Subgroups — C₆₇×#33030 #30712281

Order 17152.

1098 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1926	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1879	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	—	2	2	no	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	—	2	1	no	—
17	#476×C₆₇	—	2	4	no	—
18	#459×C₆₇	—	2	4	no	—
19	C₂×#106×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	4	yes	—
25	#390×C₆₇	—	2	1	no	—
26	#390×C₆₇	—	2	1	no	—
27	#390×C₆₇	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
31	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
39	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	2	no	—
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂×D₈×C₆₇	—	2	4	no	—
43	C₂×D₈×C₆₇	—	2	8	no	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	#127×C₆₇	—	2	2	no	—
49	#127×C₆₇	—	2	2	no	—
50	#124×C₆₇	—	4	8	no	—
51	C₄×D₄×C₆₇	—	2	4	no	—
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	D₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	—	4	4	no	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	2	yes	—
68	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#106×C₆₇	—	2	4	no	—
72	C₄⋊C₈×C₆₇	—	2	2	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	4	no	—
75	#102×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	8	no	—
77	#95×C₆₇	—	2	2	no	—
78	#95×C₆₇	—	2	1	no	—
79	C₂⁴×C₆₇	—	2	1	no	—
80	C₂⁴×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	4	4	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	2	no	—
93	C₂×D₄×C₆₇	—	4	2	no	—
94	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×D₄×C₆₇	#41	1	4	yes	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	4	4	no	—
102	C₂²×C₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	4	4	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	D₈×C₆₇	—	4	8	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₄⋊C₄×C₆₇	#41	1	4	yes	—
122	C₂²⋊C₄×C₆₇	—	2	8	no	—
123	C₂²⋊C₄×C₆₇	—	2	4	no	—
124	C₂²⋊C₄×C₆₇	—	4	8	no	—
125	C₂²⋊C₄×C₆₇	—	4	8	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	4	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	4	4	no	—
132	C₄²×C₆₇	—	4	2	no	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₄²×C₆₇	#41	1	2	yes	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	4	no	—
137	C₂³×C₆₇	—	4	1	no	—
138	C₂³×C₆₇	—	4	1	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	4	no	—
142	C₂³×C₆₇	—	4	2	no	—
143	C₂³×C₆₇	—	4	4	no	—
144	C₂³×C₆₇	—	4	2	no	—
145	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₆₇	—	8	4	no	—
147	D₄×C₆₇	—	2	4	no	—
148	D₄×C₆₇	—	2	2	no	—
149	D₄×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	4	4	no	—
151	D₄×C₆₇	—	4	2	no	—
152	D₄×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	8	4	no	—
154	D₄×C₆₇	#124	1	4	yes	complemented
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	4	16	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	#121	1	2	yes	—
172	C₂×C₄×C₆₇	#121	1	2	yes	—
173	C₂×C₄×C₆₇	#135	1	2	yes	—
174	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
175	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
180	C₈×C₆₇	—	4	2	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	8	2	no	—
188	C₂²×C₆₇	—	8	4	no	—
189	C₂²×C₆₇	—	8	4	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
193	C₂²×C₆₇	#525	1	2	yes	—
194	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
195	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
196	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	4	2	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	8	2	no	—
203	C₄×C₆₇	#463	1	4	yes	—
204	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
205	C₄×C₆₇	#529	1	2	yes	—
206	C₄×C₆₇	#532	1	1	yes	characteristic
207	#33030	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₆₇	—	4	4	no	—
209	C₂×C₆₇	—	16	2	no	—
210	C₂×C₆₇	#2850	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
212	C₂×C₆₇	#3038	1	1	yes	characteristic
213	C₂×C₆₇	#3106	1	2	yes	—
214	C₂×C₆₇	#3109	1	2	yes	—
215	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
217	#2858	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1926	C₂×C₆₇	1	4	yes	complemented
222	#1879	C₂×C₆₇	1	4	yes	complemented
223	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
224	C₆₇	#33030	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
226	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×#124	—	2	2	no	—
229	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
230	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂²×C₄²	—	2	1	no	—
232	#476	—	2	4	no	—
233	#459	—	2	4	no	—
234	C₂×#106	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#102	C₂²×C₆₇	1	2	yes	—
238	C₂×#102	C₂²×C₆₇	1	4	yes	—
239	C₂×#102	C₂²×C₆₇	1	2	yes	—
240	#390	—	2	1	no	—
241	#390	—	2	1	no	—
242	#390	C₂²×C₆₇	1	4	yes	—
243	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
244	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
245	C₄×C₂²⋊C₄	—	2	2	no	—
246	C₄×C₂²⋊C₄	—	2	1	no	—
247	C₄³	C₂²×C₆₇	1	1	yes	characteristic
248	#340	C₂²×C₆₇	1	4	yes	—
249	#336	C₂²×C₆₇	1	4	yes	—
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
254	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
255	C₂³×C₄	—	2	2	no	—
256	C₂³×C₄	—	2	1	no	—
257	C₂×D₈	—	2	4	no	—
258	C₂×D₈	—	2	8	no	—
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	—	2	2	no	—
263	#127	—	2	2	no	—
264	#127	—	2	2	no	—
265	#124	—	4	8	no	—
266	C₄×D₄	—	2	4	no	—
267	C₄×D₄	—	2	4	no	—
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	D₄×C₆₇	1	4	yes	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	1	no	—
276	C₂×C₂²⋊C₄	—	2	4	no	—
277	C₂×C₂²⋊C₄	—	2	4	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	—	4	4	no	—
281	C₂×C₄²	D₄×C₆₇	1	2	yes	—
282	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
286	#106	—	2	4	no	—
287	C₄⋊C₈	—	2	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	#102	—	2	4	no	—
290	#102	—	2	4	no	—
291	#102	—	2	8	no	—
292	#95	—	2	2	no	—
293	#95	—	2	1	no	—
294	C₂⁴	—	2	1	no	—
295	C₂⁴	—	2	2	no	—
296	C₂×Q₈	—	4	4	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	1	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	4	4	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂×D₄	#41×C₆₇	1	4	yes	—
311	C₂²×C₄	—	2	8	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	4	no	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
325	D₈	—	4	8	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	2	4	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₄⋊C₄	#41×C₆₇	1	4	yes	—
337	C₂²⋊C₄	—	2	8	no	—
338	C₂²⋊C₄	—	2	4	no	—
339	C₂²⋊C₄	—	4	8	no	—
340	C₂²⋊C₄	—	4	8	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	4	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	4	4	no	—
347	C₄²	—	4	2	no	—
348	C₄²	C₂×D₄×C₆₇	1	2	yes	—
349	C₄²	#41×C₆₇	1	2	yes	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	4	no	—
352	C₂³	—	4	1	no	—
353	C₂³	—	4	1	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	4	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	4	no	—
359	C₂³	—	4	2	no	—
360	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
361	Q₈	—	8	4	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	4	no	—
368	D₄	—	8	4	no	—
369	D₄	#124×C₆₇	1	4	yes	complemented
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	16	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	8	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	#121×C₆₇	1	2	yes	—
387	C₂×C₄	#121×C₆₇	1	2	yes	—
388	C₂×C₄	#135×C₆₇	1	2	yes	—
389	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
390	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
393	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
395	C₈	—	4	2	no	—
396	C₈	—	4	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	4	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	4	no	—
406	C₂²	—	8	4	no	—
407	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
408	C₂²	#525×C₆₇	1	2	yes	—
409	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
410	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
411	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
412	C₄	—	2	2	no	—
413	C₄	—	2	4	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	4	no	—
417	C₄	—	8	2	no	—
418	C₄	#463×C₆₇	1	4	yes	—
419	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
420	C₄	#529×C₆₇	1	2	yes	—
421	C₄	#532×C₆₇	1	1	yes	characteristic
422	C₂	—	4	4	no	—
423	C₂	—	16	2	no	—
424	C₂	C₆₇×#2850	1	1	yes	characteristic
425	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
426	C₂	C₆₇×#3038	1	1	yes	characteristic
427	C₂	C₆₇×#3106	1	2	yes	—
428	C₂	C₆₇×#3109	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#476×C₆₇	C₂	semidirect
3. C₆₇×#2858	C₂	semidirect
4. C₄²×D₄×C₆₇	C₂	semidirect
5. C₆₇×#1987	C₂	semidirect
6. C₆₇×#1982	C₂	semidirect
7. C₆₇×#1926	C₂	semidirect
8. C₆₇×#1879	C₂	semidirect
9. C₆₇×#1831	C₂	semidirect
154. D₄×C₆₇	#124	semidirect
207. #33030	C₆₇	direct (N × H)
215. C₂×#476	C₂×C₆₇	semidirect
217. #2858	C₂×C₆₇	semidirect
218. C₄²×D₄	C₂×C₆₇	semidirect
219. #1987	C₂×C₆₇	semidirect
220. #1982	C₂×C₆₇	semidirect
221. #1926	C₂×C₆₇	semidirect
222. #1879	C₂×C₆₇	semidirect
223. #1831	C₂×C₆₇	semidirect
224. C₆₇	#33030	direct (N × H)
369. D₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#476×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#2858	2
C₄²×D₄×C₆₇	2
C₆₇×#1987	2
C₆₇×#1982	2
C₆₇×#1926	2
C₆₇×#1879	2
C₆₇×#1831	2
#33030	67

Subgroups — C67×#33030 #30712281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33030 #30712281