Subgroups — C₂×C₆₇×#1677 #30698357

Order 17152.

810 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1677	C₂	1	8	yes	maximal, direct product factor
6	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#331×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	—	2	1	no	—
10	C₂³×C₈×C₆₇	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#486×C₆₇	—	2	2	no	—
15	#486×C₆₇	C₂²	1	4	yes	complemented
16	#476×C₆₇	—	2	2	no	—
17	#476×C₆₇	C₂²	1	4	yes	complemented
18	#459×C₆₇	—	2	2	no	—
19	#459×C₆₇	C₂²	1	4	yes	—
20	C₈×D₄×C₆₇	—	2	2	no	—
21	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
22	C₂×#106×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	—	2	1	no	—
27	C₂×#102×C₆₇	—	2	1	no	—
28	C₂×#102×C₆₇	—	2	1	no	—
29	C₂×#102×C₆₇	—	2	1	no	—
30	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#334×C₆₇	C₂²	1	4	yes	complemented
33	#333×C₆₇	C₂²	1	4	yes	complemented
34	#331×C₆₇	C₂²	1	4	yes	complemented
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	—	2	1	no	—
39	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	C₂×D₈×C₆₇	—	2	4	no	—
42	C₂×D₈×C₆₇	—	4	4	no	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	4	no	—
47	C₂²×C₈×C₆₇	—	4	4	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
54	#127×C₆₇	—	2	1	no	—
55	#127×C₆₇	—	2	1	no	—
56	#127×C₆₇	—	2	1	no	—
57	#127×C₆₇	C₂³	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	D₄	1	2	yes	—
61	C₄×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#106×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	1	no	—
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	#102×C₆₇	—	2	4	no	—
75	#102×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	C₂²⋊C₈×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	D₄	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂⁴×C₆₇	—	2	1	no	—
82	C₂⁴×C₆₇	—	4	1	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	—	8	2	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	4	1	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂×D₄×C₆₇	—	4	2	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	8	2	no	—
99	C₂×D₄×C₆₇	SD₁₆	1	4	yes	—
100	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	#41	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	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	4	8	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	4	1	no	—
126	C₄⋊C₄×C₆₇	—	4	1	no	—
127	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₂²⋊C₄×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₆₇	#41	1	1	yes	characteristic
133	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	4	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	2	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	8	2	no	—
140	C₂³×C₆₇	—	8	2	no	—
141	C₂³×C₆₇	—	8	2	no	—
142	C₂³×C₆₇	#121	1	1	yes	characteristic
143	Q₈×C₆₇	—	8	2	no	—
144	Q₈×C₆₇	—	8	2	no	—
145	D₄×C₆₇	—	2	2	no	—
146	D₄×C₆₇	—	2	4	no	—
147	D₄×C₆₇	—	4	2	no	—
148	D₄×C₆₇	—	4	2	no	—
149	D₄×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	4	2	no	—
151	D₄×C₆₇	—	8	2	no	—
152	D₄×C₆₇	—	8	2	no	—
153	D₄×C₆₇	—	8	2	no	—
154	D₄×C₆₇	—	8	2	no	—
155	D₄×C₆₇	C₂×SD₁₆	1	4	yes	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	4	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	8	2	no	—
164	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₆₇	#135	1	2	yes	—
169	C₂×C₄×C₆₇	#135	1	2	yes	—
170	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	2	4	no	—
180	C₂²×C₆₇	—	4	8	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	4	2	no	—
183	C₂²×C₆₇	—	8	2	no	—
184	C₂²×C₆₇	—	8	2	no	—
185	C₂²×C₆₇	—	8	2	no	—
186	C₂²×C₆₇	—	16	4	no	—
187	C₂²×C₆₇	#463	1	2	yes	—
188	C₂²×C₆₇	#465	1	2	yes	—
189	C₂²×C₆₇	#472	1	2	yes	—
190	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	8	2	no	—
195	C₄×C₆₇	#463	1	2	yes	—
196	C₄×C₆₇	#465	1	2	yes	—
197	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
198	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
199	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
200	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂×#1677	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	4	4	no	—
203	C₂×C₆₇	—	16	2	no	—
204	C₂×C₆₇	#1677	1	4	yes	direct product factor
205	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
206	C₂×C₆₇	C₂×#465	1	1	yes	characteristic
207	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
208	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
209	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
211	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1677	C₂×C₆₇	1	8	yes	direct product factor
213	C₂×#334	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₂×#333	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#331	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₆₇	C₂×#1677	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×D₈	—	2	1	no	—
218	C₂³×C₈	—	2	1	no	—
219	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
222	#486	—	2	2	no	—
223	#486	C₂²×C₆₇	1	4	yes	complemented
224	#476	—	2	2	no	—
225	#476	C₂²×C₆₇	1	4	yes	complemented
226	#459	—	2	2	no	—
227	#459	C₂²×C₆₇	1	4	yes	—
228	C₈×D₄	—	2	2	no	—
229	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
230	C₂×#106	—	2	1	no	—
231	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#102	—	2	1	no	—
235	C₂×#102	—	2	1	no	—
236	C₂×#102	—	2	1	no	—
237	C₂×#102	—	2	1	no	—
238	C₂×C₂²⋊C₈	—	2	1	no	—
239	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
240	#334	C₂²×C₆₇	1	4	yes	complemented
241	#333	C₂²×C₆₇	1	4	yes	complemented
242	#331	C₂²×C₆₇	1	4	yes	complemented
243	C₂²×Q₈	—	2	1	no	—
244	C₂²×D₄	—	2	1	no	—
245	C₂²×D₄	—	2	1	no	—
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
248	C₂³×C₄	—	2	1	no	—
249	C₂×D₈	—	2	4	no	—
250	C₂×D₈	—	4	4	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	2	4	no	—
255	C₂²×C₈	—	4	4	no	—
256	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
262	#127	—	2	1	no	—
263	#127	—	2	1	no	—
264	#127	—	2	1	no	—
265	#127	C₂³×C₆₇	1	2	yes	—
266	C₄×D₄	—	2	1	no	—
267	C₄×D₄	—	2	1	no	—
268	C₄×D₄	D₄×C₆₇	1	2	yes	—
269	C₄×D₄	C₂³×C₆₇	1	2	yes	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
275	#106	—	2	4	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
280	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
281	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
282	#102	—	2	4	no	—
283	#102	—	2	4	no	—
284	#102	—	2	4	no	—
285	#102	—	2	4	no	—
286	C₂²⋊C₈	—	2	4	no	—
287	C₄×C₈	D₄×C₆₇	1	2	yes	—
288	C₄×C₈	C₂³×C₆₇	1	2	yes	—
289	C₂⁴	—	2	1	no	—
290	C₂⁴	—	4	1	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	4	1	no	—
293	C₂×Q₈	—	8	2	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	8	2	no	—
307	C₂×D₄	SD₁₆×C₆₇	1	4	yes	—
308	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	2	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
318	D₈	—	4	8	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	8	no	—
325	C₂×C₈	—	4	2	no	—
326	C₂×C₈	—	4	2	no	—
327	C₂×C₈	—	4	2	no	—
328	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
329	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₄²	#41×C₆₇	1	1	yes	characteristic
341	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
342	C₂³	—	2	4	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	8	2	no	—
348	C₂³	—	8	2	no	—
349	C₂³	—	8	2	no	—
350	C₂³	#121×C₆₇	1	1	yes	characteristic
351	Q₈	—	8	2	no	—
352	Q₈	—	8	2	no	—
353	D₄	—	2	4	no	—
354	D₄	—	2	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	8	2	no	—
360	D₄	—	8	2	no	—
361	D₄	—	8	2	no	—
362	D₄	—	8	2	no	—
363	D₄	C₂×SD₁₆×C₆₇	1	4	yes	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	8	2	no	—
372	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
373	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
374	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
376	C₂×C₄	#135×C₆₇	1	2	yes	—
377	C₂×C₄	#135×C₆₇	1	2	yes	—
378	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
379	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	2	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₂²	—	2	4	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	8	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	16	4	no	—
395	C₂²	#463×C₆₇	1	2	yes	—
396	C₂²	#465×C₆₇	1	2	yes	—
397	C₂²	#472×C₆₇	1	2	yes	—
398	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	#463×C₆₇	1	2	yes	—
404	C₄	#465×C₆₇	1	2	yes	—
405	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
406	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
407	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
408	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
409	C₂	—	4	4	no	—
410	C₂	—	16	2	no	—
411	C₂	C₆₇×#1677	1	4	yes	direct product factor
412	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
413	C₂	C₂×#465×C₆₇	1	1	yes	characteristic
414	C₂	C₂×#472×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#486×C₆₇	C₂	semidirect
2. C₂×#476×C₆₇	C₂	semidirect
4. C₂×C₈×D₄×C₆₇	C₂	semidirect
5. C₆₇×#1677	C₂	direct (N × H)
6. C₂×#334×C₆₇	C₂	semidirect
7. C₂×#333×C₆₇	C₂	semidirect
8. C₂×#331×C₆₇	C₂	semidirect
15. #486×C₆₇	C₂²	semidirect
17. #476×C₆₇	C₂²	semidirect
21. C₈×D₄×C₆₇	C₂²	semidirect
32. #334×C₆₇	C₂²	semidirect
33. #333×C₆₇	C₂²	semidirect
34. #331×C₆₇	C₂²	semidirect
201. C₂×#1677	C₆₇	direct (N × H)
204. C₂×C₆₇	#1677	direct (N × H)
208. C₂×#486	C₂×C₆₇	semidirect
209. C₂×#476	C₂×C₆₇	semidirect
211. C₂×C₈×D₄	C₂×C₆₇	semidirect
212. #1677	C₂×C₆₇	direct (N × H)
213. C₂×#334	C₂×C₆₇	semidirect
214. C₂×#333	C₂×C₆₇	semidirect
215. C₂×#331	C₂×C₆₇	semidirect
216. C₆₇	C₂×#1677	direct (N × H)
223. #486	C₂²×C₆₇	semidirect
225. #476	C₂²×C₆₇	semidirect
229. C₈×D₄	C₂²×C₆₇	semidirect
240. #334	C₂²×C₆₇	semidirect
241. #333	C₂²×C₆₇	semidirect
242. #331	C₂²×C₆₇	semidirect
411. C₂	C₆₇×#1677	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×#486×C₆₇	2
C₂×#476×C₆₇	2
C₂×#459×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₆₇×#1677	2
C₂×#334×C₆₇	2
C₂×#333×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1677	67

Subgroups — C2×C67×#1677 #30698357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1677 #30698357