	no	—
56	C₂×C₁₆×C₁₀₇	—	2	4	no	—
57	C₂×C₁₆×C₁₀₇	C₂³	1	4	yes	—
58	#107×C₁₀₇	—	2	1	no	—
59	#107×C₁₀₇	C₂³	1	2	yes	—
60	#103×C₁₀₇	—	2	4	no	—
61	C₂⁴×C₁₀₇	D₈	1	1	yes	characteristic
62	C₂×Q₈×C₁₀₇	—	2	16	no	—
63	C₂×Q₈×C₁₀₇	—	2	8	no	—
64	C₂×Q₈×C₁₀₇	—	2	4	no	—
65	C₂×Q₈×C₁₀₇	—	2	2	no	—
66	C₂×Q₈×C₁₀₇	—	4	8	no	—
67	C₂×Q₈×C₁₀₇	—	4	8	no	—
68	C₂×Q₈×C₁₀₇	—	4	16	no	—
69	C₂×Q₈×C₁₀₇	—	4	4	no	—
70	C₂×Q₈×C₁₀₇	—	4	2	no	—
71	C₂×Q₈×C₁₀₇	—	4	1	no	—
72	C₂×Q₈×C₁₀₇	—	4	4	no	—
73	C₂×Q₈×C₁₀₇	—	4	2	no	—
74	C₂×Q₈×C₁₀₇	—	4	2	no	—
75	C₂²×C₄×C₁₀₇	—	2	2	no	—
76	C₂²×C₄×C₁₀₇	—	2	2	no	—
77	C₂²×C₄×C₁₀₇	—	4	8	no	—
78	C₂²×C₄×C₁₀₇	—	4	4	no	—
79	C₂²×C₄×C₁₀₇	—	4	2	no	—
80	C₂²×C₄×C₁₀₇	—	4	1	no	—
81	C₂²×C₄×C₁₀₇	—	4	1	no	—
82	C₂²×C₄×C₁₀₇	D₈	1	2	yes	—
83	C₂²×C₄×C₁₀₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₁₀₇	—	2	16	no	—
87	Q₁₆×C₁₀₇	—	2	8	no	—
88	Q₁₆×C₁₀₇	—	2	4	no	—
89	Q₁₆×C₁₀₇	—	2	4	no	—
90	Q₁₆×C₁₀₇	—	4	2	no	—
91	Q₁₆×C₁₀₇	C₂×D₄	1	8	yes	—
92	C₂×C₈×C₁₀₇	—	2	2	no	—
93	C₂×C₈×C₁₀₇	—	2	4	no	—
94	C₂×C₈×C₁₀₇	—	2	4	no	—
95	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₀₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₇	—	2	2	no	—
100	C₄⋊C₄×C₁₀₇	—	4	1	no	—
101	C₄⋊C₄×C₁₀₇	—	4	4	no	—
102	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
103	C₁₆×C₁₀₇	—	2	4	no	—
104	C₂³×C₁₀₇	—	2	2	no	—
105	C₂³×C₁₀₇	—	2	2	no	—
106	C₂³×C₁₀₇	Q₃₂	1	4	yes	complemented
107	C₂³×C₁₀₇	C₂×D₈	1	2	yes	—
108	C₂³×C₁₀₇	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₁₀₇	—	2	8	no	—
110	Q₈×C₁₀₇	—	4	16	no	—
111	Q₈×C₁₀₇	—	4	4	no	—
112	Q₈×C₁₀₇	—	4	8	no	—
113	Q₈×C₁₀₇	—	4	2	no	—
114	Q₈×C₁₀₇	—	4	2	no	—
115	Q₈×C₁₀₇	—	8	2	no	—
116	C₂×C₄×C₁₀₇	—	2	2	no	—
117	C₂×C₄×C₁₀₇	—	2	2	no	—
118	C₂×C₄×C₁₀₇	—	2	2	no	—
119	C₂×C₄×C₁₀₇	—	2	4	no	—
120	C₂×C₄×C₁₀₇	—	4	16	no	—
121	C₂×C₄×C₁₀₇	—	4	4	no	—
122	C₂×C₄×C₁₀₇	—	4	8	no	—
123	C₂×C₄×C₁₀₇	—	4	2	no	—
124	C₂×C₄×C₁₀₇	—	4	2	no	—
125	C₂×C₄×C₁₀₇	—	8	2	no	—
126	C₂×C₄×C₁₀₇	—	8	2	no	—
127	C₂×C₄×C₁₀₇	#120	1	2	yes	—
128	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₀₇	—	2	2	no	—
133	C₈×C₁₀₇	#120	1	2	yes	—
134	C₈×C₁₀₇	C₂²×D₄	1	2	yes	—
135	C₂²×C₁₀₇	—	2	2	no	—
136	C₂²×C₁₀₇	—	2	8	no	—
137	C₂²×C₁₀₇	—	2	2	no	—
138	C₂²×C₁₀₇	#447	1	2	yes	—
139	C₂²×C₁₀₇	C₂×Q₃₂	1	2	yes	—
140	C₂²×C₁₀₇	C₂×Q₃₂	1	4	yes	complemented
141	C₂²×C₁₀₇	#510	1	2	yes	—
142	C₂²×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₁₀₇	—	2	2	no	—
144	C₄×C₁₀₇	—	4	8	no	—
145	C₄×C₁₀₇	—	8	2	no	—
146	C₄×C₁₀₇	—	8	2	no	—
147	C₄×C₁₀₇	#447	1	2	yes	—
148	C₄×C₁₀₇	C₂×#120	1	1	yes	characteristic
149	C₄×C₁₀₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×#2188	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₇	—	2	4	no	—
152	C₂×C₁₀₇	#2188	1	4	yes	direct product factor
153	C₂×C₁₀₇	C₂×#447	1	1	yes	characteristic
154	C₂×C₁₀₇	C₂²×Q₃₂	1	1	yes	characteristic
155	C₂×C₁₀₇	C₂×#510	1	1	yes	characteristic
156	C₂³×Q₁₆	C₂×C₁₀₇	1	1	yes	characteristic
157	C₂²×Q₃₂	C₂×C₁₀₇	1	2	yes	complemented
158	C₂×#467	C₂×C₁₀₇	1	1	yes	characteristic
159	#2188	C₂×C₁₀₇	1	8	yes	direct product factor
160	C₂×#358	C₂×C₁₀₇	1	2	yes	complemented
161	C₂×C₂²⋊C₁₆	C₂×C₁₀₇	1	1	yes	characteristic
162	C₁₀₇	C₂×#2188	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³×Q₈	—	2	1	no	—
164	C₂²×Q₁₆	—	2	8	no	—
165	C₂²×Q₁₆	—	2	1	no	—
166	C₂²×Q₁₆	—	2	2	no	—
167	C₂²×Q₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
168	C₂²×Q₁₆	C₂²×C₁₀₇	1	4	yes	—
169	C₂²×Q₁₆	C₂²×C₁₀₇	1	2	yes	—
170	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
171	C₂×C₂²⋊Q₈	—	2	1	no	—
172	C₂×Q₃₂	—	2	8	no	—
173	C₂×Q₃₂	C₂²×C₁₀₇	1	8	yes	complemented
174	C₂²×C₁₆	C₂²×C₁₀₇	1	2	yes	—
175	#467	—	2	2	no	—
176	#467	C₂²×C₁₀₇	1	4	yes	—
177	C₂×#107	C₂²×C₁₀₇	1	1	yes	characteristic
178	C₂×#103	—	2	1	no	—
179	#358	C₂²×C₁₀₇	1	8	yes	complemented
180	C₂²⋊C₁₆	C₂²×C₁₀₇	1	4	yes	—
181	C₂²×Q₈	—	2	8	no	—
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	2	1	no	—
185	C₂²×Q₈	—	4	4	no	—
186	C₂²×Q₈	—	4	1	no	—
187	C₂²×Q₈	—	4	2	no	—
188	C₂³×C₄	—	4	1	no	—
189	C₂³×C₄	D₄×C₁₀₇	1	1	yes	characteristic
190	C₂×Q₁₆	—	2	8	no	—
191	C₂×Q₁₆	—	2	16	no	—
192	C₂×Q₁₆	—	2	16	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	2	2	no	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	D₄×C₁₀₇	1	8	yes	—
199	C₂×Q₁₆	C₂³×C₁₀₇	1	2	yes	—
200	C₂×Q₁₆	C₂³×C₁₀₇	1	4	yes	—
201	C₂²×C₈	—	2	4	no	—
202	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
203	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
204	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
205	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
206	C₂²⋊Q₈	—	2	4	no	—
207	C₂²⋊Q₈	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₂²⋊C₄	—	4	1	no	—
211	Q₃₂	—	2	16	no	—
212	C₂×C₁₆	—	2	4	no	—
213	C₂×C₁₆	C₂³×C₁₀₇	1	4	yes	—
214	#107	—	2	1	no	—
215	#107	C₂³×C₁₀₇	1	2	yes	—
216	#103	—	2	4	no	—
217	C₂⁴	D₈×C₁₀₇	1	1	yes	characteristic
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	8	no	—
223	C₂×Q₈	—	4	16	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	D₈×C₁₀₇	1	1	yes	characteristic
239	C₂²×C₄	D₈×C₁₀₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
242	Q₁₆	—	2	16	no	—
243	Q₁₆	—	2	8	no	—
244	Q₁₆	—	2	4	no	—
245	Q₁₆	—	2	4	no	—
246	Q₁₆	—	4	2	no	—
247	Q₁₆	C₂×D₄×C₁₀₇	1	8	yes	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₁₀₇	1	4	yes	—
254	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₁₆	—	2	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	Q₃₂×C₁₀₇	1	4	yes	complemented
263	C₂³	C₂×D₈×C₁₀₇	1	2	yes	—
264	C₂³	C₂×D₈×C₁₀₇	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	Q₈	—	4	16	no	—
267	Q₈	—	4	8	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	16	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	#120×C₁₀₇	1	2	yes	—
284	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
286	C₂×C₄	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	#120×C₁₀₇	1	2	yes	—
290	C₈	C₂²×D₄×C₁₀₇	1	2	yes	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	#447×C₁₀₇	1	2	yes	—
295	C₂²	C₂×Q₃₂×C₁₀₇	1	2	yes	—
296	C₂²	C₂×Q₃₂×C₁₀₇	1	4	yes	complemented
297	C₂²	#510×C₁₀₇	1	2	yes	—
298	C₂²	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	8	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	#447×C₁₀₇	1	2	yes	—
304	C₄	C₂×#120×C₁₀₇	1	1	yes	characteristic
305	C₄	C₂×AGL(1,ℤ/8)×C₁₀₇	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	C₁₀₇×#2188	1	4	yes	direct product factor
308	C₂	C₂×#447×C₁₀₇	1	1	yes	characteristic
309	C₂	C₂²×Q₃₂×C₁₀₇	1	1	yes	characteristic
310	C₂	C₂×#510×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
prime radical	C₂⁴×C₁₀₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ C₂³×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₃₂×C₁₀₇	C₂	semidirect
4. C₁₀₇×#2188	C₂	direct (N × H)
5. C₂×#358×C₁₀₇	C₂	semidirect
17. C₂×Q₃₂×C₁₀₇	C₂²	semidirect
23. #358×C₁₀₇	C₂²	semidirect
106. C₂³×C₁₀₇	Q₃₂	semidirect
140. C₂²×C₁₀₇	C₂×Q₃₂	semidirect
150. C₂×#2188	C₁₀₇	direct (N × H)
152. C₂×C₁₀₇	#2188	direct (N × H)
157. C₂²×Q₃₂	C₂×C₁₀₇	semidirect
159. #2188	C₂×C₁₀₇	direct (N × H)
160. C₂×#358	C₂×C₁₀₇	semidirect
162. C₁₀₇	C₂×#2188	direct (N × H)
173. C₂×Q₃₂	C₂²×C₁₀₇	semidirect
179. #358	C₂²×C₁₀₇	semidirect
262. C₂³	Q₃₂×C₁₀₇	semidirect
296. C₂²	C₂×Q₃₂×C₁₀₇	semidirect
307. C₂	C₁₀₇×#2188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₀₇	2
C₂²×Q₃₂×C₁₀₇	2
C₂×#467×C₁₀₇	2
C₁₀₇×#2188	2
C₂×#358×C₁₀₇	2
C₂×C₂²⋊C₁₆×C₁₀₇	2
C₂×#2188	107

Subgroups — C2×C107×#2188 #41059477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#2188 #41059477