Subgroups — C₆₇×#25934 #30705185

Order 17152.

1078 conjugacy classes of subgroups.

The subgroup lattice has 418 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	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2026	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1877	C₂	1	4	yes	maximal
9	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	#554×C₆₇	—	2	4	no	—
12	#554×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
22	#487×C₆₇	—	2	4	no	—
23	#461×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	—	2	2	no	—
30	#391×C₆₇	—	2	1	no	—
31	#391×C₆₇	C₂²	1	4	yes	—
32	#382×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	4	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₆₇	—	2	1	no	—
38	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	4	no	—
44	C₂×SD₁₆×C₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
51	#125×C₆₇	—	2	4	no	—
52	#125×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	16	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	4	no	—
58	C₄×D₄×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	D₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	8	no	—
80	#102×C₆₇	—	2	4	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	4	no	—
84	#95×C₆₇	—	2	4	no	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	4	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	—	8	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×D₄×C₆₇	#41	1	4	yes	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	8	4	no	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	SD₁₆×C₆₇	—	4	8	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	4	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	2	8	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₄⋊C₄×C₆₇	—	2	4	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	—	2	4	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	4	2	no	—
137	C₄²×C₆₇	C₂×D₄	1	2	yes	—
138	C₄²×C₆₇	C₂×D₄	1	2	yes	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	1	no	—
143	C₂³×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₆₇	—	4	4	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	4	no	—
149	Q₈×C₆₇	—	8	4	no	—
150	D₄×C₆₇	—	2	2	no	—
151	D₄×C₆₇	—	2	4	no	—
152	D₄×C₆₇	#124	1	4	yes	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	4	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	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	4	8	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	4	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	8	8	no	—
171	C₂×C₄×C₆₇	#120	1	2	yes	—
172	C₂×C₄×C₆₇	#120	1	2	yes	—
173	C₂×C₄×C₆₇	#135	1	2	yes	—
174	C₂×C₄×C₆₇	#137	1	2	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
178	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
179	C₈×C₆₇	—	4	2	no	—
180	C₈×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	4	4	no	—
183	C₂²×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
187	C₂²×C₆₇	#548	1	2	yes	—
188	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
190	C₂²×C₆₇	#577	1	2	yes	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	4	4	no	—
196	C₄×C₆₇	—	8	2	no	—
197	C₄×C₆₇	—	8	2	no	—
198	C₄×C₆₇	#452	1	4	yes	—
199	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
200	C₄×C₆₇	#536	1	1	yes	characteristic
201	C₄×C₆₇	#548	1	2	yes	—
202	#25934	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	4	4	no	—
204	C₂×C₆₇	#2436	1	1	yes	characteristic
205	C₂×C₆₇	C₂×#452	1	1	yes	characteristic
206	C₂×C₆₇	#3009	1	1	yes	characteristic
207	C₂×C₆₇	#3360	1	2	yes	—
208	C₂×C₆₇	#3372	1	2	yes	—
209	C₂×#554	C₂×C₆₇	1	1	yes	characteristic
210	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
211	C₂×#461	C₂×C₆₇	1	1	yes	characteristic
212	#2568	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#2026	C₂×C₆₇	1	4	yes	complemented
214	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1877	C₂×C₆₇	1	4	yes	—
217	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
218	C₆₇	#25934	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
220	#554	—	2	4	no	—
221	#554	—	2	4	no	—
222	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#125	C₂²×C₆₇	1	2	yes	—
225	C₂×C₂²⋊Q₈	—	2	1	no	—
226	C₂×C₂²⋊Q₈	—	2	2	no	—
227	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂²×C₄⋊C₄	—	2	1	no	—
231	#487	—	2	4	no	—
232	#461	—	2	4	no	—
233	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#103	C₂²×C₆₇	1	4	yes	—
235	C₂×#103	C₂²×C₆₇	1	2	yes	—
236	C₂×#102	C₂²×C₆₇	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
238	#400	—	2	2	no	—
239	#391	—	2	1	no	—
240	#391	C₂²×C₆₇	1	4	yes	—
241	#382	—	2	1	no	—
242	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
243	#340	C₂²×C₆₇	1	4	yes	—
244	#340	C₂²×C₆₇	1	2	yes	—
245	#336	C₂²×C₆₇	1	4	yes	—
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
248	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
249	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂³×C₄	—	2	2	no	—
252	C₂×SD₁₆	—	2	4	no	—
253	C₂×SD₁₆	—	2	8	no	—
254	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
255	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
256	C₄⋊Q₈	—	2	2	no	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	—	2	2	no	—
259	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
260	#125	—	2	4	no	—
261	#125	—	2	4	no	—
262	C₂²⋊Q₈	—	2	8	no	—
263	C₂²⋊Q₈	—	2	16	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	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	8	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
286	C₄⋊C₈	—	2	2	no	—
287	#103	—	2	4	no	—
288	#103	—	2	8	no	—
289	#102	—	2	4	no	—
290	C₄×C₈	—	2	2	no	—
291	#95	—	2	1	no	—
292	#95	—	2	4	no	—
293	#95	—	2	4	no	—
294	C₂⁴	—	2	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	4	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	8	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
307	C₂×D₄	#41×C₆₇	1	4	yes	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	8	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	—	8	4	no	—
317	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
322	SD₁₆	—	4	8	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	2	no	—
328	C₄⋊C₄	—	2	8	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
340	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	2	no	—
346	C₄²	C₂×D₄×C₆₇	1	2	yes	—
347	C₄²	C₂×D₄×C₆₇	1	2	yes	—
348	C₂³	—	2	2	no	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	1	no	—
351	C₂³	—	4	1	no	—
352	C₂³	—	4	2	no	—
353	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
354	Q₈	—	4	4	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	4	4	no	—
358	Q₈	—	8	4	no	—
359	D₄	—	2	2	no	—
360	D₄	—	2	4	no	—
361	D₄	#124×C₆₇	1	4	yes	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	8	8	no	—
380	C₂×C₄	#120×C₆₇	1	2	yes	—
381	C₂×C₄	#120×C₆₇	1	2	yes	—
382	C₂×C₄	#135×C₆₇	1	2	yes	—
383	C₂×C₄	#137×C₆₇	1	2	yes	—
384	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
387	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
388	C₈	—	4	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	4	no	—
394	C₂²	—	4	2	no	—
395	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
396	C₂²	#548×C₆₇	1	2	yes	—
397	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
398	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
399	C₂²	#577×C₆₇	1	2	yes	—
400	C₄	—	2	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	2	no	—
407	C₄	#452×C₆₇	1	4	yes	—
408	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
409	C₄	#536×C₆₇	1	1	yes	characteristic
410	C₄	#548×C₆₇	1	2	yes	—
411	C₂	—	4	4	no	—
412	C₂	C₆₇×#2436	1	1	yes	characteristic
413	C₂	C₂×#452×C₆₇	1	1	yes	characteristic
414	C₂	C₆₇×#3009	1	1	yes	characteristic
415	C₂	C₆₇×#3360	1	2	yes	—
416	C₂	C₆₇×#3372	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#487×C₆₇	C₂	semidirect
4. C₆₇×#2568	C₂	semidirect
5. C₆₇×#2026	C₂	semidirect
6. C₆₇×#1987	C₂	semidirect
7. C₆₇×#1984	C₂	semidirect
202. #25934	C₆₇	direct (N × H)
210. C₂×#487	C₂×C₆₇	semidirect
212. #2568	C₂×C₆₇	semidirect
213. #2026	C₂×C₆₇	semidirect
214. #1987	C₂×C₆₇	semidirect
215. #1984	C₂×C₆₇	semidirect
218. C₆₇	#25934	direct (N × H)

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
C₂×#487×C₆₇	2
C₂×#461×C₆₇	2
C₆₇×#2568	2
C₆₇×#2026	2
C₆₇×#1987	2
C₆₇×#1984	2
C₆₇×#1877	2
C₄×#102×C₆₇	2
#25934	67

Subgroups — C67×#25934 #30705185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25934 #30705185