Subgroups — C₆₇×#59660 #30738911

Order 17152.

2300 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2728	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2582	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2537	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2536	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2489	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2442	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2427	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2361	C₂	1	4	yes	maximal, complemented
10	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
12	#554×C₆₇	—	2	16	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#125×C₆₇	C₂²	1	4	yes	—
15	C₂×#124×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
25	#400×C₆₇	C₂²	1	8	yes	—
26	#398×C₆₇	C₂²	1	4	yes	—
27	#398×C₆₇	C₂²	1	4	yes	—
28	#397×C₆₇	C₂²	1	8	yes	—
29	#394×C₆₇	C₂²	1	4	yes	—
30	#394×C₆₇	C₂²	1	4	yes	—
31	#391×C₆₇	C₂²	1	2	yes	—
32	#391×C₆₇	C₂²	1	2	yes	—
33	#391×C₆₇	C₂²	1	4	yes	—
34	#390×C₆₇	C₂²	1	2	yes	—
35	#389×C₆₇	C₂²	1	4	yes	—
36	#389×C₆₇	C₂²	1	4	yes	—
37	#388×C₆₇	C₂²	1	4	yes	—
38	#388×C₆₇	C₂²	1	8	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#386×C₆₇	C₂²	1	8	yes	—
41	#385×C₆₇	C₂²	1	8	yes	—
42	#384×C₆₇	C₂²	1	1	yes	characteristic
43	#382×C₆₇	C₂²	1	4	yes	—
44	#382×C₆₇	C₂²	1	2	yes	—
45	#380×C₆₇	C₂²	1	8	yes	—
46	#380×C₆₇	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
48	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
50	C₄³×C₆₇	C₂²	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
57	C₄⋊Q₈×C₆₇	D₄	1	8	yes	complemented
58	#125×C₆₇	—	2	16	no	—
59	#124×C₆₇	—	2	8	no	—
60	C₂²⋊Q₈×C₆₇	—	2	32	no	—
61	C₂²⋊Q₈×C₆₇	—	2	16	no	—
62	#121×C₆₇	—	2	16	no	—
63	C₄×Q₈×C₆₇	—	2	8	no	—
64	C₄×Q₈×C₆₇	—	4	4	no	—
65	C₄×D₄×C₆₇	—	2	16	no	—
66	C₄×D₄×C₆₇	—	2	8	no	—
67	C₄×D₄×C₆₇	—	4	8	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	#95×C₆₇	C₂³	1	4	yes	—
94	#95×C₆₇	C₂³	1	4	yes	—
95	#95×C₆₇	C₂³	1	4	yes	—
96	#95×C₆₇	C₂³	1	4	yes	—
97	#95×C₆₇	C₂³	1	8	yes	—
98	#95×C₆₇	C₂³	1	4	yes	—
99	#95×C₆₇	C₂³	1	8	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
102	C₂×Q₈×C₆₇	—	2	4	no	—
103	C₂×Q₈×C₆₇	—	2	4	no	—
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	—	4	2	no	—
106	C₂×Q₈×C₆₇	—	4	2	no	—
107	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
108	C₂×D₄×C₆₇	—	2	16	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	4	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	#41	1	8	yes	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	8	no	—
115	C₂²×C₄×C₆₇	—	2	8	no	—
116	C₂²×C₄×C₆₇	—	2	16	no	—
117	C₂²×C₄×C₆₇	—	4	8	no	—
118	C₂²×C₄×C₆₇	—	4	8	no	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	16	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	16	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	4	4	no	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
139	C₂²⋊C₄×C₆₇	—	2	16	no	—
140	C₂²⋊C₄×C₆₇	—	2	8	no	—
141	C₂²⋊C₄×C₆₇	—	2	8	no	—
142	C₂²⋊C₄×C₆₇	—	2	16	no	—
143	C₂²⋊C₄×C₆₇	—	2	8	no	—
144	C₂²⋊C₄×C₆₇	—	2	8	no	—
145	C₂²⋊C₄×C₆₇	—	4	4	no	—
146	C₄²×C₆₇	—	2	8	no	—
147	C₄²×C₆₇	—	2	4	no	—
148	C₄²×C₆₇	—	2	8	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	4	no	—
151	C₄²×C₆₇	—	4	1	no	—
152	C₄²×C₆₇	C₂×D₄	1	4	yes	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	4	no	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	C₂×#41	1	8	yes	—
160	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
161	Q₈×C₆₇	—	4	8	no	—
162	Q₈×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	2	16	no	—
164	D₄×C₆₇	—	4	16	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	8	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	4	16	no	—
187	C₂×C₄×C₆₇	—	8	16	no	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
191	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
192	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
193	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
194	C₂²×C₆₇	—	2	8	no	—
195	C₂²×C₆₇	—	2	8	no	—
196	C₂²×C₆₇	—	2	8	no	—
197	C₂²×C₆₇	—	4	4	no	—
198	C₂²×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
202	C₂²×C₆₇	C₂²×#41	1	2	yes	—
203	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
205	C₂²×C₆₇	#585	1	1	yes	characteristic
206	C₄×C₆₇	—	2	8	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	2	no	—
209	C₄×C₆₇	—	4	4	no	—
210	C₄×C₆₇	—	4	4	no	—
211	C₄×C₆₇	—	4	1	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	—	4	2	no	—
215	C₄×C₆₇	#548	1	8	yes	—
216	#59660	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	4	8	no	—
218	C₂×C₆₇	C₂×#548	1	2	yes	—
219	C₂×C₆₇	#3481	1	1	yes	characteristic
220	C₂×C₆₇	#3496	1	1	yes	characteristic
221	C₂×C₆₇	#3515	1	2	yes	—
222	C₂×C₆₇	#3544	1	1	yes	characteristic
223	C₂×#554	C₂×C₆₇	1	2	yes	complemented
224	#2728	C₂×C₆₇	1	4	yes	complemented
225	#2582	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#2537	C₂×C₆₇	1	1	yes	characteristic
227	#2536	C₂×C₆₇	1	4	yes	complemented
228	#2489	C₂×C₆₇	1	4	yes	complemented
229	#2442	C₂×C₆₇	1	4	yes	complemented
230	#2427	C₂×C₆₇	1	4	yes	complemented
231	#2361	C₂×C₆₇	1	4	yes	complemented
232	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₄×#124	C₂×C₆₇	1	2	yes	complemented
234	C₆₇	#59660	1	1	yes	characteristic, direct product factor, Sylow
235	#554	—	2	16	no	—
236	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#125	C₂²×C₆₇	1	4	yes	—
238	C₂×#124	C₂²×C₆₇	1	2	yes	—
239	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
240	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	—
241	C₂×#121	C₂²×C₆₇	1	4	yes	—
242	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
243	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
244	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
245	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
247	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
248	#400	C₂²×C₆₇	1	8	yes	—
249	#398	C₂²×C₆₇	1	4	yes	—
250	#398	C₂²×C₆₇	1	4	yes	—
251	#397	C₂²×C₆₇	1	8	yes	—
252	#394	C₂²×C₆₇	1	4	yes	—
253	#394	C₂²×C₆₇	1	4	yes	—
254	#391	C₂²×C₆₇	1	2	yes	—
255	#391	C₂²×C₆₇	1	4	yes	—
256	#391	C₂²×C₆₇	1	2	yes	—
257	#390	C₂²×C₆₇	1	2	yes	—
258	#389	C₂²×C₆₇	1	4	yes	—
259	#389	C₂²×C₆₇	1	4	yes	—
260	#388	C₂²×C₆₇	1	8	yes	—
261	#388	C₂²×C₆₇	1	4	yes	—
262	#388	C₂²×C₆₇	1	4	yes	—
263	#386	C₂²×C₆₇	1	8	yes	—
264	#385	C₂²×C₆₇	1	8	yes	—
265	#384	C₂²×C₆₇	1	1	yes	characteristic
266	#382	C₂²×C₆₇	1	2	yes	—
267	#382	C₂²×C₆₇	1	4	yes	—
268	#380	C₂²×C₆₇	1	8	yes	—
269	#380	C₂²×C₆₇	1	4	yes	—
270	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
271	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
272	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
273	C₄³	C₂²×C₆₇	1	1	yes	characteristic
274	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
275	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
276	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
277	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
278	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
279	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
280	C₄⋊Q₈	D₄×C₆₇	1	8	yes	complemented
281	#125	—	2	16	no	—
282	#124	—	2	8	no	—
283	C₂²⋊Q₈	—	2	32	no	—
284	C₂²⋊Q₈	—	2	16	no	—
285	#121	—	2	16	no	—
286	C₄×Q₈	—	2	8	no	—
287	C₄×Q₈	—	4	4	no	—
288	C₄×D₄	—	2	16	no	—
289	C₄×D₄	—	2	8	no	—
290	C₄×D₄	—	4	8	no	—
291	C₂×C₄⋊C₄	—	2	16	no	—
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
309	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
310	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
313	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
314	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
316	#95	C₂³×C₆₇	1	4	yes	—
317	#95	C₂³×C₆₇	1	4	yes	—
318	#95	C₂³×C₆₇	1	4	yes	—
319	#95	C₂³×C₆₇	1	4	yes	—
320	#95	C₂³×C₆₇	1	8	yes	—
321	#95	C₂³×C₆₇	1	4	yes	—
322	#95	C₂³×C₆₇	1	8	yes	—
323	#95	C₂³×C₆₇	1	4	yes	—
324	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
325	C₂×Q₈	—	2	4	no	—
326	C₂×Q₈	—	2	4	no	—
327	C₂×Q₈	—	2	4	no	—
328	C₂×Q₈	—	4	2	no	—
329	C₂×Q₈	—	4	2	no	—
330	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
331	C₂×D₄	—	2	16	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	4	4	no	—
335	C₂×D₄	#41×C₆₇	1	8	yes	—
336	C₂²×C₄	—	2	8	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	16	no	—
340	C₂²×C₄	—	4	8	no	—
341	C₂²×C₄	—	4	8	no	—
342	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
343	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
344	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
345	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	16	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	2	16	no	—
357	C₄⋊C₄	—	2	8	no	—
358	C₄⋊C₄	—	4	4	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	4	no	—
361	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
362	C₂²⋊C₄	—	2	16	no	—
363	C₂²⋊C₄	—	2	8	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	2	16	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	4	4	no	—
369	C₄²	—	2	8	no	—
370	C₄²	—	2	4	no	—
371	C₄²	—	2	8	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	4	no	—
374	C₄²	—	4	1	no	—
375	C₄²	C₂×D₄×C₆₇	1	4	yes	—
376	C₂³	—	2	4	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	4	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	4	no	—
382	C₂³	C₂×#41×C₆₇	1	8	yes	—
383	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
384	Q₈	—	4	8	no	—
385	Q₈	—	4	8	no	—
386	D₄	—	2	16	no	—
387	D₄	—	4	16	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	4	16	no	—
410	C₂×C₄	—	8	16	no	—
411	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
413	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
414	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
415	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
416	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
417	C₂²	—	2	8	no	—
418	C₂²	—	2	8	no	—
419	C₂²	—	2	8	no	—
420	C₂²	—	4	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
425	C₂²	C₂²×#41×C₆₇	1	2	yes	—
426	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
427	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
428	C₂²	#585×C₆₇	1	1	yes	characteristic
429	C₄	—	2	2	no	—
430	C₄	—	2	8	no	—
431	C₄	—	2	2	no	—
432	C₄	—	4	4	no	—
433	C₄	—	4	4	no	—
434	C₄	—	4	4	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	2	no	—
438	C₄	#548×C₆₇	1	8	yes	—
439	C₂	—	4	8	no	—
440	C₂	C₂×#548×C₆₇	1	2	yes	—
441	C₂	C₆₇×#3481	1	1	yes	characteristic
442	C₂	C₆₇×#3496	1	1	yes	characteristic
443	C₂	C₆₇×#3515	1	2	yes	—
444	C₂	C₆₇×#3544	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₆₇×#2537	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
2. C₆₇×#2728	C₂	semidirect
3. C₆₇×#2582	C₂	semidirect
5. C₆₇×#2536	C₂	semidirect
6. C₆₇×#2489	C₂	semidirect
7. C₆₇×#2442	C₂	semidirect
8. C₆₇×#2427	C₂	semidirect
9. C₆₇×#2361	C₂	semidirect
10. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
11. C₄×#124×C₆₇	C₂	semidirect
57. C₄⋊Q₈×C₆₇	D₄	semidirect
216. #59660	C₆₇	direct (N × H)
223. C₂×#554	C₂×C₆₇	semidirect
224. #2728	C₂×C₆₇	semidirect
225. #2582	C₂×C₆₇	semidirect
227. #2536	C₂×C₆₇	semidirect
228. #2489	C₂×C₆₇	semidirect
229. #2442	C₂×C₆₇	semidirect
230. #2427	C₂×C₆₇	semidirect
231. #2361	C₂×C₆₇	semidirect
232. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
233. C₄×#124	C₂×C₆₇	semidirect
234. C₆₇	#59660	direct (N × H)
280. C₄⋊Q₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₆₇×#2728	2
C₆₇×#2582	2
C₆₇×#2537	2
C₆₇×#2536	2
C₆₇×#2489	2
C₆₇×#2442	2
C₆₇×#2427	2
C₆₇×#2361	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#124×C₆₇	2
#59660	67

Subgroups — C67×#59660 #30738911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59660 #30738911