Subgroups — C₆₇×#20397 #30699648

Order 17152.

1518 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	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#413×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2873	C₂	1	2	yes	maximal, complemented
4	C₄×#101×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₄×#100×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#343×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
9	#520×C₆₇	—	2	8	no	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₆₇	C₄	1	4	yes	—
12	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂×#117×C₆₇	C₄	1	2	yes	—
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#413×C₆₇	C₄	1	16	yes	complemented
18	C₂×#101×C₆₇	C₂²	1	4	yes	—
19	C₂×#100×C₆₇	C₂²	1	4	yes	—
20	#405×C₆₇	—	2	8	no	—
21	C₄×M₁₆×C₆₇	C₂²	1	8	yes	—
22	#343×C₆₇	C₂²	1	8	yes	—
23	C₂×#41×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	—	2	4	no	—
25	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
26	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	4	no	—
34	C₂×M₁₆×C₆₇	—	2	8	no	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
36	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₄×Q₈×C₆₇	—	2	8	no	—
39	C₄×D₄×C₆₇	—	2	16	no	—
40	C₄×D₄×C₆₇	—	2	8	no	—
41	#117×C₆₇	—	2	8	no	—
42	#117×C₆₇	D₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	#101×C₆₇	C₂×C₄	1	16	yes	complemented
55	#100×C₆₇	C₂×C₄	1	16	yes	complemented
56	#97×C₆₇	—	2	8	no	—
57	C₄×C₈×C₆₇	—	2	8	no	—
58	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	#41×C₆₇	—	2	16	no	—
61	#41×C₆₇	—	2	16	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂×D₄×C₆₇	—	2	4	no	—
66	C₂×D₄×C₆₇	—	2	4	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	C₄²	1	8	yes	—
70	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	8	no	—
75	C₂²×C₄×C₆₇	—	8	4	no	—
76	C₂²×C₄×C₆₇	—	8	2	no	—
77	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₆₇	#41	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	16	no	—
88	M₁₆×C₆₇	—	2	8	no	—
89	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
90	C₂×C₈×C₆₇	—	2	16	no	—
91	C₂×C₈×C₆₇	—	2	8	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	—	2	8	no	—
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
96	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
97	C₂²⋊C₄×C₆₇	—	2	8	no	—
98	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
99	C₄²×C₆₇	—	2	8	no	—
100	C₄²×C₆₇	C₂×D₄	1	8	yes	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	4	1	no	—
104	C₂³×C₆₇	—	4	1	no	—
105	C₂³×C₆₇	—	4	2	no	—
106	C₂³×C₆₇	—	4	2	no	—
107	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
108	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
109	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₆₇	#117	1	2	yes	—
111	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	Q₈×C₆₇	—	2	8	no	—
113	D₄×C₆₇	—	2	16	no	—
114	D₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	16	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	—	8	8	no	—
123	C₂×C₄×C₆₇	—	8	4	no	—
124	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
125	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	#117	1	2	yes	—
130	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
131	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
132	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
133	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
135	C₈×C₆₇	—	2	16	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	—	4	4	no	—
144	C₂²×C₆₇	—	4	4	no	—
145	C₂²×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
148	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
151	C₄×C₆₇	—	2	4	no	—
152	C₄×C₆₇	—	2	4	no	—
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	4	4	no	—
155	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
156	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
157	#20397	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	—	2	1	no	—
160	C₂×C₆₇	—	2	4	no	—
161	C₂×C₆₇	—	4	4	no	—
162	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₆₇	#2888	1	1	yes	characteristic
164	C₂×C₆₇	#2889	1	1	yes	characteristic
165	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
166	C₂×#413	C₂×C₆₇	1	2	yes	—
167	#2873	C₂×C₆₇	1	2	yes	complemented
168	C₄×#101	C₂×C₆₇	1	4	yes	complemented
169	C₄×#100	C₂×C₆₇	1	4	yes	complemented
170	C₂×#343	C₂×C₆₇	1	2	yes	complemented
171	C₆₇	#20397	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
173	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
174	#520	—	2	8	no	—
175	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
176	C₂×C₄×D₄	C₄×C₆₇	1	4	yes	—
177	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
178	C₂×#117	C₄×C₆₇	1	2	yes	—
179	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
182	#413	C₄×C₆₇	1	16	yes	complemented
183	C₂×#101	C₂²×C₆₇	1	4	yes	—
184	C₂×#100	C₂²×C₆₇	1	4	yes	—
185	#405	—	2	8	no	—
186	C₄×M₁₆	C₂²×C₆₇	1	8	yes	—
187	#343	C₂²×C₆₇	1	8	yes	—
188	C₂×#41	—	2	4	no	—
189	C₂×#41	—	2	4	no	—
190	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
191	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
192	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
193	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
194	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
195	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
196	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
197	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	8	no	—
200	C₂×M₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
201	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
202	C₂²×C₈	—	2	4	no	—
203	C₄×Q₈	—	2	8	no	—
204	C₄×D₄	—	2	16	no	—
205	C₄×D₄	—	2	8	no	—
206	#117	—	2	8	no	—
207	#117	D₄×C₆₇	1	8	yes	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	8	no	—
210	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
212	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
215	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
216	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
217	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
218	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
219	#101	C₂×C₄×C₆₇	1	16	yes	complemented
220	#100	C₂×C₄×C₆₇	1	16	yes	complemented
221	#97	—	2	8	no	—
222	C₄×C₈	—	2	8	no	—
223	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
224	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	#41	—	2	16	no	—
226	#41	—	2	16	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₄²×C₆₇	1	8	yes	—
235	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	8	4	no	—
241	C₂²×C₄	—	8	2	no	—
242	C₂²×C₄	C₄²×C₆₇	1	8	yes	—
243	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
245	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
249	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
250	C₂²×C₄	#41×C₆₇	1	4	yes	—
251	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
252	M₁₆	—	2	16	no	—
253	M₁₆	—	2	8	no	—
254	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
255	C₂×C₈	—	2	16	no	—
256	C₂×C₈	—	2	8	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
261	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
264	C₄²	—	2	8	no	—
265	C₄²	C₂×D₄×C₆₇	1	8	yes	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₄²×C₆₇	1	4	yes	—
273	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
274	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
275	C₂³	#117×C₆₇	1	2	yes	—
276	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	D₄	—	2	16	no	—
279	D₄	—	2	8	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	16	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	8	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
290	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
292	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
294	C₂×C₄	#117×C₆₇	1	2	yes	—
295	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
296	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
297	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
298	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
299	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
300	C₈	—	2	16	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
313	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
314	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
315	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
316	C₄	—	2	4	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	4	4	no	—
320	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
321	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	4	no	—
325	C₂	—	4	4	no	—
326	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
327	C₂	C₆₇×#2888	1	1	yes	characteristic
328	C₂	C₆₇×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2873	C₂	semidirect
4. C₄×#101×C₆₇	C₂	semidirect
5. C₄×#100×C₆₇	C₂	semidirect
6. C₂×#343×C₆₇	C₂	semidirect
17. #413×C₆₇	C₄	semidirect
35. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
54. #101×C₆₇	C₂×C₄	semidirect
55. #100×C₆₇	C₂×C₄	semidirect
157. #20397	C₆₇	direct (N × H)
167. #2873	C₂×C₆₇	semidirect
168. C₄×#101	C₂×C₆₇	semidirect
169. C₄×#100	C₂×C₆₇	semidirect
170. C₂×#343	C₂×C₆₇	semidirect
171. C₆₇	#20397	direct (N × H)
182. #413	C₄×C₆₇	semidirect
200. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
219. #101	C₂×C₄×C₆₇	semidirect
220. #100	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#520×C₆₇	2
C₂×#413×C₆₇	2
C₆₇×#2873	2
C₄×#101×C₆₇	2
C₄×#100×C₆₇	2
C₂×#343×C₆₇	2
#20397	67

Subgroups — C67×#20397 #30699648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20397 #30699648