Subgroups — C₆₇×#20504 #30699755

Order 17152.

1130 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#440×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2275	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1776	C₂	1	2	yes	maximal, complemented
6	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1763	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic
9	C₄×#103×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₄×#102×C₆₇	C₂	1	2	yes	maximal
11	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	#440×C₆₇	C₄	1	16	yes	complemented
18	C₄×SD₁₆×C₆₇	C₄	1	8	yes	complemented
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
37	#376×C₆₇	—	2	1	no	—
38	#376×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#95×C₆₇	—	2	1	no	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
55	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₄×D₄×C₆₇	—	2	4	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	C₂×C₄	1	8	yes	complemented
77	#106×C₆₇	C₂×C₄	1	4	yes	complemented
78	#103×C₆₇	C₂×C₄	1	4	yes	complemented
79	#103×C₆₇	C₂×C₄	1	8	yes	complemented
80	#102×C₆₇	C₂×C₄	1	4	yes	—
81	#102×C₆₇	C₂×C₄	1	8	yes	—
82	#97×C₆₇	—	2	2	no	—
83	#97×C₆₇	C₂×C₄	1	8	yes	complemented
84	C₄×C₈×C₆₇	—	2	4	no	—
85	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	1	no	—
90	#95×C₆₇	D₄	1	1	yes	characteristic
91	#95×C₆₇	D₄	1	1	yes	characteristic
92	#95×C₆₇	D₄	1	2	yes	—
93	C₂⁴×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	8	2	no	—
110	C₂²×C₄×C₆₇	—	8	2	no	—
111	C₂²×C₄×C₆₇	—	8	2	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₆₇	C₄²	1	16	yes	complemented
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
125	C₄⋊C₄×C₆₇	—	2	4	no	—
126	C₄⋊C₄×C₆₇	—	2	8	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
133	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	8	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₄²×C₆₇	—	2	8	no	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	8	1	no	—
140	C₄²×C₆₇	—	8	1	no	—
141	C₄²×C₆₇	—	8	2	no	—
142	C₄²×C₆₇	—	8	2	no	—
143	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
144	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
145	C₄²×C₆₇	#41	1	4	yes	—
146	C₄²×C₆₇	#41	1	2	yes	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	4	1	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₆₇	—	2	2	no	—
154	Q₈×C₆₇	—	2	4	no	—
155	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	C₂×C₄²	1	4	yes	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	8	8	no	—
180	C₂×C₄×C₆₇	—	8	4	no	—
181	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
182	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
185	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
186	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
188	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
189	C₈×C₆₇	—	2	2	no	—
190	C₈×C₆₇	—	2	4	no	—
191	C₈×C₆₇	C₂×C₄²	1	4	yes	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
198	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂²×C₆₇	#576	1	1	yes	characteristic
200	C₂²×C₆₇	#577	1	2	yes	—
201	C₂²×C₆₇	#578	1	1	yes	characteristic
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	4	no	—
207	C₄×C₆₇	—	4	4	no	—
208	C₄×C₆₇	—	4	4	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
212	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
213	C₄×C₆₇	C₄×#41	1	2	yes	—
214	#20504	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₆₇	—	4	4	no	—
216	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
217	C₂×C₆₇	#2946	1	2	yes	—
218	C₂×C₆₇	#2947	1	2	yes	—
219	C₂×C₆₇	#2949	1	1	yes	characteristic
220	C₂×C₆₇	#2951	1	1	yes	characteristic
221	C₂×#440	C₂×C₆₇	1	2	yes	—
222	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
223	#2276	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#2275	C₂×C₆₇	1	1	yes	characteristic
225	#1776	C₂×C₆₇	1	2	yes	complemented
226	C₄×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1763	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1762	C₂×C₆₇	1	1	yes	characteristic
229	C₄×#103	C₂×C₆₇	1	2	yes	complemented
230	C₄×#102	C₂×C₆₇	1	2	yes	—
231	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₆₇	#20504	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
236	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
238	#440	C₄×C₆₇	1	16	yes	complemented
239	C₄×SD₁₆	C₄×C₆₇	1	8	yes	complemented
240	C₂×#107	C₂²×C₆₇	1	2	yes	—
241	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#103	C₂²×C₆₇	1	2	yes	—
244	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#97	C₂²×C₆₇	1	2	yes	—
248	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
249	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₄×C₄⋊C₄	—	2	1	no	—
251	C₄×C₄⋊C₄	—	2	2	no	—
252	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
254	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
256	C₄×C₂²⋊C₄	—	2	1	no	—
257	C₄×C₂²⋊C₄	—	2	2	no	—
258	#376	—	2	1	no	—
259	#376	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#95	—	2	1	no	—
261	#340	C₂²×C₆₇	1	2	yes	—
262	#340	C₂²×C₆₇	1	2	yes	—
263	#340	C₂²×C₆₇	1	2	yes	—
264	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂³×C₄	—	2	1	no	—
267	C₂³×C₄	—	2	2	no	—
268	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
269	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄×Q₈	—	2	4	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
276	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
277	C₄×D₄	—	2	4	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
280	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	—	2	2	no	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
297	#107	C₂×C₄×C₆₇	1	8	yes	complemented
298	#106	C₂×C₄×C₆₇	1	4	yes	complemented
299	#103	C₂×C₄×C₆₇	1	4	yes	complemented
300	#103	C₂×C₄×C₆₇	1	8	yes	complemented
301	#102	C₂×C₄×C₆₇	1	4	yes	—
302	#102	C₂×C₄×C₆₇	1	8	yes	—
303	#97	—	2	2	no	—
304	#97	C₂×C₄×C₆₇	1	8	yes	complemented
305	C₄×C₈	—	2	4	no	—
306	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
307	#95	—	2	1	no	—
308	#95	—	2	2	no	—
309	#95	—	2	1	no	—
310	#95	—	2	2	no	—
311	#95	D₄×C₆₇	1	1	yes	characteristic
312	#95	D₄×C₆₇	1	1	yes	characteristic
313	#95	D₄×C₆₇	1	2	yes	—
314	C₂⁴	—	2	1	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
318	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
322	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
338	SD₁₆	C₄²×C₆₇	1	16	yes	complemented
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
343	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
344	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
345	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	8	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
353	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
354	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
355	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₄²	—	2	8	no	—
359	C₄²	—	2	4	no	—
360	C₄²	—	8	1	no	—
361	C₄²	—	8	1	no	—
362	C₄²	—	8	2	no	—
363	C₄²	—	8	2	no	—
364	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
365	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
366	C₄²	#41×C₆₇	1	4	yes	—
367	C₄²	#41×C₆₇	1	2	yes	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	2	4	no	—
376	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
377	D₄	—	2	2	no	—
378	D₄	—	2	4	no	—
379	D₄	C₂×C₄²×C₆₇	1	4	yes	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	8	8	no	—
401	C₂×C₄	—	8	4	no	—
402	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
403	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
404	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
405	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
406	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
407	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
409	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
410	C₈	—	2	2	no	—
411	C₈	—	2	4	no	—
412	C₈	C₂×C₄²×C₆₇	1	4	yes	—
413	C₂²	—	2	4	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
419	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
420	C₂²	#576×C₆₇	1	1	yes	characteristic
421	C₂²	#577×C₆₇	1	2	yes	—
422	C₂²	#578×C₆₇	1	1	yes	characteristic
423	C₄	—	2	4	no	—
424	C₄	—	2	4	no	—
425	C₄	—	2	4	no	—
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	4	4	no	—
429	C₄	—	4	4	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
433	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
434	C₄	C₄×#41×C₆₇	1	2	yes	—
435	C₂	—	4	4	no	—
436	C₂	C₄²×D₄×C₆₇	1	1	yes	characteristic
437	C₂	C₆₇×#2946	1	2	yes	—
438	C₂	C₆₇×#2947	1	2	yes	—
439	C₂	C₆₇×#2949	1	1	yes	characteristic
440	C₂	C₆₇×#2951	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2276	C₂	semidirect
5. C₆₇×#1776	C₂	semidirect
6. C₄×#107×C₆₇	C₂	semidirect
7. C₆₇×#1763	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
11. C₆₇×#1726	C₂	semidirect
17. #440×C₆₇	C₄	semidirect
18. C₄×SD₁₆×C₆₇	C₄	semidirect
54. C₄×Q₈×C₆₇	C₂×C₄	semidirect
55. C₄×Q₈×C₆₇	C₂×C₄	semidirect
76. #107×C₆₇	C₂×C₄	semidirect
77. #106×C₆₇	C₂×C₄	semidirect
78. #103×C₆₇	C₂×C₄	semidirect
79. #103×C₆₇	C₂×C₄	semidirect
83. #97×C₆₇	C₂×C₄	semidirect
85. C₄×C₈×C₆₇	C₂×C₄	semidirect
117. SD₁₆×C₆₇	C₄²	semidirect
214. #20504	C₆₇	direct (N × H)
223. #2276	C₂×C₆₇	semidirect
225. #1776	C₂×C₆₇	semidirect
226. C₄×#107	C₂×C₆₇	semidirect
227. #1763	C₂×C₆₇	semidirect
229. C₄×#103	C₂×C₆₇	semidirect
231. #1726	C₂×C₆₇	semidirect
232. C₆₇	#20504	direct (N × H)
238. #440	C₄×C₆₇	semidirect
239. C₄×SD₁₆	C₄×C₆₇	semidirect
275. C₄×Q₈	C₂×C₄×C₆₇	semidirect
276. C₄×Q₈	C₂×C₄×C₆₇	semidirect
297. #107	C₂×C₄×C₆₇	semidirect
298. #106	C₂×C₄×C₆₇	semidirect
299. #103	C₂×C₄×C₆₇	semidirect
300. #103	C₂×C₄×C₆₇	semidirect
304. #97	C₂×C₄×C₆₇	semidirect
306. C₄×C₈	C₂×C₄×C₆₇	semidirect
338. SD₁₆	C₄²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#440×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#2276	2
C₆₇×#2275	2
C₆₇×#1776	2
C₄×#107×C₆₇	2
C₆₇×#1763	2
C₆₇×#1762	2
C₄×#103×C₆₇	2
C₄×#102×C₆₇	2
C₆₇×#1726	2
#20504	67

Subgroups — C67×#20504 #30699755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20504 #30699755