Subgroups — C₁₀₇×#18693 #41044657

Order 27392.

868 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₇×#3550	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#3255	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#3131	C₂	1	1	yes	maximal, characteristic
4	C₁₀₇×#1645	C₂	1	4	yes	maximal, complemented
5	C₁₀₇×#1640	C₂	1	4	yes	maximal, complemented
6	C₁₀₇×#1560	C₂	1	1	yes	maximal, characteristic
7	C₁₀₇×#1493	C₂	1	2	yes	maximal, complemented
8	#563×C₁₀₇	—	2	2	no	—
9	#559×C₁₀₇	C₂²	1	2	yes	—
10	Q₈²×C₁₀₇	C₂²	1	2	yes	—
11	#556×C₁₀₇	—	2	2	no	—
12	D₄×Q₈×C₁₀₇	—	2	2	no	—
13	#542×C₁₀₇	—	2	4	no	—
14	#536×C₁₀₇	—	2	2	no	—
15	#533×C₁₀₇	C₂²	1	1	yes	characteristic
16	#533×C₁₀₇	C₂²	1	2	yes	—
17	#532×C₁₀₇	C₂²	1	1	yes	characteristic
18	#488×C₁₀₇	—	2	2	no	—
19	#487×C₁₀₇	C₂²	1	2	yes	—
20	#486×C₁₀₇	C₂²	1	2	yes	—
21	#475×C₁₀₇	—	2	2	no	—
22	#475×C₁₀₇	C₂²	1	4	yes	—
23	#470×C₁₀₇	—	2	4	no	—
24	#462×C₁₀₇	—	2	2	no	—
25	#462×C₁₀₇	C₂²	1	4	yes	—
26	#423×C₁₀₇	C₂²	1	2	yes	—
27	#416×C₁₀₇	—	2	2	no	—
28	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
29	#333×C₁₀₇	C₂²	1	2	yes	—
30	#331×C₁₀₇	C₂²	1	2	yes	—
31	Q₈⋊C₈×C₁₀₇	C₂²	1	8	yes	—
32	C₂×#41×C₁₀₇	D₄	1	2	yes	—
33	C₂²×Q₈×C₁₀₇	—	2	2	no	—
34	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
35	C₂×M₁₆×C₁₀₇	—	2	2	no	—
36	C₂²×C₈×C₁₀₇	—	2	1	no	—
37	C₄⋊Q₈×C₁₀₇	—	2	4	no	—
38	C₄⋊Q₈×C₁₀₇	—	2	4	no	—
39	C₄⋊Q₈×C₁₀₇	D₄	1	2	yes	—
40	C₄⋊Q₈×C₁₀₇	D₄	1	2	yes	—
41	C₄⋊Q₈×C₁₀₇	C₂³	1	2	yes	—
42	C₄⋊Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
43	#127×C₁₀₇	C₂³	1	1	yes	characteristic
44	#126×C₁₀₇	—	2	8	no	—
45	#125×C₁₀₇	—	2	2	no	—
46	#125×C₁₀₇	—	2	1	no	—
47	#124×C₁₀₇	—	2	4	no	—
48	#124×C₁₀₇	—	2	4	no	—
49	#124×C₁₀₇	D₄	1	2	yes	—
50	#123×C₁₀₇	—	2	4	no	—
51	#123×C₁₀₇	—	2	4	no	—
52	C₂²⋊Q₈×C₁₀₇	—	2	4	no	—
53	C₂²⋊Q₈×C₁₀₇	—	2	8	no	—
54	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
55	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
56	C₂²⋊Q₈×C₁₀₇	D₄	1	4	yes	—
57	#121×C₁₀₇	—	2	2	no	—
58	C₄×Q₈×C₁₀₇	—	2	4	no	—
59	C₄×Q₈×C₁₀₇	—	2	2	no	—
60	C₄×Q₈×C₁₀₇	—	2	2	no	—
61	C₄×Q₈×C₁₀₇	C₂³	1	4	yes	—
62	C₄×D₄×C₁₀₇	—	2	4	no	—
63	C₄×D₄×C₁₀₇	—	2	2	no	—
64	#117×C₁₀₇	—	2	2	no	—
65	#117×C₁₀₇	—	2	2	no	—
66	#117×C₁₀₇	—	2	1	no	—
67	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
68	#106×C₁₀₇	—	2	4	no	—
69	C₄⋊C₈×C₁₀₇	—	2	2	no	—
70	C₄⋊C₈×C₁₀₇	C₂³	1	4	yes	—
71	#103×C₁₀₇	—	2	4	no	—
72	#103×C₁₀₇	—	2	4	no	—
73	#103×C₁₀₇	—	2	4	no	—
74	#102×C₁₀₇	—	2	4	no	—
75	C₄×C₈×C₁₀₇	—	2	1	no	—
76	C₄×C₈×C₁₀₇	C₂³	1	2	yes	—
77	#41×C₁₀₇	—	4	4	no	—
78	C₂×Q₈×C₁₀₇	—	2	4	no	—
79	C₂×Q₈×C₁₀₇	—	2	2	no	—
80	C₂×Q₈×C₁₀₇	—	2	2	no	—
81	C₂×Q₈×C₁₀₇	—	8	4	no	—
82	C₂×Q₈×C₁₀₇	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₁₀₇	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₁₀₇	—	2	2	no	—
85	C₂×D₄×C₁₀₇	—	2	2	no	—
86	C₂×D₄×C₁₀₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₀₇	—	2	4	no	—
88	C₂²×C₄×C₁₀₇	—	2	2	no	—
89	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₁₀₇	—	8	4	no	—
92	M₁₆×C₁₀₇	—	8	2	no	—
93	C₂×C₈×C₁₀₇	—	2	2	no	—
94	C₂×C₈×C₁₀₇	—	2	4	no	—
95	C₂×C₈×C₁₀₇	—	4	2	no	—
96	C₄⋊C₄×C₁₀₇	—	2	8	no	—
97	C₄⋊C₄×C₁₀₇	—	2	2	no	—
98	C₄⋊C₄×C₁₀₇	—	2	2	no	—
99	C₄⋊C₄×C₁₀₇	—	2	2	no	—
100	C₄⋊C₄×C₁₀₇	—	2	4	no	—
101	C₄⋊C₄×C₁₀₇	—	2	2	no	—
102	C₄⋊C₄×C₁₀₇	—	2	4	no	—
103	C₄⋊C₄×C₁₀₇	—	2	2	no	—
104	C₄⋊C₄×C₁₀₇	—	2	2	no	—
105	C₄⋊C₄×C₁₀₇	—	4	2	no	—
106	C₄⋊C₄×C₁₀₇	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
108	C₂²⋊C₄×C₁₀₇	—	2	8	no	—
109	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
110	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
111	C₂²⋊C₄×C₁₀₇	—	4	1	no	—
112	C₄²×C₁₀₇	—	2	4	no	—
113	C₄²×C₁₀₇	—	2	4	no	—
114	C₄²×C₁₀₇	—	4	1	no	—
115	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
117	C₄²×C₁₀₇	C₂⁴	1	1	yes	characteristic
118	C₂³×C₁₀₇	—	2	2	no	—
119	C₂³×C₁₀₇	#120	1	1	yes	characteristic
120	Q₈×C₁₀₇	—	2	4	no	—
121	Q₈×C₁₀₇	—	2	4	no	—
122	Q₈×C₁₀₇	—	4	2	no	—
123	Q₈×C₁₀₇	—	4	4	no	—
124	Q₈×C₁₀₇	—	4	2	no	—
125	Q₈×C₁₀₇	—	4	2	no	—
126	Q₈×C₁₀₇	—	8	4	no	—
127	D₄×C₁₀₇	—	4	2	no	—
128	D₄×C₁₀₇	—	4	2	no	—
129	D₄×C₁₀₇	—	4	2	no	—
130	D₄×C₁₀₇	—	8	2	no	—
131	C₂×C₄×C₁₀₇	—	2	2	no	—
132	C₂×C₄×C₁₀₇	—	2	2	no	—
133	C₂×C₄×C₁₀₇	—	2	2	no	—
134	C₂×C₄×C₁₀₇	—	2	4	no	—
135	C₂×C₄×C₁₀₇	—	2	2	no	—
136	C₂×C₄×C₁₀₇	—	2	4	no	—
137	C₂×C₄×C₁₀₇	—	4	4	no	—
138	C₂×C₄×C₁₀₇	—	4	2	no	—
139	C₂×C₄×C₁₀₇	—	8	4	no	—
140	C₂×C₄×C₁₀₇	#120	1	2	yes	—
141	C₂×C₄×C₁₀₇	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
144	C₈×C₁₀₇	—	4	2	no	—
145	C₈×C₁₀₇	—	4	4	no	—
146	C₂²×C₁₀₇	—	2	2	no	—
147	C₂²×C₁₀₇	—	2	1	no	—
148	C₂²×C₁₀₇	—	4	4	no	—
149	C₂²×C₁₀₇	—	4	2	no	—
150	C₂²×C₁₀₇	C₂×#120	1	1	yes	characteristic
151	C₄×C₁₀₇	—	2	2	no	—
152	C₄×C₁₀₇	—	2	1	no	—
153	C₄×C₁₀₇	—	4	4	no	—
154	C₄×C₁₀₇	—	4	2	no	—
155	C₄×C₁₀₇	—	4	4	no	—
156	C₄×C₁₀₇	—	8	2	no	—
157	C₄×C₁₀₇	#455	1	2	yes	—
158	C₄×C₁₀₇	#577	1	2	yes	—
159	C₄×C₁₀₇	#578	1	2	yes	—
160	#18693	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₁₀₇	—	4	1	no	—
162	C₂×C₁₀₇	—	8	2	no	—
163	C₂×C₁₀₇	#3013	1	2	yes	—
164	C₂×C₁₀₇	C₂×#455	1	1	yes	characteristic
165	#3550	C₂×C₁₀₇	1	1	yes	characteristic
166	#3255	C₂×C₁₀₇	1	2	yes	complemented
167	#3131	C₂×C₁₀₇	1	1	yes	characteristic
168	#1645	C₂×C₁₀₇	1	4	yes	complemented
169	#1640	C₂×C₁₀₇	1	4	yes	complemented
170	#1560	C₂×C₁₀₇	1	1	yes	characteristic
171	#1493	C₂×C₁₀₇	1	2	yes	complemented
172	C₁₀₇	#18693	1	1	yes	characteristic, direct product factor, Sylow
173	#563	—	2	2	no	—
174	#559	C₂²×C₁₀₇	1	2	yes	—
175	Q₈²	C₂²×C₁₀₇	1	2	yes	—
176	#556	—	2	2	no	—
177	D₄×Q₈	—	2	2	no	—
178	#542	—	2	4	no	—
179	#536	—	2	2	no	—
180	#533	C₂²×C₁₀₇	1	1	yes	characteristic
181	#533	C₂²×C₁₀₇	1	2	yes	—
182	#532	C₂²×C₁₀₇	1	1	yes	characteristic
183	#488	—	2	2	no	—
184	#487	C₂²×C₁₀₇	1	2	yes	—
185	#486	C₂²×C₁₀₇	1	2	yes	—
186	#475	—	2	2	no	—
187	#475	C₂²×C₁₀₇	1	4	yes	—
188	#470	—	2	4	no	—
189	#462	—	2	2	no	—
190	#462	C₂²×C₁₀₇	1	4	yes	—
191	#423	C₂²×C₁₀₇	1	2	yes	—
192	#416	—	2	2	no	—
193	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
194	#333	C₂²×C₁₀₇	1	2	yes	—
195	#331	C₂²×C₁₀₇	1	2	yes	—
196	Q₈⋊C₈	C₂²×C₁₀₇	1	8	yes	—
197	C₂×#41	D₄×C₁₀₇	1	2	yes	—
198	C₂²×Q₈	—	2	2	no	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	D₄×C₁₀₇	1	2	yes	—
205	C₄⋊Q₈	D₄×C₁₀₇	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₁₀₇	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₁₀₇	1	1	yes	characteristic
208	#127	C₂³×C₁₀₇	1	1	yes	characteristic
209	#126	—	2	8	no	—
210	#125	—	2	2	no	—
211	#125	—	2	1	no	—
212	#124	—	2	4	no	—
213	#124	—	2	4	no	—
214	#124	D₄×C₁₀₇	1	2	yes	—
215	#123	—	2	4	no	—
216	#123	—	2	4	no	—
217	C₂²⋊Q₈	—	2	4	no	—
218	C₂²⋊Q₈	—	2	8	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	D₄×C₁₀₇	1	4	yes	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	2	4	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	C₂³×C₁₀₇	1	4	yes	—
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	—	2	2	no	—
231	#117	—	2	1	no	—
232	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
233	#106	—	2	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	C₂³×C₁₀₇	1	4	yes	—
236	#103	—	2	4	no	—
237	#103	—	2	4	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	C₂³×C₁₀₇	1	2	yes	—
242	#41	—	4	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	8	4	no	—
247	C₂×Q₈	C₂×D₄×C₁₀₇	1	4	yes	—
248	C₂×Q₈	C₂×D₄×C₁₀₇	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂×D₄×C₁₀₇	1	2	yes	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
256	Q₁₆	—	8	4	no	—
257	M₁₆	—	8	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₂×D₄×C₁₀₇	1	4	yes	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	2	8	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
282	C₄²	C₂⁴×C₁₀₇	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	#120×C₁₀₇	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	#120×C₁₀₇	1	2	yes	—
306	C₂×C₄	#120×C₁₀₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₀₇	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×#120×C₁₀₇	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	#455×C₁₀₇	1	2	yes	—
323	C₄	#577×C₁₀₇	1	2	yes	—
324	C₄	#578×C₁₀₇	1	2	yes	—
325	C₂	—	4	1	no	—
326	C₂	—	8	2	no	—
327	C₂	C₁₀₇×#3013	1	2	yes	—
328	C₂	C₂×#455×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₇
prime radical	#532×C₁₀₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₀₇×#3255	C₂	semidirect
4. C₁₀₇×#1645	C₂	semidirect
5. C₁₀₇×#1640	C₂	semidirect
7. C₁₀₇×#1493	C₂	semidirect
160. #18693	C₁₀₇	direct (N × H)
166. #3255	C₂×C₁₀₇	semidirect
168. #1645	C₂×C₁₀₇	semidirect
169. #1640	C₂×C₁₀₇	semidirect
171. #1493	C₂×C₁₀₇	semidirect
172. C₁₀₇	#18693	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₇×#3550	2
C₁₀₇×#3255	2
C₁₀₇×#3131	2
C₁₀₇×#1645	2
C₁₀₇×#1640	2
C₁₀₇×#1560	2
C₁₀₇×#1493	2
#18693	107

Subgroups — C107×#18693 #41044657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#18693 #41044657