Subgroups — C₆₇×#33590 #30712841

Order 17152.

694 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2210	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2209	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2208	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#358×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	—	2	1	no	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	#536×C₆₇	—	2	1	no	—
17	#520×C₆₇	—	2	1	no	—
18	#510×C₆₇	—	2	4	no	—
19	#508×C₆₇	—	2	4	no	—
20	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
25	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	#497×C₆₇	—	2	2	no	—
29	#495×C₆₇	C₂²	1	2	yes	—
30	#494×C₆₇	C₂²	1	2	yes	—
31	#440×C₆₇	—	2	2	no	—
32	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
33	C₄×D₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
35	#417×C₆₇	—	2	1	no	—
36	#405×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
38	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
39	#358×C₆₇	C₂²	1	2	yes	—
40	#358×C₆₇	C₂²	1	2	yes	—
41	#358×C₆₇	C₂²	1	2	yes	—
42	#357×C₆₇	C₂²	1	2	yes	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	#137×C₆₇	—	4	4	no	—
47	#135×C₆₇	—	2	2	no	—
48	#135×C₆₇	—	2	2	no	—
49	#135×C₆₇	—	2	2	no	—
50	#135×C₆₇	D₄	1	4	yes	—
51	C₂×Q₁₆×C₆₇	—	2	2	no	—
52	C₂×Q₁₆×C₆₇	—	2	2	no	—
53	C₂×Q₁₆×C₆₇	—	2	1	no	—
54	C₂×Q₁₆×C₆₇	—	2	2	no	—
55	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×SD₁₆×C₆₇	—	2	1	no	—
59	C₂×SD₁₆×C₆₇	—	2	2	no	—
60	C₂×SD₁₆×C₆₇	—	2	1	no	—
61	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	#124×C₆₇	—	2	2	no	—
67	#123×C₆₇	—	4	2	no	—
68	C₂²⋊Q₈×C₆₇	—	4	1	no	—
69	C₂²⋊Q₈×C₆₇	—	4	1	no	—
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	—	4	2	no	—
73	#117×C₆₇	—	4	1	no	—
74	#117×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
77	Q₃₂×C₆₇	—	2	2	no	—
78	Q₃₂×C₆₇	—	2	4	no	—
79	SD₃₂×C₆₇	—	2	2	no	—
80	SD₃₂×C₆₇	—	2	2	no	—
81	SD₃₂×C₆₇	—	2	4	no	—
82	D₁₆×C₆₇	—	2	2	no	—
83	M₃₂×C₆₇	—	2	2	no	—
84	C₂×C₁₆×C₆₇	—	2	2	no	—
85	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
88	#107×C₆₇	—	2	1	no	—
89	#107×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	—	2	2	no	—
91	#102×C₆₇	—	2	2	no	—
92	#97×C₆₇	D₄	1	2	yes	—
93	C₄×C₈×C₆₇	C₂³	1	2	yes	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	4	2	no	—
96	#41×C₆₇	—	4	2	no	—
97	#41×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	2	1	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×Q₈×C₆₇	—	4	1	no	—
102	C₂×Q₈×C₆₇	—	4	2	no	—
103	C₂×Q₈×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	—	4	1	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	—	4	2	no	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	Q₁₆×C₆₇	—	2	2	no	—
115	Q₁₆×C₆₇	—	2	1	no	—
116	Q₁₆×C₆₇	—	2	1	no	—
117	Q₁₆×C₆₇	—	2	2	no	—
118	Q₁₆×C₆₇	—	2	1	no	—
119	Q₁₆×C₆₇	—	4	4	no	—
120	Q₁₆×C₆₇	—	4	2	no	—
121	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
122	SD₁₆×C₆₇	—	2	4	no	—
123	SD₁₆×C₆₇	—	4	4	no	—
124	SD₁₆×C₆₇	—	4	2	no	—
125	D₈×C₆₇	—	2	1	no	—
126	D₈×C₆₇	—	2	1	no	—
127	D₈×C₆₇	—	2	1	no	—
128	D₈×C₆₇	C₂×D₄	1	2	yes	—
129	M₁₆×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	#41	1	2	yes	—
134	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	4	1	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
141	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₂²⋊C₄×C₆₇	—	4	2	no	—
144	C₂²⋊C₄×C₆₇	—	4	2	no	—
145	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
146	C₄²×C₆₇	—	4	2	no	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₁₆×C₆₇	—	2	1	no	—
149	C₁₆×C₆₇	—	2	1	no	—
150	C₁₆×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	1	no	—
153	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	4	1	no	—
156	Q₈×C₆₇	—	4	1	no	—
157	Q₈×C₆₇	—	4	2	no	—
158	Q₈×C₆₇	—	4	2	no	—
159	Q₈×C₆₇	—	4	1	no	—
160	Q₈×C₆₇	—	8	2	no	—
161	D₄×C₆₇	—	2	2	no	—
162	D₄×C₆₇	—	4	4	no	—
163	D₄×C₆₇	—	4	1	no	—
164	D₄×C₆₇	—	4	1	no	—
165	D₄×C₆₇	—	4	1	no	—
166	D₄×C₆₇	—	8	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	4	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	8	2	no	—
173	C₂×C₄×C₆₇	—	8	2	no	—
174	C₂×C₄×C₆₇	—	16	2	no	—
175	C₂×C₄×C₆₇	—	16	1	no	—
176	C₂×C₄×C₆₇	#121	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	#121	1	2	yes	—
184	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	2	1	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	8	1	no	—
190	C₂²×C₆₇	—	8	1	no	—
191	C₂²×C₆₇	—	8	1	no	—
192	C₂²×C₆₇	—	8	1	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	#459	1	2	yes	—
195	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	8	2	no	—
200	C₄×C₆₇	—	8	1	no	—
201	C₄×C₆₇	—	8	2	no	—
202	C₄×C₆₇	#459	1	2	yes	—
203	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
204	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	#33590	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	2	2	no	—
207	C₂×C₆₇	—	8	2	no	—
208	C₂×C₆₇	—	16	1	no	—
209	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
210	C₂×C₆₇	#3416	1	1	yes	characteristic
211	C₂×C₆₇	#3417	1	1	yes	characteristic
212	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₂×#508	C₂×C₆₇	1	1	yes	characteristic
214	#3156	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#2951	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#2211	C₂×C₆₇	1	2	yes	complemented
217	#2210	C₂×C₆₇	1	2	yes	complemented
218	#2209	C₂×C₆₇	1	2	yes	complemented
219	#2208	C₂×C₆₇	1	2	yes	complemented
220	#2151	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#2139	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₂×#358	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₆₇	#33590	1	1	yes	characteristic, direct product factor, Sylow
224	C₂×#137	—	2	1	no	—
225	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
227	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
228	#536	—	2	1	no	—
229	#520	—	2	1	no	—
230	#510	—	2	4	no	—
231	#508	—	2	4	no	—
232	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
234	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
237	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
240	#497	—	2	2	no	—
241	#495	C₂²×C₆₇	1	2	yes	—
242	#494	C₂²×C₆₇	1	2	yes	—
243	#440	—	2	2	no	—
244	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
245	C₄×D₈	C₂²×C₆₇	1	2	yes	—
246	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
247	#417	—	2	1	no	—
248	#405	C₂²×C₆₇	1	1	yes	characteristic
249	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
250	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
251	#358	C₂²×C₆₇	1	2	yes	—
252	#358	C₂²×C₆₇	1	2	yes	—
253	#358	C₂²×C₆₇	1	2	yes	—
254	#357	C₂²×C₆₇	1	2	yes	—
255	C₂×#41	—	2	1	no	—
256	C₂×#41	—	2	1	no	—
257	C₂²×Q₈	—	2	1	no	—
258	#137	—	4	4	no	—
259	#135	—	2	2	no	—
260	#135	—	2	2	no	—
261	#135	—	2	2	no	—
262	#135	D₄×C₆₇	1	4	yes	—
263	C₂×Q₁₆	—	2	2	no	—
264	C₂×Q₁₆	—	2	1	no	—
265	C₂×Q₁₆	—	2	2	no	—
266	C₂×Q₁₆	—	2	2	no	—
267	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
268	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
269	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
270	C₂×SD₁₆	—	2	1	no	—
271	C₂×SD₁₆	—	2	1	no	—
272	C₂×SD₁₆	—	2	2	no	—
273	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
274	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
275	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₄⋊Q₈	—	2	2	no	—
278	#124	—	2	2	no	—
279	#123	—	4	2	no	—
280	C₂²⋊Q₈	—	4	1	no	—
281	C₂²⋊Q₈	—	4	1	no	—
282	C₄×Q₈	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	—	4	2	no	—
285	#117	—	4	1	no	—
286	#117	D₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	4	1	no	—
289	Q₃₂	—	2	2	no	—
290	Q₃₂	—	2	4	no	—
291	SD₃₂	—	2	2	no	—
292	SD₃₂	—	2	2	no	—
293	SD₃₂	—	2	4	no	—
294	D₁₆	—	2	2	no	—
295	M₃₂	—	2	2	no	—
296	C₂×C₁₆	—	2	2	no	—
297	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
298	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
299	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
300	#107	—	2	1	no	—
301	#107	C₂³×C₆₇	1	2	yes	—
302	#103	—	2	2	no	—
303	#102	—	2	2	no	—
304	#97	D₄×C₆₇	1	2	yes	—
305	C₄×C₈	C₂³×C₆₇	1	2	yes	—
306	#41	—	2	4	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	4	2	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	1	no	—
313	C₂×Q₈	—	4	1	no	—
314	C₂×Q₈	—	4	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂²×C₄	—	4	1	no	—
321	C₂²×C₄	—	4	1	no	—
322	C₂²×C₄	—	4	1	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	1	no	—
328	Q₁₆	—	2	1	no	—
329	Q₁₆	—	2	2	no	—
330	Q₁₆	—	2	1	no	—
331	Q₁₆	—	4	4	no	—
332	Q₁₆	—	4	2	no	—
333	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
334	SD₁₆	—	2	4	no	—
335	SD₁₆	—	4	4	no	—
336	SD₁₆	—	4	2	no	—
337	D₈	—	2	1	no	—
338	D₈	—	2	1	no	—
339	D₈	—	2	1	no	—
340	D₈	C₂×D₄×C₆₇	1	2	yes	—
341	M₁₆	—	2	2	no	—
342	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
343	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
345	C₂×C₈	#41×C₆₇	1	2	yes	—
346	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
353	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	4	2	no	—
357	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
358	C₄²	—	4	2	no	—
359	C₄²	C₂×D₄×C₆₇	1	2	yes	—
360	C₁₆	—	2	1	no	—
361	C₁₆	—	2	1	no	—
362	C₁₆	—	2	2	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
366	Q₈	—	2	2	no	—
367	Q₈	—	4	1	no	—
368	Q₈	—	4	1	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	1	no	—
372	Q₈	—	8	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	4	no	—
375	D₄	—	4	1	no	—
376	D₄	—	4	1	no	—
377	D₄	—	4	1	no	—
378	D₄	—	8	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	16	2	no	—
387	C₂×C₄	—	16	1	no	—
388	C₂×C₄	#121×C₆₇	1	2	yes	—
389	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
390	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
391	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
392	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
393	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	#121×C₆₇	1	2	yes	—
396	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
397	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
398	C₂²	—	2	1	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	8	1	no	—
402	C₂²	—	8	1	no	—
403	C₂²	—	8	1	no	—
404	C₂²	—	8	1	no	—
405	C₂²	—	8	1	no	—
406	C₂²	#459×C₆₇	1	2	yes	—
407	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	1	no	—
412	C₄	—	8	2	no	—
413	C₄	—	8	2	no	—
414	C₄	#459×C₆₇	1	2	yes	—
415	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
416	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
417	C₂	—	2	2	no	—
418	C₂	—	8	2	no	—
419	C₂	—	16	1	no	—
420	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
421	C₂	C₆₇×#3416	1	1	yes	characteristic
422	C₂	C₆₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#508×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#510×C₆₇	C₂	semidirect
3. C₆₇×#3156	C₂	semidirect
4. C₆₇×#2951	C₂	semidirect
5. C₆₇×#2211	C₂	semidirect
6. C₆₇×#2210	C₂	semidirect
7. C₆₇×#2209	C₂	semidirect
8. C₆₇×#2208	C₂	semidirect
9. C₆₇×#2151	C₂	semidirect
10. C₆₇×#2139	C₂	semidirect
11. C₂×#358×C₆₇	C₂	semidirect
205. #33590	C₆₇	direct (N × H)
212. C₂×#510	C₂×C₆₇	semidirect
214. #3156	C₂×C₆₇	semidirect
215. #2951	C₂×C₆₇	semidirect
216. #2211	C₂×C₆₇	semidirect
217. #2210	C₂×C₆₇	semidirect
218. #2209	C₂×C₆₇	semidirect
219. #2208	C₂×C₆₇	semidirect
220. #2151	C₂×C₆₇	semidirect
221. #2139	C₂×C₆₇	semidirect
222. C₂×#358	C₂×C₆₇	semidirect
223. C₆₇	#33590	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×#510×C₆₇	2
C₂×#508×C₆₇	2
C₆₇×#3156	2
C₆₇×#2951	2
C₆₇×#2211	2
C₆₇×#2210	2
C₆₇×#2209	2
C₆₇×#2208	2
C₆₇×#2151	2
C₆₇×#2139	2
C₂×#358×C₆₇	2
#33590	67

Subgroups — C67×#33590 #30712841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33590 #30712841